From a95604f79e381e8d95740a35453fdaadf35c5260 Mon Sep 17 00:00:00 2001 From: Patricia Wollstadt Date: Mon, 28 Feb 2022 18:03:37 +0100 Subject: [PATCH] Update sphinx documentation --- docs/conf.py | 4 +- docs/doctrees/environment.pickle | Bin 720989 -> 837363 bytes docs/doctrees/idtxl.doctree | Bin 1459388 -> 1798393 bytes docs/doctrees/idtxl_data_class.doctree | Bin 165589 -> 170574 bytes docs/doctrees/idtxl_estimators.doctree | Bin 381072 -> 392700 bytes docs/doctrees/idtxl_helper.doctree | Bin 156288 -> 160025 bytes .../doctrees/idtxl_network_comparison.doctree | Bin 50355 -> 52001 bytes docs/doctrees/idtxl_network_inference.doctree | Bin 207769 -> 211790 bytes docs/doctrees/idtxl_process_analysis.doctree | Bin 96075 -> 380588 bytes docs/doctrees/idtxl_results_class.doctree | Bin 116053 -> 144760 bytes docs/doctrees/index.doctree | Bin 8333 -> 8581 bytes docs/doctrees/modules.doctree | Bin 2538 -> 2666 bytes docs/html/.buildinfo | 2 +- .../embedding_optimization_ais_Rudelt.html | 1145 ++++++++ .../_modules/idtxl/estimators_Rudelt.html | 1217 +++++++++ docs/html/_modules/idtxl/results.html | 210 +- docs/html/_modules/index.html | 19 +- docs/html/_sources/idtxl.rst.txt | 24 +- .../_sources/idtxl_process_analysis.rst.txt | 13 + .../html/_sources/idtxl_results_class.rst.txt | 3 + docs/html/_static/basic.css | 114 +- docs/html/_static/doctools.js | 7 +- docs/html/_static/documentation_options.js | 2 +- docs/html/_static/epub.css | 2 +- docs/html/_static/language_data.js | 2 +- docs/html/_static/pygments.css | 7 +- docs/html/_static/pyramid.css | 2 +- docs/html/_static/searchtools.js | 17 +- docs/html/_static/underscore-1.13.1.js | 2042 ++++++++++++++ docs/html/_static/underscore.js | 8 +- docs/html/genindex.html | 238 +- docs/html/idtxl.html | 2405 ++++++++++++----- docs/html/idtxl_data_class.html | 105 +- docs/html/idtxl_estimators.html | 289 +- docs/html/idtxl_helper.html | 162 +- docs/html/idtxl_network_comparison.html | 59 +- docs/html/idtxl_network_inference.html | 117 +- docs/html/idtxl_process_analysis.html | 1240 ++++++++- docs/html/idtxl_results_class.html | 223 +- docs/html/index.html | 55 +- docs/html/modules.html | 26 +- docs/html/objects.inv | Bin 2001 -> 2648 bytes docs/html/py-modindex.html | 25 +- docs/html/search.html | 29 +- docs/html/searchindex.js | 2 +- docs/idtxl.rst | 24 +- docs/idtxl_process_analysis.rst | 13 + docs/idtxl_results_class.rst | 3 + idtxl/embedding_optimization_ais_Rudelt.py | 94 +- idtxl/estimators_Rudelt.py | 54 +- idtxl/results.py | 116 +- 51 files changed, 8647 insertions(+), 1472 deletions(-) create mode 100644 docs/html/_modules/idtxl/embedding_optimization_ais_Rudelt.html create mode 100644 docs/html/_modules/idtxl/estimators_Rudelt.html create mode 100644 docs/html/_static/underscore-1.13.1.js diff --git a/docs/conf.py b/docs/conf.py index 51aa3426..4dccc255 100644 --- a/docs/conf.py +++ b/docs/conf.py @@ -69,9 +69,9 @@ # built documents. # # The short X.Y version. -version = u'1.2.1' +version = u'1.3' # The full version, including alpha/beta/rc tags. -release = u'1.2.1' +release = version # The language for content autogenerated by Sphinx. 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 + +
+
+
+
+ +

Source code for idtxl.embedding_optimization_ais_Rudelt

+""" Optimization of embedding parameters of spike times using the history dependence estimators """
+
+import numpy as np
+from scipy.optimize import newton
+from sys import stderr
+from idtxl.estimators_Rudelt import RudeltBBCEstimator, RudeltShufflingEstimator
+import idtxl.hde_utils as utl
+from idtxl.results import DotDict, ResultsSingleProcessRudelt
+from pathlib import Path
+from idtxl.data_spiketime import Data_spiketime
+
+
+# noinspection PyAttributeOutsideInit
+
[docs]class OptimizationRudelt():
+    """
+    Optimization of embedding parameters of spike times using the history dependence estimators
+
+    References:
+
+        [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
+            optimization reveals long-lasting history dependence in
+            neural spiking activity, 2021, PLOS Computational Biology, 17(6)
+
+        [2]: https://github.com/Priesemann-Group/hdestimator
+
+    implemented in idtxl by Michael Lindner, Göttingen 2021
+
+    Args:
+        settings : dict
+            - estimation_method : string
+                The method to be used to estimate the history dependence 'bbc' or 'shuffling'.
+            - embedding_step_size : float
+                Step size delta t (in seconds) with which the window is slid through the data.
+                (default: 0.005)
+            - embedding_number_of_bins_set : list of integer values
+                Set of values for d, the number of bins in the embedding.
+                (default: [1, 2, 3, 4, 5])
+            - embedding_past_range_set : list of floating-point values
+                Set of values for T, the past range (in seconds) to be used for embeddings.
+                (default: [0.005, 0.00561, 0.00629, 0.00706, 0.00792, 0.00889,
+                0.00998, 0.01119, 0.01256, 0.01409, 0.01581, 0.01774, 0.01991,
+                0.02233, 0.02506, 0.02812, 0.03155, 0.0354, 0.03972, 0.04456,
+                0.05, 0.0561, 0.06295, 0.07063, 0.07924, 0.08891, 0.09976,
+                0.11194, 0.12559, 0.14092, 0.15811, 0.17741, 0.19905, 0.22334,
+                0.25059, 0.28117, 0.31548, 0.35397, 0.39716, 0.44563, 0.5,
+                0.56101, 0.62946, 0.70627, 0.79245, 0.88914, 0.99763, 1.11936,
+                1.25594, 1.40919, 1.58114, 1.77407, 1.99054, 2.23342, 2.50594,
+                2.81171, 3.15479, 3.53973, 3.97164, 4.45625, 5.0])
+            - embedding_scaling_exponent_set : dict
+                Set of values for kappa, the scaling exponent for the bins in the embedding.
+                Should be a python-dictionary with the three entries 'number_of_scalings', 'min_first_bin_size' and
+                'min_step_for_scaling'.
+                defaults: {'number_of_scalings': 10, 'min_first_bin_size': 0.005, 'min_step_for_scaling': 0.01})
+            - bbc_tolerance : float
+                The tolerance for the Bayesian Bias Criterion. Influences which embeddings are
+                discarded from the analysis.
+                (default: 0.05)
+            - return_averaged_R : bool
+                Return R_tot as the average over R(T) for T in [T_D, T_max], instead of R_tot = R(T_D).
+                If set to True, the setting for number_of_bootstraps_R_tot (see below) is ignored and set to 0
+                and CI bounds are not calculated.
+                (default: True)
+            - timescale_minimum_past_range : float
+                Minimum past range T_0 (in seconds) to take into consideration for the estimation of the
+                information timescale tau_R.
+                (default: 0.01)
+            - number_of_bootstraps_R_max : int
+                The number of bootstrap re-shuffles that should be used to determine the optimal
+                embedding. (Bootstrap the estimates of R_max to determine R_tot.)
+                These are computed during the 'history-dependence' task because they are essential
+                to obtain R_tot.
+                (default: 250)
+            - number_of_bootstraps_R_tot : int
+                The number of bootstrap re-shuffles that should be used to estimate the confidence
+                interval of the optimal embedding. (Bootstrap the estimates of R_tot = R(T_D) to
+                obtain a confidence interval for R_tot.).
+                These are computed during the 'confidence-intervals' task.
+                The setting return_averaged_R (see above) needs to be set to False for this setting
+                to take effect.
+                (default: 250)
+            - number_of_bootstraps_nonessential : int
+                The number of bootstrap re-shuffles that should be used to estimate the confidence
+                intervals for embeddings other than the optimal one. (Bootstrap the estimates of
+                R(T) for all other T.)
+                (These are not necessary for the main analysis and therefore default to 0.)
+            - symbol_block_length : int
+                The number of symbols that should be drawn in each block for bootstrap resampling
+                If it is set to None (recommended), the length is automatically chosen, based
+                on heuristics
+                (default: None)
+            - bootstrap_CI_use_sd : bool
+                Most of the time we observed normally-distributed bootstrap replications,
+                so it is sufficient (and more efficient) to compute confidence intervals
+                based on the standard deviation
+                (default: True)
+            - bootstrap_CI_percentile_lo : float
+                The lower percentile for the confidence interval.
+                This has no effect if bootstrap_CI_use_sd is set to True
+                (default: 2.5)
+            - bootstrap_CI_percentile_hi : float
+                The upper percentiles for the confidence interval.
+                This has no effect if bootstrap_CI_use_sd is set to True
+                (default: 97.5)
+            - analyse_auto_MI : bool
+                perform calculation of auto mutual information of the spike train
+                (default: True)
+                If set to True:
+
+                - auto_MI_bin_size_set : list of floating-point values
+                    Set of values for the sizes of the bins (in seconds).
+                    (default: [0.005, 0.01, 0.025, 0.05, 0.25, 0.5])
+                - auto_MI_max_delay : int
+                    The maximum delay (in seconds) between the past bin and the response.
+                    (default: 5)
+
+            - visualization : bool
+                create .eps output image showing the optimization values and graphs for
+                the history dependence and the auto mutual information
+                (default: False)
+                if set to True:
+
+                - output_path : String
+                    Path where the .eps images should be saved
+                - output_prefix : String
+                    Prefix of the output images
+                    e.g. <output_prefix>_process0.eps
+
+            - debug: bool
+                show values while calculating
+                (default: False)
+    """
+
+    def __init__(self, settings=None):
+        settings = self._check_settings(settings)
+        self.settings = settings.copy()
+
+        self.settings.setdefault('embedding_step_size', 0.005)
+        self.settings.setdefault('embedding_past_range_set', [0.005, 0.00561, 0.00629, 0.00706, 0.00792, 0.00889,
+                                                              0.00998, 0.01119, 0.01256, 0.01409, 0.01581, 0.01774,
+                                                              0.01991, 0.02233, 0.02506, 0.02812, 0.03155, 0.0354,
+                                                              0.03972, 0.04456, 0.05, 0.0561, 0.06295, 0.07063,
+                                                              0.07924, 0.08891, 0.09976, 0.11194, 0.12559, 0.14092,
+                                                              0.15811, 0.17741, 0.19905, 0.22334, 0.25059, 0.28117,
+                                                              0.31548, 0.35397, 0.39716, 0.44563, 0.5, 0.56101, 0.62946,
+                                                              0.70627, 0.79245, 0.88914, 0.99763, 1.11936, 1.25594,
+                                                              1.40919, 1.58114, 1.77407, 1.99054, 2.23342, 2.50594,
+                                                              2.81171, 3.15479, 3.53973, 3.97164, 4.45625, 5.0])
+        self.settings.setdefault('embedding_number_of_bins_set', [1, 2, 3, 4, 5])
+        self.settings.setdefault('embedding_scaling_exponent_set',
+                                 {'number_of_scalings': 10,
+                                  'min_first_bin_size': 0.005,
+                                  'min_step_for_scaling': 0.01})
+
+        self.settings.setdefault('bbc_tolerance', 0.05)
+        self.settings.setdefault('return_averaged_R', True)
+        self.settings.setdefault('timescale_minimum_past_range', 0.01)
+
+        self.settings.setdefault('analyse_auto_MI', True)
+        self.settings.setdefault('auto_MI_bin_size_set', [0.005, 0.01, 0.025, 0.05, 0.25, 0.5])
+        self.settings.setdefault('auto_MI_max_delay', 5)
+
+        self.settings.setdefault('number_of_bootstraps_R_max', 250)
+        self.settings.setdefault('number_of_bootstraps_R_tot', 250)
+        self.settings.setdefault('number_of_bootstraps_nonessential', 0)
+        self.settings.setdefault('symbol_block_length', None)
+        self.settings.setdefault('bootstrap_CI_use_sd', True)
+        self.settings.setdefault('bootstrap_CI_percentile_lo', 2.5)
+        self.settings.setdefault('bootstrap_CI_percentile_hi', 97.5)
+        self.settings.setdefault('visualization', False)
+        self.settings.setdefault('debug', False)
+
+        self.check_inputs()                                 # ------------------------------------------------------------- TODO CHECK INPUTS
+
+        self.embeddings = self.get_embeddings(self.settings['embedding_past_range_set'],
+                                              self.settings['embedding_number_of_bins_set'],
+                                              self.settings['embedding_scaling_exponent_set'])
+
+
+    @staticmethod
+    def _check_settings(settings=None):
+        """Set default for settings dictionary.
+
+        Check if settings dictionary is None. If None, initialise an empty
+        dictionary. If not None check if type is dictionary. Function should be
+        called before setting default values.
+        """
+        if settings is None:
+            return {}
+        elif type(settings) is not dict:
+            raise TypeError('settings should be a dictionary.')
+        else:
+            return settings
+
+
[docs]    def check_inputs(self):
+        """
+        Check input settings for completeness
+
+        """
+
+        args_float = ['embedding_step_size',
+                      'timescale_minimum_past_range',
+                      'bootstrap_CI_percentile_lo',
+                      'bootstrap_CI_percentile_hi',
+                      'bbc_tolerance']
+        for key in args_float:
+            assert (key in self.settings), \
+                (key + ' has to be specified (see help)!')
+            assert (isinstance(self.settings[key], float)), \
+                ("Error: setting '" + key + "' needs to an floating point value (see help!!). Aborting.")
+
+        args_int = ['number_of_bootstraps_R_max',
+                    'number_of_bootstraps_R_tot',
+                    'number_of_bootstraps_nonessential']
+        for key in args_int:
+            assert (key in self.settings), \
+                (key + ' has to be specified (see help)!')
+            assert (isinstance(self.settings[key], int)), \
+                ("Error: setting '" + key + "' needs to an integer value (see help!!). Aborting.")
+
+        args_bool = ['return_averaged_R',
+                     'bootstrap_CI_use_sd',
+                     'debug']
+        for key in args_bool:
+            assert (key in self.settings), \
+                (key + ' has to be specified (see help)!')
+            assert (isinstance(self.settings[key], bool)), \
+                ("Error: setting '" + key + "' needs to a boolean expression (see help!!). Aborting.")
+
+        assert('embedding_past_range_set' in self.settings), \
+            'embedding_past_range_set has to be specified (see help)!'
+        assert (isinstance(self.settings['embedding_past_range_set'], list)), \
+            "Error: setting 'embedding_past_range_set' needs to be a list but is defined as {0}. " \
+            "Aborting.".format(type(self.settings['embedding_past_range_set']))
+
+        assert ('embedding_number_of_bins_set' in self.settings), \
+            'embedding_number_of_bins_set has to be specified (see help)!'
+        assert (isinstance(self.settings['embedding_number_of_bins_set'], list)), \
+            "Error: setting 'embedding_number_of_bins_set' needs to a list but is defined as {0}. " \
+            "Aborting.".format(type(self.settings['embedding_number_of_bins_set']))
+
+        assert ('embedding_scaling_exponent_set' in self.settings), \
+            'embedding_scaling_exponent_set has to be specified (see help)!'
+
+        assert ('number_of_scalings' in self.settings['embedding_scaling_exponent_set']), \
+            "'number_of_scalings' have to be specified in self.settings['embedding_scaling_exponent_set'] (see help)!"
+        assert (isinstance(self.settings['embedding_scaling_exponent_set']['number_of_scalings'], int)), \
+            "Error: setting 'number_of_scalings' needs to be an integer value but is defined as {0}. " \
+            "Aborting.".format(type(self.settings['number_of_scalings']))
+        assert ('min_first_bin_size' in self.settings['embedding_scaling_exponent_set']), \
+            "'min_first_bin_size' have to be specified in self.settings['embedding_scaling_exponent_set'] (see help)!"
+        assert (isinstance(self.settings['embedding_scaling_exponent_set']['min_first_bin_size'], float)), \
+            "Error: setting 'min_first_bin_sizes' needs to be a floating point value but is defined as {0}. " \
+            "Aborting.".format(type(self.settings['min_first_bin_sizes']))
+        assert ('min_step_for_scaling' in self.settings['embedding_scaling_exponent_set']), \
+            "'min_step_for_scaling' have to be specified in self.settings['embedding_scaling_exponent_set'] (see help)!"
+        assert (isinstance(self.settings['embedding_scaling_exponent_set']['min_step_for_scaling'], float)), \
+            "Error: setting 'min_step_for_scaling' needs to be a floating point value but is defined as {0}. " \
+            "Aborting.".format(type(self.settings['min_step_for_scaling']))
+
+        assert ('symbol_block_length' in self.settings), \
+            'symbol_block_length has to be specified (see help)!'
+        if self.settings['symbol_block_length'] is not None:
+            assert (isinstance(self.settings['symbol_block_length'], int)), \
+                "Error: setting 'symbol_block_length' needs to an integer value but is defined as {0}. " \
+                "Aborting.".format(type(self.settings['symbol_block_length']))
+
+        if self.settings['analyse_auto_MI']:
+            assert (isinstance(self.settings['analyse_auto_MI'], bool)), \
+                "Error: setting 'analyse_auto_MI' needs to be a boolean expression but is defined as {0}. " \
+                "Aborting.".format(type(self.settings['analyse_auto_MI']))
+            assert ('auto_MI_bin_size_set' in self.settings), \
+                'If analyse_auto_MI is set to True, auto_MI_bin_size_set has to be specified (see help)!'
+            assert (isinstance(self.settings['auto_MI_bin_size_set'], list)), \
+                "Error: setting 'auto_MI_bin_size_set' needs to a list but is defined as {0}. " \
+                "Aborting.".format(type(self.settings['auto_MI_bin_size_set']))
+            assert ('auto_MI_max_delay' in self.settings), \
+                'If analyse_auto_MI is set to True, auto_MI_max_delay has to be specified (see help)!'
+            assert (isinstance(self.settings['auto_MI_max_delay'], int)), \
+                "Error: setting 'auto_MI_max_delay' needs to be an integer value but is defined as {0}. " \
+                "Aborting.".format(type(self.settings['auto_MI_max_delay']))
+
+        if self.settings['visualization']:
+            assert (isinstance(self.settings['visualization'], bool)), \
+                "Error: setting 'visualization' needs to be boolean but is defined as {0}. " \
+                "Aborting.".format(type(self.settings['visualization']))
+            assert ('output_path' in self.settings), \
+                'If visualization is set to True an output path has to be specified (see help)!'
+            assert ('output_prefix' in self.settings), \
+                'If visualization is set to True an output prefix has to be specified (see help)!'
+
+        # Cython implementation uses 64bit unsigned integers for the symbols,
+        # we allow up to 62 bins (window has 1 bin more..)
+        assert (max(self.settings['embedding_number_of_bins_set']) <= 62), \
+            'Error: Max number of bins too large; use less than 63. Aborting.'
+
+        # If R_tot is computed as an average over Rs, no confidence interval can be estimated
+        if self.settings['return_averaged_R']:
+            self.settings['number_of_bootstraps_R_tot'] = 0

+
+
[docs]    def get_embeddings(self,
+                       embedding_past_range_set,
+                       embedding_number_of_bins_set,
+                       embedding_scaling_exponent_set):
+        """
+        Get all combinations of parameters T, d, k, based on the
+        sets of selected parameters.
+        """
+
+        embeddings = []
+        for past_range_T in embedding_past_range_set:
+            for number_of_bins_d in embedding_number_of_bins_set:
+                if not isinstance(number_of_bins_d, int) or number_of_bins_d < 1:
+                    print("Error: number of bins {} is not a positive integer. Skipping.".format(number_of_bins_d),
+                          file=stderr, flush=True)
+                    continue
+
+                if type(embedding_scaling_exponent_set) == dict:
+                    scaling_set_given_T_and_d = self.get_set_of_scalings(past_range_T,
+                                                                         number_of_bins_d,
+                                                                         **embedding_scaling_exponent_set)
+                else:
+                    scaling_set_given_T_and_d = embedding_scaling_exponent_set
+
+                for scaling_k in scaling_set_given_T_and_d:
+                    embeddings += [(past_range_T, number_of_bins_d, scaling_k)]
+
+        return embeddings

+
+
[docs]    def get_set_of_scalings(self, past_range_T,
+                            number_of_bins_d,
+                            number_of_scalings,
+                            min_first_bin_size,
+                            min_step_for_scaling):
+        """
+        Get scaling exponents such that the uniform embedding as well as
+        the embedding for which the first bin has a length of
+        min_first_bin_size (in seconds), as well as linearly spaced
+        scaling factors in between, such that in total
+        number_of_scalings scalings are obtained.
+        """
+
+        min_scaling = 0
+        if past_range_T / number_of_bins_d <= min_first_bin_size or number_of_bins_d == 1:
+            max_scaling = 0
+        else:
+            # for the initial guess assume the largest bin dominates, so k is approx. log(T) / d
+
+            max_scaling = newton(lambda scaling: self.get_past_range(number_of_bins_d,
+                                                                     min_first_bin_size,
+                                                                     scaling) - past_range_T,
+                                 np.log10(past_range_T / min_first_bin_size) / (number_of_bins_d - 1),
+                                 tol=1e-04, maxiter=500)
+
+        while np.linspace(min_scaling, max_scaling,
+                          number_of_scalings, retstep=True)[1] < min_step_for_scaling:
+            number_of_scalings -= 1
+
+        return np.linspace(min_scaling, max_scaling, number_of_scalings)

+
+
[docs]    def get_past_range(self, number_of_bins_d, first_bin_size, scaling_k):
+        """
+        Get the past range T of the embedding, based on the parameters d, tau_1 and k.
+        """
+
+        return np.sum([first_bin_size * 10 ** ((number_of_bins_d - i) * scaling_k)
+                       for i in range(1, number_of_bins_d + 1)])

+
+
[docs]    def get_history_dependence(self, data, process):
+        """
+            Estimate the history dependence for each embedding to all given processes.
+        """
+
+        # load estimators
+        if self.settings['estimation_method'] == 'bbc':
+            estbbc = RudeltBBCEstimator()
+        elif self.settings['estimation_method'] == 'shuffling':
+            estshu = RudeltShufflingEstimator()
+
+        # get history dependence
+        history_dependence = np.empty(shape=(len(self.embeddings)))
+        if self.settings['estimation_method'] == 'bbc':
+            bbc_term = np.empty(shape=(len(self.embeddings)))
+
+        embedding_count = 0
+        for embedding in self.embeddings:
+
+            if self.settings['debug']:
+                print("Embedding: " + str(embedding[0]) + ", " + str(embedding[1]) + ", " + str(embedding[2]))
+
+            symbol_array, past_symbol_array, current_symbol_array, symbol_array_length = \
+                    data.get_realisations_symbols(process,
+                                                  embedding[0],
+                                                  embedding[1],
+                                                  embedding[2],
+                                                  self.settings['embedding_step_size'],
+                                                  output_spike_times=False)
+
+            if self.settings['estimation_method'] == 'bbc':
+                I_bbc, R_bbc, bbc_t = \
+                            estbbc.estimate(symbol_array[0], past_symbol_array[0], current_symbol_array[0])
+                history_dependence[embedding_count] = R_bbc
+                bbc_term[embedding_count] = bbc_t
+
+                if self.settings['debug']:
+                    print("\tHD: " + str(R_bbc) + " BBC: " + str(bbc_t))
+
+            elif self.settings['estimation_method'] == 'shuffling':
+                I_sh, R_sh = estshu.estimate(symbol_array[0])
+                history_dependence[embedding_count] = R_sh
+
+                if self.settings['debug']:
+                    print("\tHD: " + str(R_sh))
+
+            embedding_count += 1
+
+        if self.settings['estimation_method'] == 'bbc':
+            return history_dependence, bbc_term
+        elif self.settings['estimation_method'] == 'shuffling':
+            return history_dependence

+
+
[docs]    def get_bootstrap_history_dependence(self,
+                                         data,
+                                         embedding,
+                                         number_of_bootstraps,
+                                         symbol_block_length=None):
+        """
+        For a given embedding, return bootstrap replications for R.
+        """
+        estbbc = RudeltBBCEstimator()
+        estshu = RudeltShufflingEstimator()
+
+        if symbol_block_length is not None:
+            symbol_block_length = int(symbol_block_length)
+
+        # compute the bootstrap replications
+        bs_Rs = np.zeros(number_of_bootstraps)
+
+        for rep in range(number_of_bootstraps):
+            bs_symbol_array, bs_past_symbol_array, bs_current_symbol_array = \
+                data.get_bootstrap_realisations_symbols(self.process,
+                                                        embedding[0],
+                                                        embedding[1],
+                                                        embedding[2],
+                                                        self.settings['embedding_step_size'],
+                                                        symbol_block_length=symbol_block_length)
+
+            if self.settings['estimation_method'] == 'bbc':
+                I_bbc, R_bbc, bbc_t = \
+                    estbbc.estimate(bs_symbol_array[0], bs_past_symbol_array[0], bs_current_symbol_array[0])
+                bs_Rs[rep] = R_bbc
+
+                if self.settings['debug']:
+                    print("\tHD: " + str(R_bbc) + " BBC: " + str(bbc_t))
+
+            elif self.settings['estimation_method'] == 'shuffling':
+                I_sh, R_sh = estshu.estimate(bs_symbol_array[0])
+                bs_Rs[rep] = R_sh
+
+                if self.settings['debug']:
+                    print("\tHD: " + str(R_sh))
+
+        return bs_Rs

+
+
[docs]    def get_temporal_depth_T_D(self, get_R_thresh=False):
+        """
+        Get the temporal depth T_D, the past range for the
+        'optimal' embedding parameters.
+
+        Given the maximal history dependence R at each past range T,
+        (cf get_embeddings_that_maximise_R), first find the smallest T at
+        which R is maximised (cf get_max_R_T).  If bootstrap replications
+        for this R are available, get the smallest T at which this R minus
+        one standard deviation of the bootstrap estimates is attained.
+        """
+
+        # load data
+        embedding_maximising_R_at_T, max_Rs \
+            = self.get_embeddings_that_maximise_R()
+
+        Ts = sorted([key for key in max_Rs.keys()])
+        Rs = [max_Rs[T] for T in Ts]
+
+        # first get the max history dependence, and if available its bootstrap replications
+        max_R, max_R_T = utl.get_max_R_T(max_Rs)
+
+        number_of_bins_d, scaling_k = embedding_maximising_R_at_T[max_R_T]
+
+        embindex = self.embeddings.index((max_R_T, number_of_bins_d, scaling_k))
+
+        bs_Rs = self.bs_history_dependence[embindex]
+
+        if isinstance(bs_Rs, np.ndarray):
+            max_R_sd = np.std(bs_Rs)
+        else:
+            max_R_sd = 0
+
+        R_tot_thresh = max_R - max_R_sd
+
+        T_D = min(Ts)
+        for R, T in zip(Rs, Ts):
+            if R >= R_tot_thresh:
+                T_D = T
+                break
+
+        if not get_R_thresh:
+            return T_D
+        else:
+            return T_D, R_tot_thresh

+
+
[docs]    def get_embeddings_that_maximise_R(self,
+                                       bbc_tolerance=None,
+                                       dependent_var="T",
+                                       get_as_list=False):
+        """
+        For each T (or d), get the embedding for which R is maximised.
+
+        For the bbc estimator, here the bbc_tolerance is applied, ie
+        get the unbiased embeddings that maximise R.
+        """
+
+        assert dependent_var in ["T", "d"]
+
+        if bbc_tolerance is None:
+            bbc_tolerance = np.inf
+
+        max_Rs = {}
+        embeddings_that_maximise_R = {}
+
+        for i in range(len(self.embeddings)):
+            embedding = self.embeddings[i]
+            past_range_T = float(embedding[0])
+            number_of_bins_d = int(float(embedding[1]))
+            scaling_k = float(embedding[2])
+            history_dependence = self.history_dependence[i]
+
+            if self.settings['estimation_method'] == "bbc":
+                if self.bbc_term[i] >= self.settings['bbc_tolerance']:                          # ----------------------- TODO check
+                    continue
+
+            if dependent_var == "T":
+                if not past_range_T in embeddings_that_maximise_R \
+                                or history_dependence > max_Rs[past_range_T]:
+                    max_Rs[past_range_T] = history_dependence
+                    embeddings_that_maximise_R[past_range_T] = (number_of_bins_d, scaling_k)
+            elif dependent_var == "d":
+                if not number_of_bins_d in embeddings_that_maximise_R \
+                                or history_dependence > max_Rs[number_of_bins_d]:
+                    max_Rs[number_of_bins_d] = history_dependence
+                    embeddings_that_maximise_R[number_of_bins_d] = (past_range_T, scaling_k)
+
+        if get_as_list:
+            embeddings = []
+            if dependent_var == "T":
+                for past_range_T in embeddings_that_maximise_R:
+                    number_of_bins_d, scaling_k = embeddings_that_maximise_R[past_range_T]
+                    embeddings += [(past_range_T, number_of_bins_d, scaling_k)]
+            elif dependent_var == "d":
+                for number_of_bins_d in embeddings_that_maximise_R:
+                    past_range_T, scaling_k = embeddings_that_maximise_R[number_of_bins_d]
+                    embeddings += [(past_range_T, number_of_bins_d, scaling_k)]
+            return embeddings
+        else:
+            return embeddings_that_maximise_R, max_Rs

+
+
[docs]    def get_information_timescale_tau_R(self):
+        """
+        Get the information timescale tau_R, a characteristic
+        timescale of history dependence similar to an autocorrelation
+        time.
+        """
+
+        max_Rs = self.max_Rs
+
+        Ts = np.array(sorted([key for key in max_Rs.keys()]))
+        Rs = np.array([max_Rs[T] for T in Ts])
+
+        R_tot = self.get_R_tot()
+
+        T_0 = self.settings["timescale_minimum_past_range"]
+
+        # get dRs
+        dRs = []
+        R_prev = 0.
+
+        # No values higher than R_tot are allowed,
+        # otherwise the information timescale might be
+        # misestimated because of spurious contributions
+        # at large T
+        for R, T in zip(Rs[Rs <= R_tot], Ts[Rs <= R_tot]):
+
+            # No negative increments are allowed
+            dRs += [np.amax([0.0, R - R_prev])]
+
+            # The increment is taken with respect to the highest previous value of R
+            if R > R_prev:
+                R_prev = R
+
+        dRs = np.pad(dRs, (0, len(Rs) - len(dRs)),
+                     mode='constant', constant_values=0)
+
+        # compute tau_R
+        Ts_0 = np.append([0], Ts)
+        dRs_0 = dRs[Ts_0[:-1] >= T_0]
+
+        # Only take into considerations contributions beyond T_0
+        Ts_0 = Ts_0[Ts_0 >= T_0]
+        norm = np.sum(dRs_0)
+
+        if norm == 0.:
+            tau = 0.0
+        else:
+            Ts_0 -= Ts_0[0]
+            tau = np.dot(((Ts_0[:-1] + Ts_0[1:]) / 2), dRs_0) / norm
+        return tau

+
+
[docs]    def get_R_tot(self,
+                  return_averaged_R=False,
+                  **kwargs):
+
+        max_Rs = self.max_Rs
+
+        if return_averaged_R:
+            T_D, R_tot_thresh = self.get_temporal_depth_T_D(get_R_thresh=True)
+
+            Ts = sorted([key for key in max_Rs.keys()])
+            Rs = [max_Rs[T] for T in Ts]
+
+            T_max = T_D
+            for R, T in zip(Rs, Ts):
+                if T < T_D:
+                    continue
+                T_max = T
+                if R < R_tot_thresh:
+                    break
+
+            return np.average([R for R, T in zip(Rs, Ts) if T_D <= T < T_max])
+
+        else:
+
+            temporal_depth_T_D = self.get_temporal_depth_T_D()
+
+            return max_Rs[temporal_depth_T_D]

+
+
[docs]    def compute_CIs(self,
+                    data,
+                    target_R='R_max',
+                    symbol_block_length=None):
+        """
+        Compute bootstrap replications of the history dependence estimate
+        which can be used to obtain confidence intervals.
+
+        Args:
+            data : data_spiketime object
+                Input data
+            target_R : String
+                One of 'R_max', 'R_tot' or 'nonessential'.
+                If set to R_max, replications of R are produced for the T at which
+                R is maximised.
+                If set to R_tot, replications of R are produced for T = T_D (cf
+                get_temporal_depth_T_D).
+                If set to nonessential, replications of R are produced for each T
+                (one embedding per T, cf get_embeddings_that_maximise_R).  These
+                are not otherwise used in the analysis and are probably only useful
+                if the resulting plot is visually inspected, so in most cases it can
+                be set to zero.
+            symbol_block_length : int
+                The number of symbols that should be drawn in each block for bootstrap resampling
+                If it is set to None (recommended), the length is automatically chosen, based
+                on heuristics
+        """
+
+        assert target_R in ['nonessential', 'R_max', 'R_tot']
+
+        number_of_bootstraps = self.settings['number_of_bootstraps_{}'.format(target_R)]
+
+        if number_of_bootstraps == 0:
+            return
+
+        embedding_maximising_R_at_T, max_Rs \
+            = self.get_embeddings_that_maximise_R()
+        self.embedding_maximising_R_at_T = embedding_maximising_R_at_T
+        self.max_Rs = max_Rs
+
+        if target_R == 'nonessential':
+            # bootstrap R for unessential Ts (not required for the main analysis)
+            embeddings = []
+            for past_range_T in embedding_maximising_R_at_T:
+                number_of_bins_d, scaling_k = embedding_maximising_R_at_T[past_range_T]
+                embeddings += [(past_range_T, number_of_bins_d, scaling_k)]
+
+        elif target_R == 'R_max':
+            # bootstrap R for the max R, to get a good estimate for the standard deviation
+            # which is used to determine R_tot
+            max_R, max_R_T = utl.get_max_R_T(max_Rs)
+            self.max_R = max_R
+            self.max_R_T = max_R_T
+            number_of_bins_d, scaling_k = embedding_maximising_R_at_T[max_R_T]
+            embeddings = [(max_R_T, number_of_bins_d, scaling_k)]
+
+        elif target_R == 'R_tot':
+            T_D = self.get_temporal_depth_T_D()
+            number_of_bins_d, scaling_k = embedding_maximising_R_at_T[T_D]
+
+            embeddings = [(T_D, number_of_bins_d, scaling_k)]
+
+        for embedding in embeddings:
+
+            embindex = self.embeddings.index(embedding)
+
+            if hasattr(self, "bs_history_dependence"):
+                stored_bs_Rs = self.bs_history_dependence[embindex]
+            else:
+                self.bs_history_dependence = dict()
+                stored_bs_Rs = None
+
+            if isinstance(stored_bs_Rs, np.ndarray):
+                number_of_stored_bootstraps = len(stored_bs_Rs)
+            else:
+                number_of_stored_bootstraps = 0
+
+            if not number_of_bootstraps > number_of_stored_bootstraps:
+                continue
+
+            bs_R = \
+                self.get_bootstrap_history_dependence(data,
+                                                      embedding,
+                                                      number_of_bootstraps - number_of_stored_bootstraps,
+                                                      symbol_block_length=symbol_block_length)
+
+            if stored_bs_Rs is not None:
+                bs_R = np.concatenate([stored_bs_Rs, bs_R])
+
+            self.bs_history_dependence[embindex] = bs_R

+
+
[docs]    def analyse_auto_MI(self, spike_times):
+        """
+        Get the auto MI for the spike times.  If it is available from file, load
+        it, else compute it.
+        """
+
+        auto_MI_data = {
+            "delay": [],
+            "auto_MI": []
+        }
+        auto_MI_dict = {}
+        for auto_MI_bin_size in self.settings['auto_MI_bin_size_set']:
+            number_of_delays = int(self.settings['auto_MI_max_delay'] / auto_MI_bin_size) + 1
+
+            # perform the MI analysis
+            auto_MI = self.get_auto_MI(spike_times, auto_MI_bin_size, number_of_delays)
+
+            auto_MI_data["delay"] += [number_of_delays]
+
+            auto_MI_dict[auto_MI_bin_size] = auto_MI
+
+        auto_MI_data['auto_MI'] = auto_MI_dict
+        self.auto_MI = auto_MI_data

+
+
[docs]    def get_auto_MI(self, spike_times, bin_size, number_of_delays):
+        """
+        Compute the auto mutual information in the neuron's activity, a
+        measure closely related to history dependence.
+        """
+
+        binned_neuron_activity = utl.get_binned_neuron_activity(spike_times, bin_size, relative_to_median_activity=True)
+
+        p_spike = sum(binned_neuron_activity) / len(binned_neuron_activity)
+
+        self.H_spiking = utl.get_shannon_entropy([p_spike,
+                                         1 - p_spike])
+
+        auto_MIs = np.empty(number_of_delays)
+
+        # compute auto MI
+        for delay in range(number_of_delays):
+
+            symbol_counts = []
+            number_of_symbols = len(binned_neuron_activity) - delay - 1
+            symbols = np.array([2 * binned_neuron_activity[i] + binned_neuron_activity[i + delay + 1]
+                                for i in range(number_of_symbols)])
+            symbol_counts += [dict([(unq_symbol, len(np.where(symbols == unq_symbol)[0]))
+                                    for unq_symbol in np.unique(symbols)])]
+
+            symbol_counts = utl.add_up_dicts(symbol_counts)
+            number_of_symbols = sum(symbol_counts.values())
+
+            H_joint = utl.get_shannon_entropy([number_of_occurrences / number_of_symbols
+                                           for number_of_occurrences in symbol_counts.values()])
+
+            # I(X : Y) = H(X) - H(X|Y) = H(X) - (H(X,Y) - H(Y)) = H(X) + H(Y) - H(X,Y)
+            # auto_MI = 2 * H_spiking - H_joint
+            auto_MIs[delay] = 2 - H_joint / self.H_spiking  # normalized auto MI = auto MI / H_spiking
+
+        return auto_MIs

+
+
[docs]    def optimize(self, data, processes='all'):
+        """
+        Optimize the embedding parameters of spike time data using the Rudelt history dependence estimator.
+
+        References:
+
+            [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
+                optimization reveals long-lasting history dependence in
+                neural spiking activity, 2021, PLOS Computational Biology, 17(6)
+
+            [2]: https://github.com/Priesemann-Group/hdestimator
+
+        implemented in idtxl by Michael Lindner, Göttingen 2021
+
+        Args:
+            data : Data_spiketime instance
+                raw data for analysis
+            processes : list of int
+                index of processes;
+                spike times are optimized all processes specified in the list separately.
+
+        Returns: # -------------------------------------------------------------------------------------------------------- TODO
+            ResultsSingleProcessRudelt instance
+                results of Rudelt optimization, see documentation of
+                ResultsSingleProcessRudelt()
+            if visulization in settings was set True (see class OptimizationRudelt):
+                .eps images are created for each optimized process containing:
+                    - optimized values for the process
+                    - graph for the history dependence
+                    - graph for auto mutual information (if calculated)
+
+        """
+
+        # check input data
+        if type(data) != Data_spiketime:
+            raise ValueError('Input Data nneds to be Data_spiketime object but is defined as: '
+                             '{0}.'.format(type(data)))
+
+        # check input process list
+        if processes == 'all':
+            processes = [t for t in range(data.n_processes)]
+        elif (type(processes) is list) and (type(processes[0]) is int):
+            pass
+        else:
+            raise ValueError('Processes were not specified correctly: '
+                             '{0}.'.format(processes))
+
+        if self.settings['debug']:
+            import pprint
+            pprint.pprint(self.settings, width=1)
+
+        # open result dict
+        results = ResultsSingleProcessRudelt(
+            processes=processes)
+
+        # start optimizing given processes
+        process_count = 0
+        for process in processes:
+
+            # optimize single process
+            single_result = self.optimize_single_run(data, process)
+
+            # add results of single process to result object
+            results._add_single_result(
+                process_count=process_count,
+                process=process,
+                settings=self.settings,
+                results=single_result)
+
+            process_count += 1
+
+            if self.settings['visualization'] == True:
+                filename = Path(self.settings['output_path']).joinpath(
+                     '{}_process{}.svg'.format(self.settings['output_prefix'], process))
+                utl.hde_visualize_results(results, process, filename)
+
+            # remove results from single process from self
+            self.remove_subresults_single_process()
+
+        return results

+
+
[docs]    def optimize_single_run(self, data, process):
+        """
+            optimizes a single realisation of spike time data given the process number
+
+            Args:
+                data : Data_spiketime instance
+                    raw data for analysis
+                process : int
+                    index of process;
+
+            Returns:
+                DotDict
+                    with the following keys
+
+                    Process : int
+                        Process that was optimized
+                    estimation_method : String
+                        Estimation method that was used for optimization
+                    T_D : float
+                        Estimated optimal value for the temporal depth TD
+                    tau_R :
+                        Information timescale tau_R, a characteristic timescale of history
+                        dependence similar to an autocorrelation time.
+                    R_tot : float
+                        Estimated value for the total history dependence Rtot,
+                    AIS_tot : float
+                        Estimated value for the total active information storage
+                    opt_number_of_bins_d : int
+                        Number of bins d for the embedding that yields (R̂tot ,T̂D)
+                    opt_scaling_k : int
+                        Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
+                    opt_first_bin_size : int
+                        Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
+                    history_dependence : array with floating-point values
+                        Estimated history dependence for each embedding
+                    firing_rate : float
+                        Firing rate of the neuron/ spike train
+                    recording_length : float
+                        Length of the recording (in seconds)
+                    H_spiking : float
+                        Entropy of the spike times
+
+            if analyse_auto_MI was set to True additionally:
+                auto_MI : dict
+                    numpy array of MI values for each delay
+                auto_MI_delays : list of int
+                    list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay
+        """
+
+        if (type(process) is int):
+            pass
+        else:
+            raise ValueError('Process is not specified correctly: '
+                             '{0}.'.format(process))
+
+        self.process = process
+
+        # get history dependence
+        if self.settings['debug']:
+            print("\n\nGet History dependence\n")
+
+        if self.settings['estimation_method'] == 'bbc':
+            self.history_dependence, self.bbc_term = \
+                self.get_history_dependence(data, process)
+        elif self.settings['estimation_method'] == 'shuffling':
+            self.history_dependence = \
+                self.get_history_dependence(data, process)
+
+        if self.settings['debug']:
+            print("\n\nCompute CI\n")
+
+        # get bootstrap history dependence (CI)
+        if self.settings['debug']:
+            print("R_max")
+        self.compute_CIs(data,
+                         target_R='R_max',
+                         symbol_block_length=self.settings['symbol_block_length'])
+        if self.settings['debug']:
+            print("R_tot")
+        self.compute_CIs(data,
+                         target_R='R_tot',
+                         symbol_block_length=self.settings['symbol_block_length'])
+        if self.settings['debug']:
+            print("R_nonessential")
+        self.compute_CIs(data,
+                         target_R='nonessential',
+                         symbol_block_length=self.settings['symbol_block_length'])
+
+        # analyse auto MI
+        if self.settings['analyse_auto_MI']:
+            if self.settings['debug']:
+                print("\nAnalyse auto MI\n")
+            spike_times = data.get_spike_times_single(process)
+            self.analyse_auto_MI(spike_times)
+
+
+        # get output values
+        T_D = self.get_temporal_depth_T_D()
+        tau_R = self.get_information_timescale_tau_R()
+        R_tot = self.get_R_tot()
+        opt_number_of_bins_d, opt_scaling_k = self.embedding_maximising_R_at_T[T_D]
+        max_Rs = self.max_Rs
+        mr = np.array(list(max_Rs.items()), dtype=float)
+        HD_max_R = mr[:, 1]
+        opt_first_bin_size = newton(lambda first_bin_size: self.get_past_range(opt_number_of_bins_d,
+                                                              first_bin_size,
+                                                              opt_scaling_k) - T_D,
+                                    0.005, tol=1e-03, maxiter=100)
+        firing_rate = data.get_firingrate(process, self.settings['embedding_step_size'])
+        recording_length = data.get_recording_length(process)
+        H_spiking = data.get_H_spiking(process, self.settings['embedding_step_size'])
+
+        # get CI bounds
+        if not self.settings['return_averaged_R']:
+            embedding = (T_D,
+                         opt_number_of_bins_d,
+                         opt_scaling_k)
+            emb_ind = self.embeddings.index(embedding)
+            R_tot_CI_lo, R_tot_CI_hi = utl.get_CI_bounds(R_tot,
+                                                         self.bs_history_dependence[emb_ind],
+                                                         self.settings['bootstrap_CI_use_sd'],
+                                                         self.settings['bootstrap_CI_percentile_lo'],
+                                                         self.settings['bootstrap_CI_percentile_hi'])
+        else:
+            R_tot_CI_lo = None
+            R_tot_CI_hi = None
+        R_tot_CI = [R_tot_CI_lo, R_tot_CI_hi]
+
+        if self.settings['debug']:
+            print('Process: ' + str(process))
+            print('T_D: ' + str(T_D))
+            print('tau_R: ' + str(tau_R))
+            print('R_tot: ' + str(R_tot))
+            print('R_tot_CI: ' + str(R_tot_CI))
+            print('opt_number_of_bins_d: ' + str(opt_number_of_bins_d))
+            print('opt_scaling_k: ' + str(opt_scaling_k))
+            print('opt_first_bin_size: ' + str(opt_first_bin_size))
+            print('firing_rate: ' + str(firing_rate))
+            print('recording_length: ' + str(recording_length))
+            print('H_spiking: ' + str(H_spiking))
+
+        # create output dict
+        results = {'Process': process,
+                   'estimation_method': self.settings['estimation_method'],
+                   'T_D': T_D,
+                   'tau_R': tau_R,
+                   'R_tot': R_tot,
+                   'R_tot_CI': R_tot_CI,
+                   'AIS_tot': R_tot * H_spiking,
+                   'opt_number_of_bins_d': opt_number_of_bins_d,
+                   'opt_scaling_k': opt_scaling_k,
+                   'opt_first_bin_size': opt_first_bin_size,
+                   'history_dependence': self.history_dependence,
+                   'firing_rate': firing_rate,
+                   'recording_length': recording_length,
+                   'H_spiking': H_spiking,
+                   'max_R': max_Rs}
+
+        if self.settings['analyse_auto_MI']:
+            results['auto_MI'] = self.auto_MI.get('auto_MI')
+            results['auto_MI_delays'] = self.auto_MI.get('delay')
+
+        results_d = DotDict(results)
+
+        return results_d

+
+
[docs]    def remove_subresults_single_process(self):
+        """delete results from self from single process"""
+
+        del self.bs_history_dependence
+        del self.embedding_maximising_R_at_T
+        del self.history_dependence
+        del self.max_R
+        del self.max_Rs
+        del self.max_R_T
+        del self.process
+        # del self.T_D
+        if self.settings['analyse_auto_MI']:
+            del self.auto_MI
+            del self.H_spiking
+        if self.settings['estimation_method'] == 'bbc':
+            del self.bbc_term

+
+ +
+
+
+
+ +
+
+ + + + \ No newline at end of file diff --git a/docs/html/_modules/idtxl/estimators_Rudelt.html b/docs/html/_modules/idtxl/estimators_Rudelt.html new file mode 100644 index 00000000..c4d86d52 --- /dev/null +++ b/docs/html/_modules/idtxl/estimators_Rudelt.html @@ -0,0 +1,1217 @@ + + + + + + + + idtxl.estimators_Rudelt — IDTxl 1.2.1 documentation + + + + + + + + + + + + + + + + +
+
+
+
+ +

Source code for idtxl.estimators_Rudelt

+"""Provide HDE estimators."""
+
+import logging
+import numpy as np
+from scipy.optimize import newton, minimize
+import sys
+from sys import stderr
+from idtxl.estimator import Estimator
+import idtxl.hde_utils as utl
+from collections import Counter
+import mpmath as mp
+
+FAST_EMBEDDING_AVAILABLE = True
+try:
+    import idtxl.hde_fast_embedding as fast_emb
+except:
+    FAST_EMBEDDING_AVAILABLE = False
+    print("""
+    Error importing Cython fast embedding module. Continuing with slow Python implementation.\n
+    This may take a long time.\n
+    """, file=stderr, flush=True)
+
+
+logger = logging.getLogger(__name__)
+
+
+
[docs]class RudeltAbstractEstimator(Estimator):
+    """
+    Abstract class for implementation of nsb and plugin estimators from Rudelt.
+
+    Abstract class for implementation of nsb and plugin estimators, child classes
+    implement estimators for mutual information (MI) .
+
+    References:
+
+        [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
+            optimization reveals long-lasting history dependence in
+            neural spiking activity, 2021, PLOS Computational Biology, 17(6)
+
+        [2]: https://github.com/Priesemann-Group/hdestimator
+
+    implemented in idtxl by Michael Lindner, Göttingen 2021
+
+    Args:
+        settings : dict
+            - embedding_step_size : float [optional]
+                Step size delta t (in seconds) with which the window is slid through the data
+                (default = 0.005).
+            - normalise : bool [optional]
+                rebase spike times to zero
+                (default=True)
+            - return_averaged_R : bool [optional]
+                If set to True, compute R̂tot as the average over R̂(T ) for T ∈ [T̂D, Tmax ] instead of
+                R̂tot = R(T̂D ). If set to True, the setting for number_of_bootstraps_R_tot is ignored and
+                set to 0
+                (default=True)
+    """
+
+    def __init__(self, settings=None):
+        # check settings
+        settings = self._check_settings()
+        # import given settings
+        self.settings = settings.copy()
+
+        # Get defaults for estimator settings
+        self.settings.setdefault('normalize', True)
+        self.settings.setdefault('embedding_step_size', 0.005)
+        self.settings.setdefault('return_averaged_R', True)
+
+        # check settings
+        self._check_input_settings()
+
+
[docs]    def is_parallel(self):
+        return False

+
+
[docs]    def is_analytic_null_estimator(self):
+        return False

+
+    def _check_settings(self, settings=None):
+        """Set default for settings dictionary.
+
+        Check if settings dictionary is None. If None, initialise an empty
+        dictionary. If not None check if type is dictionary. Function should be
+        called before setting default values.
+        """
+        if settings is None:
+            return {}
+        elif type(settings) is not dict:
+            raise TypeError('settings should be a dictionary.')
+        else:
+            return settings
+
+    def _check_input_settings(self):
+
+        # check that required settings are defined
+        required_settings = ['normalize',
+                             'embedding_step_size',
+                             'return_averaged_R']
+
+        # check if all settings are defined
+        for required_setting in required_settings:
+            if not required_setting in self.settings:
+                sys.exit("Error in settings file: {} is not defined. Aborting.".format(required_setting))
+
+        assert (isinstance(self.settings['normalize'], bool)), \
+                                    "Error: setting 'normalize' needs to be boolean but is defined as {0}. " \
+                                     "Aborting.".format(type(self.settings['normalize']))
+
+        assert (isinstance(self.settings['return_averaged_R'], bool)),\
+                                    "Error: setting 'return_averaged_R' needs to be boolean but is " \
+                                    "defined as {0}. Aborting.".format(type(self.settings['normalize']))
+
+        assert (isinstance(self.settings['embedding_step_size'], float)),\
+                                    "Error: setting 'embedding_step_size' " \
+                                    "needs to be float but is defined " \
+                                    "as {0}. Aborting.".format(type(self.settings['embedding_step_size']))
+
+    def _check_estimator_inputs(self,
+                                symbol_array,
+                                past_symbol_array,
+                                current_symbol_array,
+                                bbc_tolerance):
+        assert (isinstance(symbol_array, np.ndarray)), \
+            "Error: symbol_array needs to be a numpy array but is defines as {0}." \
+            "Aborting.".format(type(symbol_array))
+
+        if past_symbol_array is not None:
+            assert (isinstance(past_symbol_array, np.ndarray)), \
+                "Error: past_symbol_array needs to be a numpy array but is defines as {0}." \
+                "Aborting.".format(type(past_symbol_array))
+            assert (len(past_symbol_array) == len(symbol_array)), \
+                "Error: symbol_array and past_symbol_array need to have the same length but have:" \
+                "len(symbol_array): {0} len(past_symbol_array): {1}. " \
+                "Aborting". format(len(symbol_array), len(past_symbol_array))
+
+        if current_symbol_array is not None:
+            assert (isinstance(current_symbol_array, np.ndarray)), \
+                "Error: current_symbol_array needs to be a numpy array but is defines as {0}." \
+                "Aborting.".format(type(current_symbol_array))
+            assert (len(current_symbol_array) == len(symbol_array)), \
+                "Error: symbol_array and current_symbol_array need to have the same length but have:" \
+                "len(symbol_array): {0} len(current_symbol_array): {1}. " \
+                "Aborting".format(len(symbol_array), len(current_symbol_array))
+
+        if bbc_tolerance is not None:
+            assert (isinstance(bbc_tolerance, np.ndarray)), \
+                "Error: symbol array needs to be a numpy array but is defines as {0}." \
+                "Aborting.".format(type(current_symbol_array))
+
+    def _ensure_one_dim(self, var):
+        """
+        check if array is 1D
+        """
+        var = np.squeeze(var)
+        assert (var.ndim == 1), "Input variable needs to be one dimensional. Aborting"
+
+
[docs]    def get_past_range(self, number_of_bins_d, first_bin_size, scaling_k):
+        """
+        Get the past range T of the embedding, based on the parameters d, tau_1 and k.
+        """
+
+        return np.sum([first_bin_size * 10 ** ((number_of_bins_d - i) * scaling_k)
+                       for i in range(1, number_of_bins_d + 1)])

+
+
[docs]    def get_window_delimiters(self, number_of_bins_d, scaling_k, first_bin_size):
+        """
+        Get delimiters of the window, used to describe the embedding. The
+        window includes both the past embedding and the response.
+
+        The delimiters are times, relative to the first bin, that separate
+        two consequent bins.
+        """
+
+        bin_sizes = [first_bin_size * 10 ** ((number_of_bins_d - i) * scaling_k)
+                     for i in range(1, number_of_bins_d + 1)]
+        window_delimiters = [sum([bin_sizes[j] for j in range(i)])
+                             for i in range(1, number_of_bins_d + 1)]
+        window_delimiters.append(window_delimiters[number_of_bins_d - 1] + self.settings['embedding_step_size'])
+        return window_delimiters

+
+
[docs]    def get_median_number_of_spikes_per_bin(self, raw_symbols):
+        """
+        Given raw symbols (in which the number of spikes per bin are counted,
+        ie not necessarily binary quantity), get the median number of spikes
+        for each bin, among all symbols obtained by the embedding.
+        """
+
+        # number_of_bins here is number_of_bins_d + 1,
+        # as it here includes not only the bins of the embedding but also the response
+        number_of_bins = len(raw_symbols[0])
+
+        spike_counts_per_bin = [[] for i in range(number_of_bins)]
+
+        for raw_symbol in raw_symbols:
+            for i in range(number_of_bins):
+                spike_counts_per_bin[i] += [raw_symbol[i]]
+
+        return [np.median(spike_counts_per_bin[i]) for i in range(number_of_bins)]

+
+
[docs]    def symbol_binary_to_array(self, symbol_binary, number_of_bins_d):
+        """
+        Given a binary representation of a symbol (cf symbol_array_to_binary),
+        convert it back into its array-representation.
+        """
+
+        # assert 2 ** number_of_bins_d > symbol_binary
+
+        spikes_in_window = np.zeros(number_of_bins_d)
+        for i in range(0, number_of_bins_d):
+            b = 2 ** (number_of_bins_d - 1 - i)
+            if b <= symbol_binary:
+                spikes_in_window[i] = 1
+                symbol_binary -= b
+        return spikes_in_window

+
+
[docs]    def symbol_array_to_binary(self, spikes_in_window, number_of_bins_d):
+        """
+        Given an array of 1s and 0s, representing spikes and the absence
+        thereof, read the array as a binary number to obtain a
+        (base 10) integer.
+        """
+
+        # assert len(spikes_in_window) == number_of_bins_d
+
+        # TODO check if it makes sense to use len(spikes_in_window)
+        # directly, to avoid mismatch as well as confusion
+        # as number_of_bins_d here can also be number_of_bins
+        # as in get_median_number_of_spikes_per_bin, ie
+        # including the response
+
+        return sum([2 ** (number_of_bins_d - i - 1) * spikes_in_window[i]
+                    for i in range(0, number_of_bins_d)])

+
+
[docs]    def get_raw_symbols(self,
+                        spike_times,
+                        embedding,
+                        first_bin_size):
+        """
+        Get the raw symbols (in which the number of spikes per bin are counted,
+        ie not necessarily binary quantity), as obtained by applying the
+        embedding.
+        """
+
+        past_range_T, number_of_bins_d, scaling_k = embedding
+
+        # the window is the embedding plus the response,
+        # ie the embedding and one additional bin of size embedding_step_size
+        window_delimiters = self.get_window_delimiters(number_of_bins_d,
+                                                       scaling_k,
+                                                       first_bin_size)
+        window_length = window_delimiters[-1]
+        num_spike_times = len(spike_times)
+        last_spike_time = spike_times[-1]
+
+        num_symbols = int((last_spike_time - window_length) / self.settings['embedding_step_size'])
+
+        raw_symbols = []
+
+        time = 0
+        spike_index_lo = 0
+
+        for symbol_num in range(num_symbols):
+            while spike_index_lo < num_spike_times and spike_times[spike_index_lo] < time:
+                spike_index_lo += 1
+            spike_index_hi = spike_index_lo
+            while (spike_index_hi < num_spike_times and
+                   spike_times[spike_index_hi] < time + window_length):
+                spike_index_hi += 1
+
+            spikes_in_window = np.zeros(number_of_bins_d + 1)
+
+            embedding_bin_index = 0
+            for spike_index in range(spike_index_lo, spike_index_hi):
+                while (spike_times[spike_index] > time + window_delimiters[embedding_bin_index]):
+                    embedding_bin_index += 1
+                spikes_in_window[embedding_bin_index] += 1
+
+            raw_symbols += [spikes_in_window]
+
+            time += self.settings['embedding_step_size']
+
+        return raw_symbols

+
+
[docs]    def get_symbol_counts(self, symbol_array):
+        """
+        Count how often symbols occur
+        """
+
+        symbol_counts = Counter()
+        for symbol in np.unique(symbol_array):
+            symbol_counts[symbol] += len(np.where(symbol_array == symbol)[0])
+
+        return symbol_counts

+
+
[docs]    def get_multiplicities(self, symbol_counts, alphabet_size):
+        """
+        Get the multiplicities of some given symbol counts.
+
+        To estimate the entropy of a system, it is only important how
+        often a symbol/ event occurs (the probability that it occurs), not
+        what it represents. Therefore, computations can be simplified by
+        summarizing symbols by their frequency, as represented by the
+        multiplicities.
+        """
+
+        mk = dict(((value, 0) for value in symbol_counts.values()))
+        number_of_observed_symbols = np.count_nonzero([value for value in symbol_counts.values()])
+
+        for symbol in symbol_counts.keys():
+            mk[symbol_counts[symbol]] += 1
+
+        # the number of symbols that have not been observed in the data
+        mk[0] = alphabet_size - number_of_observed_symbols
+
+        return mk

+
+
+
[docs]class RudeltAbstractNSBEstimator(RudeltAbstractEstimator):
+    """Abstract class for implementation of NSB estimators from Rudelt.
+
+    Abstract class for implementation of Nemenman-Shafee-Bialek (NSB)
+    estimators, child classes implement nsb estimators for mutual information
+    (MI).
+
+    implemented in idtxl by Michael Lindner, Göttingen 2021
+
+    References:
+
+        [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
+            optimization reveals long-lasting history dependence in
+            neural spiking activity, 2021, PLOS Computational Biology, 17(6)
+        [2]: I. Nemenman, F. Shafee, W. Bialek: Entropy and inference,
+            revisited. In T.G. Dietterich, S. Becker, and Z. Ghahramani,
+            editors, Advances in Neural Information Processing Systems 14,
+            Cambridge, MA, 2002. MIT Press.
+
+    Args:
+        settings : dict
+            - embedding_step_size : float [optional]
+                Step size delta t (in seconds) with which the window is slid through the data
+                (default = 0.005).
+            - normalise : bool [optional]
+                rebase spike times to zero
+                (default=True)
+            - return_averaged_R : bool [optional]
+                If set to True, compute R̂tot as the average over R̂(T ) for T ∈ [T̂D, Tmax ] instead of
+                R̂tot = R(T̂D ). If set to True, the setting for number_of_bootstraps_R_tot is ignored and
+                set to 0
+                (default=True)
+    """
+
+    def __init__(self, settings=None):
+        # Set default estimator settings.
+        super().__init__(settings)
+
+
[docs]    def d_xi(self, beta, K):
+        """
+        First derivative of xi(beta).
+
+        xi(beta) is the entropy of the system when no data has been observed.
+        d_xi is the prior for the nsb estimator
+        """
+
+        return K * mp.psi(1, K * beta + 1.) - mp.psi(1, beta + 1.)

+
+
[docs]    def d2_xi(self, beta, K):
+        """
+        Second derivative of xi(beta) (cf d_xi).
+        """
+
+        return K ** 2 * mp.psi(2, K * beta + 1) - mp.psi(2, beta + 1)

+
+
[docs]    def d3_xi(self, beta, K):
+        """
+        Third derivative of xi(beta) (cf d_xi).
+        """
+
+        return K ** 3 * mp.psi(3, K * beta + 1) - mp.psi(3, beta + 1)

+
+
[docs]    def rho(self, beta, mk, K, N):
+        """
+        rho(beta, data) is the Dirichlet multinomial likelihood.
+
+        rho(beta, data) together with the d_xi(beta) make up
+        the posterior for the nsb estimator
+        """
+
+        return np.prod([mp.power(mp.rf(beta, np.double(n)), mk[n]) for n in mk]) / mp.rf(K * beta,
+                                                                                         np.double(N))

+
+
[docs]    def unnormalized_posterior(self, beta, mk, K, N):
+        """
+        The (unnormalized) posterior in the nsb estimator.
+
+        Product of the likelihood rho and the prior d_xi;
+        the normalizing factor is given by the marginal likelihood
+        """
+
+        return self.rho(beta, mk, K, N) * self.d_xi(beta, K)

+
+
[docs]    def d_log_rho(self, beta, mk, K, N):
+        """
+        First derivate of the logarithm of the Dirichlet multinomial likelihood.
+        """
+
+        return K * (mp.psi(0, K * beta) - mp.psi(0, K * beta + N)) - K * mp.psi(0, beta) \
+               + np.sum((mk[n] * mp.psi(0, n + beta) for n in mk))

+
+
[docs]    def d2_log_rho(self, beta, mk, K, N):
+        """
+        Second derivate of the logarithm of the Dirichlet multinomial likelihood.
+        """
+
+        return K ** 2 * (mp.psi(1, K * beta) - mp.psi(1, K * beta + N)) - K * mp.psi(1, beta) \
+               + np.sum((mk[n] * mp.psi(1, n + beta) for n in mk))

+
+
[docs]    def d_log_rho_xi(self, beta, mk, K, N):
+        """
+        First derivative of the logarithm of the nsb (unnormalized) posterior.
+        """
+
+        return self.d_log_rho(beta, mk, K, N) + self.d2_xi(beta, K) / self.d_xi(beta, K)

+
+
[docs]    def d2_log_rho_xi(self, beta, mk, K, N):
+        """
+        Second derivative of the logarithm of the nsb (unnormalized) posterior.
+        """
+
+        return self.d2_log_rho(beta, mk, K, N) \
+               + (self.d3_xi(beta, K) * self.d_xi(beta, K) - self.d2_xi(beta, K) ** 2) / self.d_xi(beta, K) ** 2

+
+
[docs]    def log_likelihood_DP_alpha(self, a, K1, N):
+        """
+        Alpha-dependent terms of the log-likelihood of a Dirichlet Process.
+        """
+
+        return (K1 - 1.) * mp.log(a) - mp.log(mp.rf(a + 1., N - 1.))

+
+
[docs]    def get_beta_MAP(self, mk, K, N):
+        """
+        Get the maximum a posteriori (MAP) value for beta.
+
+        Provides the location of the peak, around which we integrate.
+
+        beta_MAP is the value for beta for which the posterior of the
+        NSB estimator is maximised (or, equivalently, of the logarithm
+        thereof, as computed here).
+        """
+
+        K1 = K - mk[0]
+
+        if self.d_log_rho(10 ** 1, mk, K, N) > 0:
+            print("Warning: No ML parameter was found.", file=stderr, flush=True)
+            beta_MAP = np.float('nan')
+        else:
+            try:
+                # first guess computed via posterior of Dirichlet process
+                DP_est = self.alpha_ML(mk, K1, N) / K
+                beta_MAP = newton(lambda beta: float(self.d_log_rho_xi(beta, mk, K, N)), DP_est,
+                                  lambda beta: float(self.d2_log_rho_xi(beta, mk, K, N)),
+                                  tol=5e-08, maxiter=500)
+            except:
+                print("Warning: No ML parameter was found. (Exception caught.)", file=stderr, flush=True)
+                beta_MAP = np.float('nan')
+        return beta_MAP

+
+
[docs]    def alpha_ML(self, mk, K1, N):
+        """
+        Compute first guess for the beta_MAP (cf get_beta_MAP) parameter
+        via the posterior of a Dirichlet process.
+        """
+
+        mk = utl.remove_key(mk, 0)
+        # rnsum      = np.array([_logvarrhoi_DP(n, mk[n]) for n in mk]).sum()
+        estlist = [N * (K1 - 1.) / r / (N - K1) for r in np.arange(6., 1.5, -0.5)]
+        varrholist = {}
+        for a in estlist:
+            # varrholist[_logvarrho_DP(a, rnsum, K1, N)] = a
+            varrholist[self.log_likelihood_DP_alpha(a, K1, N)] = a
+        a_est = varrholist[max(varrholist.keys())]
+        res = minimize(lambda a: -self.log_likelihood_DP_alpha(a[0], K1, N),
+                       a_est, method='Nelder-Mead')
+        return res.x[0]

+
+
[docs]    def get_integration_bounds(self, mk, K, N):
+        """
+        Find the integration bounds for the estimator.
+
+        Typically it is a delta-like distribution so it is sufficient
+        to integrate around this peak. (If not this function is not
+        called.)
+        """
+
+        beta_MAP = self.get_beta_MAP(mk, K, N)
+        if np.isnan(beta_MAP):
+            intbounds = np.float('nan')
+        else:
+            std = np.sqrt(- self.d2_log_rho_xi(beta_MAP, mk, K, N) ** (-1))
+            intbounds = [np.float(np.amax([10 ** (-50), beta_MAP - 8 * std])),
+                         np.float(beta_MAP + 8 * std)]
+
+        return intbounds

+
+
[docs]    def H1(self, beta, mk, K, N):
+        """
+        Compute the first moment (expectation value) of the entropy H.
+
+        H is the entropy one obtains with a symmetric Dirichlet prior
+        with concentration parameter beta and a multinomial likelihood.
+        """
+
+        norm = N + beta * K
+        return mp.psi(0, norm + 1) - np.sum((mk[n] * (n + beta) *
+                                             mp.psi(0, n + beta + 1) for n in mk)) / norm

+
+
[docs]    def nsb_entropy(self, mk, K, N):
+        """
+        Estimate the entropy of a system using the NSB estimator.
+
+        :param mk: multiplicities
+        :param K:  number of possible symbols/ state space of the system
+        :param N:  total number of observed symbols
+        """
+
+        mp.pretty = True
+
+        # find the concentration parameter beta
+        # for which the posterior is maximised
+        # to integrate around this peak
+        integration_bounds = self.get_integration_bounds(mk, K, N)
+
+        if np.any(np.isnan(integration_bounds)):
+            # if no peak was found, integrate over the whole range
+            # by reformulating beta into w so that the range goes from 0 to 1
+            # instead of from 1 to infinity
+
+            integration_bounds = [0, 1]
+
+            def unnormalized_posterior_w(w, mk, K, N):
+                sbeta = w / (1 - w)
+                beta = sbeta * sbeta
+                return self.unnormalized_posterior(beta, mk, K, N) * 2 * sbeta / (1 - w) / (1 - w)
+
+            def H1_w(w, mk, K, N):
+                sbeta = w / (1 - w)
+                beta = sbeta * sbeta
+                return self.H1(w, mk, K, N)
+
+            marginal_likelihood = mp.quadgl(lambda w: unnormalized_posterior_w(w, mk, K, N),
+                                            integration_bounds)
+            H_nsb = mp.quadgl(lambda w: H1_w(w, mk, K, N) * unnormalized_posterior_w(w, mk, K, N),
+                              integration_bounds) / marginal_likelihood
+
+        else:
+            # integrate over the possible entropies, weighted such that every entropy is equally likely
+            # and normalize with the marginal likelihood
+            marginal_likelihood = mp.quadgl(lambda beta: self.unnormalized_posterior(beta, mk, K, N),
+                                            integration_bounds)
+            H_nsb = mp.quadgl(lambda beta: self.H1(beta, mk, K, N) * self.unnormalized_posterior(beta, mk, K, N),
+                              integration_bounds) / marginal_likelihood
+
+        return H_nsb

+
+
+
[docs]class RudeltNSBEstimatorSymbolsMI(RudeltAbstractNSBEstimator):
+    """History dependence NSB estimator
+
+    Calculate the mutual information (MI) of one variable depending on its past
+    using NSB estimator. See parent class for references.
+
+    implemented in idtxl by Michael Lindner, Göttingen 2021
+
+    Args:
+        settings : dict
+            - embedding_step_size : float [optional]
+                Step size delta t (in seconds) with which the window is slid through the data
+                (default = 0.005).
+            - normalise : bool [optional]
+                rebase spike times to zero
+                (default=True)
+            - return_averaged_R : bool [optional]
+                If set to True, compute R̂tot as the average over R̂(T ) for T ∈ [T̂D, Tmax ] instead of
+                R̂tot = R(T̂D ). If set to True, the setting for number_of_bootstraps_R_tot is ignored and
+                set to 0
+                (default=True)
+    """
+
+    def __init__(self, settings=None):
+        # Set default estimator settings.
+        super().__init__(settings)
+
+
[docs]    def nsb_estimator(self,
+                      symbol_counts,
+                      past_symbol_counts,
+                      alphabet_size,
+                      alphabet_size_past,
+                      H_uncond):
+        """
+        Estimate the entropy of a system using the NSB estimator.
+        """
+
+        mk = self.get_multiplicities(symbol_counts, alphabet_size)
+        mk_past = self.get_multiplicities(past_symbol_counts, alphabet_size_past)
+
+        N = sum((mk[n] * n for n in mk.keys()))
+
+        H_nsb_joint = self.nsb_entropy(mk, alphabet_size, N)
+        H_nsb_past = self.nsb_entropy(mk_past, alphabet_size_past, N)
+
+        H_nsb_cond = H_nsb_joint - H_nsb_past
+        I_nsb = H_uncond - H_nsb_cond
+        R_nsb = I_nsb / H_uncond
+
+        return I_nsb, R_nsb

+
+
[docs]    def estimate(self, symbol_array, past_symbol_array, current_symbol_array):
+        """Estimate mutual information using NSB estimator.
+
+        Args:
+            symbol_array : 1D numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+            past_symbol_array : numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+            current_symbol_array : numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+
+        Returns:
+            I (float)
+                MI (AIS)
+            R (float)
+                MI / H_uncond (History dependence)
+        """
+
+        self._check_estimator_inputs(symbol_array,
+                                     past_symbol_array,
+                                     current_symbol_array,
+                                     None)
+
+        self._ensure_one_dim(symbol_array)
+        self._ensure_one_dim(past_symbol_array)
+        self._ensure_one_dim(current_symbol_array)
+
+        symbol_counts = self.get_symbol_counts(symbol_array)
+
+        current_symbol_counts = self.get_symbol_counts(current_symbol_array)
+        H_uncond = utl.get_H_spiking(symbol_counts)
+
+        past_symbol_counts = self.get_symbol_counts(past_symbol_array)
+
+        number_of_bins_d_join = len(list(np.binary_repr(np.max(symbol_array))))
+
+        alphabet_size_past = 2 ** int(number_of_bins_d_join - 1)  # K for past activity
+        alphabet_size = alphabet_size_past * 2  # K
+
+        I, R = self.nsb_estimator(symbol_counts,
+                                  past_symbol_counts,
+                                  alphabet_size,
+                                  alphabet_size_past,
+                                  H_uncond)
+
+        return np.float(I), np.float(R)

+
+
+
[docs]class RudeltPluginEstimatorSymbolsMI(RudeltAbstractEstimator):
+    """Plugin History dependence estimator
+
+    Calculate the mutual information (MI) of one variable depending on its past
+    using plugin estimator. See parent class for references.
+
+    implemented in idtxl by Michael Lindner, Göttingen 2021
+
+    Args:
+        settings : dict
+            - embedding_step_size : float [optional] - Step size delta t (in seconds) with which the window is slid
+                            through the data (default = 0.005).
+            - normalise : bool [optional] - rebase spike times to zero (default=True)
+            - return_averaged_R : bool [optional] - rebase spike times to zero (default=True)
+
+    """
+
+
[docs]    def plugin_entropy(self, mk, N):
+        """
+        Estimate the entropy of a system using the Plugin estimator.
+
+        (In principle this is the same function as utl.get_shannon_entropy,
+        only here it is a function of the multiplicities, not the probabilities.)
+
+        :param mk: multiplicities
+        :param N:  total number of observed symbols
+        """
+
+        mk = utl.remove_key(mk, 0)
+        return - sum((mk[n] * (n / N) * np.log(n / N) for n in mk))

+
+
[docs]    def plugin_estimator(self,
+                         symbol_counts,
+                         past_symbol_counts,
+                         alphabet_size,
+                         alphabet_size_past,
+                         H_uncond):
+        """
+        Estimate the entropy of a system using the BBC estimator.
+        """
+
+        mk = self.get_multiplicities(symbol_counts, alphabet_size)
+        mk_past = self.get_multiplicities(past_symbol_counts, alphabet_size_past)
+
+        N = sum((mk[n] * n for n in mk.keys()))
+
+        H_plugin_joint = self.plugin_entropy(mk, N)
+        H_plugin_past = self.plugin_entropy(mk_past, N)
+
+        H_plugin_cond = H_plugin_joint - H_plugin_past
+        I_plugin = H_uncond - H_plugin_cond
+        R_plugin = I_plugin / H_uncond
+
+        return I_plugin, R_plugin

+
+
[docs]    def estimate(self, symbol_array, past_symbol_array, current_symbol_array):
+        """Estimate mutual information using plugin estimator.
+
+        Args:
+            symbol_array : 1D numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+            past_symbol_array : numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+            current_symbol_array : numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+
+        Returns:
+            I (float)
+                MI (AIS)
+            R (float)
+                MI / H_uncond (History dependence)
+        """
+
+        self._check_estimator_inputs(symbol_array,
+                                     past_symbol_array,
+                                     current_symbol_array,
+                                     None)
+
+        self._ensure_one_dim(symbol_array)
+        self._ensure_one_dim(past_symbol_array)
+        self._ensure_one_dim(current_symbol_array)
+
+        symbol_counts = self.get_symbol_counts(symbol_array)
+
+        current_symbol_counts = self.get_symbol_counts(current_symbol_array)
+        # H_uncond_orig = utl.get_H_spiking(symbol_counts)
+        H_uncond = utl.get_H_spiking(symbol_counts)
+
+        # past_symbol_counts = utl.get_past_symbol_counts(symbol_counts)
+        past_symbol_counts = self.get_symbol_counts(past_symbol_array)
+
+        # number_of_bins_d_join = np.array(list(np.binary_repr(np.max(past_symbol_array)))).astype(np.int8)
+        number_of_bins_d_join = len(list(np.binary_repr(np.max(symbol_array))))
+
+        alphabet_size_past = 2 ** int(number_of_bins_d_join - 1)  # K for past activity
+        alphabet_size = alphabet_size_past * 2  # K
+
+        I, R = self.plugin_estimator(symbol_counts,
+                                     past_symbol_counts,
+                                     alphabet_size,
+                                     alphabet_size_past,
+                                     H_uncond)
+
+        return np.float(I), np.float(R)

+
+
+
[docs]class RudeltBBCEstimator(RudeltAbstractEstimator):
+    """
+    Bayesian bias criterion (BBC) Estimator using NSB and Plugin estimator
+
+    Calculate the mutual information (MI) of one variable depending on its past
+    using nsb and plugin estimator and check if bias criterion is passed.
+    See parent class for references.
+
+    implemented in idtxl by Michael Lindner, Göttingen 2021
+
+    Args:
+        settings : dict
+            - embedding_step_size : float [optional]
+                Step size delta t (in seconds) with which the window is slid through the data
+                (default = 0.005).
+            - normalise : bool [optional]
+                rebase spike times to zero
+                (default=True)
+            - return_averaged_R : bool [optional]
+                If set to True, compute R̂tot as the average over R̂(T ) for T ∈ [T̂D, Tmax ] instead of
+                R̂tot = R(T̂D ). If set to True, the setting for number_of_bootstraps_R_tot is ignored and
+                set to 0
+                (default=True)
+    """
+
+
[docs]    def bayesian_bias_criterion(self, R_nsb, R_plugin, bbc_tolerance):
+        """
+        Get whether the Bayesian bias criterion (bbc) is passed.
+
+        :param R_nsb: history dependence computed with NSB estimator
+        :param R_plugin: history dependence computed with plugin estimator
+        :param bbc_tolerance: tolerance for the Bayesian bias criterion
+        """
+
+        if self.get_bbc_term(R_nsb, R_plugin) < bbc_tolerance:
+            return 1
+        else:
+            return 0

+
+
[docs]    def get_bbc_term(self, R_nsb, R_plugin):
+        """
+        Get the bbc tolerance-independent term of the Bayesian bias
+        criterion (bbc).
+
+        :param R_nsb: history dependence computed with NSB estimator
+        :param R_plugin: history dependence computed with plugin estimator
+        """
+
+        if R_nsb > 0:
+            return np.abs(R_nsb - R_plugin) / R_nsb
+        else:
+            return np.inf

+
+
[docs]    def estimate(self, symbol_array, past_symbol_array, current_symbol_array, bbc_tolerance=None):
+        """
+        Calculate the mutual information (MI) of one variable depending on its past
+        using nsb and plugin estimator and check if bias criterion is passed/
+
+        Args:
+            symbol_array : 1D numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+            past_symbol_array : numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+            current_symbol_array : numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+
+        Returns:
+            I (float)
+                MI (AIS)
+            R (float)
+                MI / H_uncond (History dependence)
+            bbc_term (float)
+                bbc tolerance-independent term of the Bayesian bias
+                criterion (bbc)
+        """
+
+        self._check_estimator_inputs(symbol_array,
+                                     past_symbol_array,
+                                     current_symbol_array,
+                                     bbc_tolerance)
+
+        self._ensure_one_dim(symbol_array)
+        self._ensure_one_dim(past_symbol_array)
+        self._ensure_one_dim(current_symbol_array)
+
+        estnsb = RudeltNSBEstimatorSymbolsMI()
+        I_nsb, R_nsb = estnsb.estimate(symbol_array, past_symbol_array, current_symbol_array)
+
+        estplugin = RudeltPluginEstimatorSymbolsMI()
+        I_plugin, R_plugin = estplugin.estimate(symbol_array, past_symbol_array, current_symbol_array)
+
+        if not bbc_tolerance == None:
+            if self.bayesian_bias_criterion(R_nsb, R_plugin, bbc_tolerance):
+                return np.float(I_nsb), np.float(R_nsb)
+            else:
+                return None
+        else:
+            return np.float(I_nsb), np.float(R_nsb), np.float(self.get_bbc_term(R_nsb,
+                                                  R_plugin))

+
+
+
[docs]class RudeltShufflingEstimator(RudeltAbstractEstimator):
+    """
+    Estimate the history dependence in a spike train using the shuffling estimator.
+
+    See parent class for references.
+
+    implemented in idtxl by Michael Lindner, Göttingen 2021
+    """
+
+
[docs]    def get_P_X_uncond(self, number_of_symbols):
+        """
+        Compute P(X), the probability of the current activity using
+        the plug-in estimator.
+        """
+
+        return [number_of_symbols[0] / sum(number_of_symbols),
+                number_of_symbols[1] / sum(number_of_symbols)]

+
+
[docs]    def get_P_X_past_uncond(self, past_symbol_counts, number_of_symbols):
+        """
+        Compute P(X_past), the probability of the past activity using
+        the plug-in estimator.
+        """
+
+        P_X_past_uncond = {}
+        for response in [0, 1]:
+            for symbol in past_symbol_counts[response]:
+                if symbol in P_X_past_uncond:
+                    P_X_past_uncond[symbol] += past_symbol_counts[response][symbol]
+                else:
+                    P_X_past_uncond[symbol] = past_symbol_counts[response][symbol]
+        number_of_symbols_uncond = sum(number_of_symbols)
+
+        for symbol in P_X_past_uncond:
+            P_X_past_uncond[symbol] /= number_of_symbols_uncond
+        return P_X_past_uncond

+
+
[docs]    def get_P_X_past_cond_X(self, past_symbol_counts, number_of_symbols):
+        """
+        Compute P(X_past | X), the probability of the past activity conditioned
+        on the response X using the plug-in estimator.
+        """
+
+        P_X_past_cond_X = [{}, {}]
+        for response in [0, 1]:
+            for symbol in past_symbol_counts[response]:
+                P_X_past_cond_X[response][symbol] \
+                    = past_symbol_counts[response][symbol] / number_of_symbols[response]
+        return P_X_past_cond_X

+
+
[docs]    def get_H0_X_past_cond_X_eq_x(self, marginal_probabilities, number_of_bins_d):
+        """
+        Compute H_0(X_past | X = x), cf get_H0_X_past_cond_X.
+        """
+        return utl.get_shannon_entropy(marginal_probabilities) \
+               + utl.get_shannon_entropy(1 - marginal_probabilities)

+
+
[docs]    def get_H0_X_past_cond_X(self, marginal_probabilities, number_of_bins_d, P_X_uncond):
+        """
+        Compute H_0(X_past | X), the estimate of the entropy for the past
+        symbols given a response, under the assumption that activity in
+        the past contributes independently towards the response.
+        """
+        H0_X_past_cond_X_eq_x = [0, 0]
+        for response in [0, 1]:
+            H0_X_past_cond_X_eq_x[response] \
+                = self.get_H0_X_past_cond_X_eq_x(marginal_probabilities[response],
+                                            number_of_bins_d)
+        return sum([P_X_uncond[response] * H0_X_past_cond_X_eq_x[response] for response in [0, 1]])

+
+
[docs]    def get_H_X_past_uncond(self, P_X_past_uncond):
+        """
+        Compute H(X_past), the plug-in estimate of the entropy for the past symbols, given
+        their probabilities.
+        """
+
+        return utl.get_shannon_entropy(P_X_past_uncond.values())

+
+
[docs]    def get_H_X_past_cond_X(self, P_X_uncond, P_X_past_cond_X):
+        """
+        Compute H(X_past | X), the plug-in estimate of the conditional entropy for the past
+        symbols, conditioned on the response X,  given their probabilities.
+        """
+
+        return sum((P_X_uncond[response] * self.get_H_X_past_uncond(P_X_past_cond_X[response])
+                    for response in [0, 1]))

+
+
[docs]    def get_marginal_frequencies_of_spikes_in_bins(self, symbol_counts, number_of_bins_d):
+        """
+        Compute for each past bin 1...d the sum of spikes found in that bin across all
+        observed symbols.
+        """
+        return np.array(sum((self.symbol_binary_to_array(symbol, number_of_bins_d)
+                             * symbol_counts[symbol]
+                             for symbol in symbol_counts)), dtype=int)

+
+
[docs]    def get_shuffled_symbol_counts(self, symbol_counts, past_symbol_counts, number_of_bins_d,
+                                   number_of_symbols):
+        """
+        Simulate new data by, for each past bin 1...d, permutating the activity
+        across all observed past_symbols (for a given response X). The marginal
+        probability of observing a spike given the response is thus preserved for
+        each past bin.
+        """
+        number_of_spikes = sum(past_symbol_counts[1].values())
+
+        marginal_frequencies = [self.get_marginal_frequencies_of_spikes_in_bins(past_symbol_counts[response],
+                                                                           number_of_bins_d)
+                                for response in [0, 1]]
+
+        shuffled_past_symbols = [np.zeros(number_of_symbols[response]) for response in [0, 1]]
+
+        for i in range(0, number_of_bins_d):
+            for response in [0, 1]:
+                shuffled_past_symbols[response] \
+                    += 2 ** (number_of_bins_d - i - 1) \
+                       * np.random.permutation(np.hstack((np.ones(marginal_frequencies[response][i]),
+                                                          np.zeros(number_of_symbols[response] \
+                                                                   - marginal_frequencies[response][i]))))
+
+        for response in [0, 1]:
+            shuffled_past_symbols[response] = np.array(shuffled_past_symbols[response], dtype=int)
+
+        shuffled_past_symbol_counts = [Counter(), Counter()]
+
+        for response in [0, 1]:
+            for past_symbol in shuffled_past_symbols[response]:
+                shuffled_past_symbol_counts[response][past_symbol] += 1
+
+        marginal_probabilities = [marginal_frequencies[response] / number_of_symbols[response]
+                                  for response in [0, 1]]
+
+        return shuffled_past_symbol_counts, marginal_probabilities

+
+
[docs]    def shuffling_MI(self, symbol_counts, number_of_bins_d):
+        """
+        Estimate the mutual information between current and past activity
+        in a spike train using the shuffling estimator.
+
+        To obtain the shuffling estimate, compute the plug-in estimate and
+        a correction term to reduce its bias.
+
+        For the plug-in estimate:
+
+        - Extract the past_symbol_counts from the symbol_counts.
+        - I_plugin = H(X_past) - H(X_past | X)
+
+        Notation:
+
+        - X: current activity, aka response
+        - X_past: past activity
+        - P_X_uncond: P(X)
+        - P_X_past_uncond: P(X_past)
+        - P_X_past_cond_X: P(X_past | X)
+        - H_X_past_uncond: H(X_past)
+        - H_X_past_cond_X: H(X_past | X)
+        - I_plugin: plugin estimate of I(X_past; X)
+
+        For the correction term:
+
+        - Simulate additional data under the assumption that activity
+            in the past contributes independently towards the current activity.
+        - Compute the entropy under the assumptions of the model, which
+            due to its simplicity is easy to sample and the estimate unbiased
+        - Compute the entropy using the plug-in estimate, whose bias is
+            similar to that of the plug-in estimate on the original data
+        - Compute the correction term as the difference between the
+            unbiased and biased terms
+
+        Notation:
+
+        - P0_sh_X_past_cond_X: P_0,sh(X_past | X), equiv. to P(X_past | X)
+          on the shuffled data
+        - H0_X_past_cond_X: H_0(X_past | X), based on the model of independent
+          contributions
+        - H0_sh_X_past_cond_X: H_0,sh(X_past | X), based on
+        - P0_sh_X_past_cond_X, ie the plug-in estimate
+        - I_corr: the correction term to reduce the bias of I_plugin
+
+        Args:
+            symbol_counts : iterable
+                the activity of a spike train is embedded into symbols,
+                whose occurrences are counted (cf emb.get_symbol_counts)
+            number_of_bins_d : int
+                the number of bins of the embedding
+        """
+
+        # plug-in estimate
+        past_symbol_counts = utl.get_past_symbol_counts(symbol_counts, merge=False)
+        number_of_symbols = [sum(past_symbol_counts[response].values()) for response in [0, 1]]
+
+        P_X_uncond = self.get_P_X_uncond(number_of_symbols)
+        P_X_past_uncond = self.get_P_X_past_uncond(past_symbol_counts, number_of_symbols)
+        P_X_past_cond_X = self.get_P_X_past_cond_X(past_symbol_counts, number_of_symbols)
+
+        H_X_past_uncond = self.get_H_X_past_uncond(P_X_past_uncond)
+        H_X_past_cond_X = self.get_H_X_past_cond_X(P_X_uncond, P_X_past_cond_X)
+
+        I_plugin = H_X_past_uncond - H_X_past_cond_X
+
+        # correction term
+        shuffled_past_symbol_counts, marginal_probabilities \
+            = self.get_shuffled_symbol_counts(symbol_counts, past_symbol_counts, number_of_bins_d,
+                                         number_of_symbols)
+
+        P0_sh_X_past_cond_X = self.get_P_X_past_cond_X(shuffled_past_symbol_counts, number_of_symbols)
+
+        H0_X_past_cond_X = self.get_H0_X_past_cond_X(marginal_probabilities, number_of_bins_d, P_X_uncond)
+        H0_sh_X_past_cond_X = self.get_H_X_past_cond_X(P_X_uncond, P0_sh_X_past_cond_X)
+
+        I_corr = H0_X_past_cond_X - H0_sh_X_past_cond_X
+
+        # shuffling estimate
+        return I_plugin - I_corr

+
+
[docs]    def estimate(self, symbol_array):
+        """
+        Estimate the history dependence in a spike train using the shuffling estimator.
+
+         Args:
+            symbol_array : 1D numpy array
+                realisations of symbols based on current and past states.
+                (first output of get_realisations_symbol from data_spiketimes object)
+
+        Returns:
+            I (float)
+                MI (AIS)
+            R (float)
+                MI / H_uncond (History dependence)
+        """
+
+        self._check_estimator_inputs(symbol_array,
+                                     None,
+                                     None,
+                                     None)
+
+        self._ensure_one_dim(symbol_array)
+
+        symbol_counts = self.get_symbol_counts(symbol_array)
+
+        # number_of_bins_d_join = np.array(list(np.binary_repr(np.max(symbol_array)))).astype(np.int8)
+        number_of_bins_d_join = len(list(np.binary_repr(np.max(symbol_array))))
+
+        H_uncond = utl.get_H_spiking(symbol_counts)
+
+        I_sh = self.shuffling_MI(symbol_counts,
+                            number_of_bins_d_join -1)
+
+        R_sh = I_sh / H_uncond
+
+        return I_sh, R_sh

+
+ +
+
+
+
+ +
+
+ + + + \ No newline at end of file diff --git a/docs/html/_modules/idtxl/results.html b/docs/html/_modules/idtxl/results.html index ac037c6c..1db75ca0 100644 --- a/docs/html/_modules/idtxl/results.html +++ b/docs/html/_modules/idtxl/results.html @@ -5,10 +5,10 @@ - idtxl.results — IDTxl 1.0 documentation - - - + idtxl.results — IDTxl 1.2.1 documentation + + + @@ -31,7 +31,7 @@

Navigation

  • modules |
    • -
  • IDTxl 1.0 documentation »
    • +
  • IDTxl 1.2.1 documentation »
  • Module code »
  • idtxl.results
    • @@ -48,13 +48,13 @@

    Source code for idtxl.results

     import warnings
 import copy as cp
 import numpy as np
-from . import idtxl_utils as utils
+from . import idtxl_utils as utils
 
 warnings.simplefilter(action='ignore', category=FutureWarning)
 MIN_INT = -sys.maxsize - 1  # minimum integer for initializing adj. matrix
 
 
-class DotDict(dict):
+
    [docs]class DotDict(dict):
     """Dictionary with dot-notation access to values.
 
     Provides the same functionality as a regular dict, but also allows
@@ -74,7 +74,7 @@ 
    Source code for idtxl.results
         __setattr__ = dict.__setitem__
     __delattr__ = dict.__delitem__
 
-    def __dir__(self):
+    def __dir__(self):
         """Return dictionary keys as list of attributes."""
         return self.keys()
 
@@ -95,13 +95,13 @@ 
    Source code for idtxl.results
     
     def __setstate__(self, state):
         # For un-pickling the object
-        self.update(state)
+        self.update(state)
    
         # self.__dict__ = self
 
 
-class AdjacencyMatrix():
+
    [docs]class AdjacencyMatrix():
     """Adjacency matrix representing inferred networks."""
-    def __init__(self, n_nodes, weight_type):
+    def __init__(self, n_nodes, weight_type):
         self._edge_matrix = np.zeros((n_nodes, n_nodes), dtype=bool)
         self._weight_matrix = np.zeros((n_nodes, n_nodes), dtype=weight_type)
         if np.issubdtype(weight_type, np.integer):
@@ -114,23 +114,23 @@ 
    Source code for idtxl.results
                 raise RuntimeError('Unknown weight data type {0}.'.format(
                 weight_type))
 
-    def n_nodes(self):
+
    [docs]    def n_nodes(self):
         """Return number of nodes."""
-        return self._edge_matrix.shape[0]
+        return self._edge_matrix.shape[0]
    
 
-    def n_edges(self):
-        return self._edge_matrix.sum()
+
    [docs]    def n_edges(self):
+        return self._edge_matrix.sum()
    
 
-    def add_edge(self, i, j, weight):
+
    [docs]    def add_edge(self, i, j, weight):
         """Add weighted edge (i, j) to adjacency matrix."""
         if not np.issubdtype(type(weight), self._weight_type):
             raise TypeError(
                 'Can not add weight of type {0} to adjacency matrix of type '
                 '{1}.'.format(type(weight), self._weight_type))
         self._edge_matrix[i, j] = True
-        self._weight_matrix[i, j] = weight
+        self._weight_matrix[i, j] = weight
    
 
-    def add_edge_list(self, i_list, j_list, weights):
+
    [docs]    def add_edge_list(self, i_list, j_list, weights):
         """Add multiple weighted edges (i, j) to adjacency matrix."""
         if len(i_list) != len(j_list):
             raise RuntimeError(
@@ -139,14 +139,14 @@ 
    Source code for idtxl.results
                 raise RuntimeError(
                 'Edge weights must have same length as edge indices.')
         for i, j, weight in zip(i_list, j_list, weights):
-            self.add_edge(i, j, weight)
+            self.add_edge(i, j, weight)
    
 
-    def print_matrix(self):
+
    [docs]    def print_matrix(self):
         """Print weight and edge matrix."""
         print(self._edge_matrix)
-        print(self._weight_matrix)
+        print(self._weight_matrix)
    
 
-    def get_edge_list(self):
+
    [docs]    def get_edge_list(self):
         """Return list of weighted edges.
 
         Returns
@@ -160,10 +160,10 @@ 
    Source code for idtxl.results
                     if self._edge_matrix[i, j]:
                     edge_list[ind] = (i, j, self._weight_matrix[i, j])
                     ind += 1
-        return edge_list
+        return edge_list
    
 
 
-class Results():
+
    [docs]class Results():
     """Parent class for results of network analysis algorithms.
 
     Provide a container for results of network analysis algorithms, e.g.,
@@ -185,7 +185,7 @@ 
    Source code for idtxl.results
                       before the estimation
     """
 
-    def __init__(self, n_nodes, n_realisations, normalised):
+    def __init__(self, n_nodes, n_realisations, normalised):
         self.settings = DotDict({})
         self.data_properties = DotDict({
             'n_nodes': n_nodes,
@@ -229,7 +229,7 @@ 
    Source code for idtxl.results
             else:
             return False
 
-    def combine_results(self, *results):
+
    [docs]    def combine_results(self, *results):
         """Combine multiple (partial) results objects.
 
         Combine a list of partial network analysis results into a single
@@ -282,7 +282,7 @@ 
    Source code for idtxl.results
                             raise AttributeError(
                             'Did not find any method attributes to combine '
                             '(.single_proces or ._single_target).')
-                self._add_single_result(p, results_to_add, r.settings)
+                self._add_single_result(p, results_to_add, r.settings)
    
 
 
    [docs]class ResultsSingleProcessAnalysis(Results):
     """Store results of single process analysis.
@@ -319,7 +319,7 @@ 
    Source code for idtxl.results
                 list of analysed processes
     """
 
-    def __init__(self, n_nodes, n_realisations, normalised):
+    def __init__(self, n_nodes, n_realisations, normalised):
         super().__init__(n_nodes, n_realisations, normalised)
         self.processes_analysed = []
         self._single_process = {}
@@ -435,9 +435,9 @@ 
    Source code for idtxl.results
             return significant_processes
    
 
 
-class ResultsNetworkAnalysis(Results):
+
    [docs]class ResultsNetworkAnalysis(Results):
 
-    def __init__(self, n_nodes, n_realisations, normalised):
+    def __init__(self, n_nodes, n_realisations, normalised):
         super().__init__(n_nodes, n_realisations, normalised)
         self._single_target = {}
         self.targets_analysed = []
@@ -459,7 +459,7 @@ 
    Source code for idtxl.results
             self._single_target[target] = DotDict(results)
         self.targets_analysed = list(self._single_target.keys())
 
-    def get_single_target(self, target, fdr=True):
+
    [docs]    def get_single_target(self, target, fdr=True):
         """Return results for a single target in the network.
 
         Return results for individual processes, contains for each process
@@ -522,9 +522,9 @@ 
    Source code for idtxl.results
                     raise RuntimeError('No results have been added.')
             except KeyError:
                 raise RuntimeError(
-                    'No results for target {0}.'.format(target))
+                    'No results for target {0}.'.format(target))
    
 
-    def get_target_sources(self, target, fdr=True):
+
    [docs]    def get_target_sources(self, target, fdr=True):
         """Return list of sources (parents) for given target.
 
         Args:
@@ -535,7 +535,7 @@ 
    Source code for idtxl.results
                     (default=True)
         """
         v = self.get_single_target(target, fdr)['selected_vars_sources']
-        return np.unique(np.array([s[0] for s in v]))
+        return np.unique(np.array([s[0] for s in v]))
    
 
 
 
    [docs]class ResultsNetworkInference(ResultsNetworkAnalysis):
@@ -572,7 +572,7 @@ 
    Source code for idtxl.results
                 list of analysed targets
     """
 
-    def __init__(self, n_nodes, n_realisations, normalised):
+    def __init__(self, n_nodes, n_realisations, normalised):
         super().__init__(n_nodes, n_realisations, normalised)
         self._single_target_fdr = DotDict()
 
@@ -757,7 +757,7 @@ 
    Source code for idtxl.results
             self._print_edge_list(adjacency_matrix, weights=weights)
    
 
 
-class ResultsPID(ResultsNetworkAnalysis):
+
    [docs]class ResultsPID(ResultsNetworkAnalysis):
     """Store results of Partial Information Decomposition (PID) analysis.
 
     Provide a container for results of Partial Information Decomposition (PID)
@@ -791,10 +791,10 @@ 
    Source code for idtxl.results
                 list of analysed targets
     """
 
-    def __init__(self, n_nodes, n_realisations, normalised):
+    def __init__(self, n_nodes, n_realisations, normalised):
         super().__init__(n_nodes, n_realisations, normalised)
 
-    def get_single_target(self, target):
+
    [docs]    def get_single_target(self, target):
         """Return results for a single target in the network.
 
         Results for single targets include for each target
@@ -823,10 +823,10 @@ 
    Source code for idtxl.results
                     (result.selected_vars_sources).
         """
         return super(ResultsPID,
-                     self).get_single_target(target, fdr=False)
+                     self).get_single_target(target, fdr=False)
    
 
 
-class ResultsMultivariatePID(ResultsNetworkAnalysis):
+
    [docs]class ResultsMultivariatePID(ResultsNetworkAnalysis):
     """Store results of Multivariate Partial Information Decomposition (PID)
 analysis.
 
@@ -861,10 +861,10 @@ 
    Source code for idtxl.results
                 list of analysed targets
     """
 
-    def __init__(self, n_nodes, n_realisations, normalised):
+    def __init__(self, n_nodes, n_realisations, normalised):
         super().__init__(n_nodes, n_realisations, normalised)
 
-    def get_single_target(self, target):
+
    [docs]    def get_single_target(self, target):
         """Return results for a single target in the network.
 
         Results for single targets include for each target
@@ -893,7 +893,7 @@ 
    Source code for idtxl.results
                     (result.selected_vars_sources).
         """
         return super(ResultsMultivariatePID,
-                     self).get_single_target(target, fdr=False)
+                     self).get_single_target(target, fdr=False)
    
 
 
 
    [docs]class ResultsNetworkComparison(ResultsNetworkAnalysis):
@@ -940,7 +940,7 @@ 
    Source code for idtxl.results
                 settings used for comparison
     """
 
-    def __init__(self, n_nodes, n_realisations, normalised):
+    def __init__(self, n_nodes, n_realisations, normalised):
         super().__init__(n_nodes, n_realisations, normalised)
 
     def _add_results(self, union_network, results, settings):
@@ -1081,6 +1081,122 @@ 
    Source code for idtxl.results
             """
         v = self.get_single_target(target)['selected_vars_sources']
         return np.unique(np.array([s[0] for s in v]))
    
+
+
+
    [docs]class ResultsSingleProcessRudelt():
+    """Store results of single process analysis.
+
+    Provides a container for the results Rudelt optimization algorithm. To
+    obtain results for individual processes, call the .get_single_process()
+    method (see docstring for details).
+
+    Note that for convenience all dictionaries in this class can additionally
+    be accessed using dot-notation:
+
+    >>> res_network.settings.estimation_method
+
+    or
+
+    >>> res_network.settings['estimation_method'].
+
+    Attributes:
+        settings : dict
+            settings used for estimation of information theoretic measures
+        data_properties : dict
+            data properties, contains
+                - n_processes : int - total number of processes analysed
+        processes_analysed : list
+            list of analysed processes
+    """
+
+    def __init__(self, processes):
+        self.settings = DotDict({})
+        self.data_properties = DotDict({
+                'n_processes': len(processes)
+        })
+        self.processes_analysed = np.zeros(shape=3, dtype=int)
+        self._single_process = {}
+        for ii in processes:
+            self._single_process[ii] = {}
+
+    @property
+    def processes_analysed(self):
+        """Get index of the current_value."""
+        return self._processes_analysed
+
+    @processes_analysed.setter
+    def processes_analysed(self, processes):
+        self._processes_analysed = processes
+
+    def _add_single_result(self, process_count, process, results, settings):
+        """Add analysis result for a single process."""
+        #self._check_result(process, settings)
+        self.settings.update(DotDict(settings))
+        self._single_process[process] = DotDict(results)
+        self.processes_analysed[process_count] = process #list(self._single_process.keys())
+
+
    [docs]    def get_single_process(self, process):
+        """Return results for a single process.
+
+        Return results for individual processes, contains for each process
+
+        Args:
+            process : int
+                process id
+
+        Returns:
+            dict
+                results for single process. Note that for convenience
+                dictionary entries can either be accessed via keywords
+                (result['selected_vars']) or via dot-notation
+                (result.selected_vars). Contains keys
+
+                Process : int
+                    Process that was optimized
+                estimation_method : String
+                    Estimation method that was used for optimization
+                T_D : float
+                    Estimated optimal value for the temporal depth TD
+                tau_R :
+                    Information timescale tau_R, a characteristic timescale of history
+                    dependence similar to an autocorrelation time.
+                R_tot : float
+                    Estimated value for the total history dependence Rtot,
+                AIS_tot : float
+                    Estimated value for the total active information storage
+                opt_number_of_bins_d : int
+                    Number of bins d for the embedding that yields (R̂tot ,T̂D)
+                opt_scaling_k : int
+                    Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
+                opt_first_bin_size : int
+                    Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
+                history_dependence : array with floating-point values
+                    Estimated history dependence for each embedding
+                firing_rate : float
+                    Firing rate of the neuron/ spike train
+                recording_length : float
+                    Length of the recording (in seconds)
+                H_spiking : float
+                    Entropy of the spike times
+
+                if analyse_auto_MI was set to True additionally:
+                auto_MI : dict
+                    numpy array of MI values for each delay
+                auto_MI_delays : list of int
+                    list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay
+
+        """
+        # Return required key from required _single_process dictionary
+        if process not in self.processes_analysed:
+            raise RuntimeError('No results for process {0}.'.format(process))
+
+        try:
+            return self._single_process[process]
+        except AttributeError:
+            raise RuntimeError('No results have been added.')
+        except KeyError:
+            raise RuntimeError(
+                'No results for process {0}.'.format(process))
    
 
    
 
             
    
@@ -1093,7 +1209,7 @@ 
    Source code for idtxl.results
       
    Quick search
    
     
    
     
    
-      
+      
       
     
    
     
    
@@ -1112,14 +1228,14 @@ 
    Navigation
    
         
  • 
           modules |
    • 
-        
  • IDTxl 1.0 documentation »
    • 
+        
  • IDTxl 1.2.1 documentation »
    • 
           
  • Module code »
    • 
         
  • idtxl.results
    •  
       
     
    
     
   
 
\ No newline at end of file
diff --git a/docs/html/_modules/index.html b/docs/html/_modules/index.html
index aea9968b..cae54e16 100644
--- a/docs/html/_modules/index.html
+++ b/docs/html/_modules/index.html
@@ -5,10 +5,10 @@
   
     
     
-    Overview: module code — IDTxl 1.2.1 documentation
-    
-    
-    
+    Overview: module code — IDTxl 1.3 documentation
+    
+    
+    
     
     
     
@@ -31,7 +31,7 @@ 
    Navigation
    
         
  • 
           modules |
    • 
-        
  • IDTxl 1.2.1 documentation »
    • 
+        
  • IDTxl 1.3 documentation »
    • 
         
  • Overview: module code
    •  
       
     
      
@@ -47,12 +47,13 @@ 
    All modules for which code is available
    
 
  • idtxl.bivariate_pid
    • 
 
  • idtxl.bivariate_te
    • 
 
  • idtxl.data
    • 
+
  • idtxl.embedding_optimization_ais_Rudelt
    • 
+
  • idtxl.estimators_Rudelt
    • 
 
  • idtxl.estimators_jidt
    • 
 
  • idtxl.estimators_multivariate_pid
    • 
 
  • idtxl.estimators_opencl
    • 
 
  • idtxl.estimators_pid
    • 
 
  • idtxl.idtxl_exceptions
    • 
-
  • idtxl.idtxl_import
    • 
 
  • idtxl.idtxl_io
    • 
 
  • idtxl.idtxl_utils
    • 
 
  • idtxl.multivariate_mi
    • 
@@ -77,7 +78,7 @@ 
    All modules for which code is available
    
   
    Quick search
    
     
    
     
    
-      
+      
       
     
    
     
    
@@ -96,13 +97,13 @@ 
    Navigation
    
         
  • 
           modules |
    • 
-        
  • IDTxl 1.2.1 documentation »
    • 
+        
  • IDTxl 1.3 documentation »
    • 
         
  • Overview: module code
    •  
       
     
    
     
   
 
\ No newline at end of file
diff --git a/docs/html/_sources/idtxl.rst.txt b/docs/html/_sources/idtxl.rst.txt
index e9d46f28..6dc6d645 100644
--- a/docs/html/_sources/idtxl.rst.txt
+++ b/docs/html/_sources/idtxl.rst.txt
@@ -69,6 +69,22 @@ idtxl.active_information_storage module
     :undoc-members:
     :show-inheritance:
 
+idtxl.embedding_optimization_ais_Rudelt module
+----------------------------------------------
+
+.. automodule:: idtxl.embedding_optimization_ais_Rudelt
+    :members:
+    :undoc-members:
+    :show-inheritance:
+
+idtxl.estimators_Rudelt module
+------------------------------
+
+.. automodule:: idtxl.estimators_Rudelt
+    :members:
+    :undoc-members:
+    :show-inheritance:
+
 idtxl.estimators_jidt module
 ----------------------------
 
@@ -102,14 +118,6 @@ idtxl.estimators_pid module
     :show-inheritance:
 
 
-idtxl.idtxl_import module
--------------------------
-
-.. automodule:: idtxl.idtxl_import
-    :members:
-    :undoc-members:
-    :show-inheritance:
-
 idtxl.idtxl_exceptions module
 -----------------------------
 
diff --git a/docs/html/_sources/idtxl_process_analysis.rst.txt b/docs/html/_sources/idtxl_process_analysis.rst.txt
index 4d58d0cd..684e1e84 100644
--- a/docs/html/_sources/idtxl_process_analysis.rst.txt
+++ b/docs/html/_sources/idtxl_process_analysis.rst.txt
@@ -18,3 +18,16 @@ Bivariate Partial Information Decomposition (PID)
 .. autoclass:: idtxl.bivariate_pid.BivariatePID
     :members:
     :noindex:
+
+History-dependence estimator for neural spiking data
+----------------------------------------------------
+Embedding optimization
+^^^^^^^^^^^^^^^^^^^^^^
+.. autoclass:: idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt
+    :members:
+    :noindex:
+Estimators
+^^^^^^^^^^
+.. automodule:: idtxl.estimators_Rudelt
+    :members:
+    :noindex:
diff --git a/docs/html/_sources/idtxl_results_class.rst.txt b/docs/html/_sources/idtxl_results_class.rst.txt
index dd16bdd0..31a48a37 100644
--- a/docs/html/_sources/idtxl_results_class.rst.txt
+++ b/docs/html/_sources/idtxl_results_class.rst.txt
@@ -24,3 +24,6 @@ Results node dynamics
 .. autoclass:: idtxl.results.ResultsMultivariatePID
     :members:
     :noindex:
+.. autoclass:: idtxl.results.ResultsSingleProcessRudelt
+    :members:
+    :noindex:
diff --git a/docs/html/_static/basic.css b/docs/html/_static/basic.css
index be19270e..bf18350b 100644
--- a/docs/html/_static/basic.css
+++ b/docs/html/_static/basic.css
@@ -4,7 +4,7 @@
  *
  * Sphinx stylesheet -- basic theme.
  *
- * :copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS.
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
  * :license: BSD, see LICENSE for details.
  *
  */
@@ -130,7 +130,7 @@ ul.search li a {
     font-weight: bold;
 }
 
-ul.search li div.context {
+ul.search li p.context {
     color: #888;
     margin: 2px 0 0 30px;
     text-align: left;
@@ -277,25 +277,25 @@ p.rubric {
     font-weight: bold;
 }
 
-img.align-left, .figure.align-left, object.align-left {
+img.align-left, figure.align-left, .figure.align-left, object.align-left {
     clear: left;
     float: left;
     margin-right: 1em;
 }
 
-img.align-right, .figure.align-right, object.align-right {
+img.align-right, figure.align-right, .figure.align-right, object.align-right {
     clear: right;
     float: right;
     margin-left: 1em;
 }
 
-img.align-center, .figure.align-center, object.align-center {
+img.align-center, figure.align-center, .figure.align-center, object.align-center {
   display: block;
   margin-left: auto;
   margin-right: auto;
 }
 
-img.align-default, .figure.align-default {
+img.align-default, figure.align-default, .figure.align-default {
   display: block;
   margin-left: auto;
   margin-right: auto;
@@ -319,7 +319,8 @@ img.align-default, .figure.align-default {
 
 /* -- sidebars -------------------------------------------------------------- */
 
-div.sidebar {
+div.sidebar,
+aside.sidebar {
     margin: 0 0 0.5em 1em;
     border: 1px solid #ddb;
     padding: 7px;
@@ -377,12 +378,14 @@ div.body p.centered {
 /* -- content of sidebars/topics/admonitions -------------------------------- */
 
 div.sidebar > :last-child,
+aside.sidebar > :last-child,
 div.topic > :last-child,
 div.admonition > :last-child {
     margin-bottom: 0;
 }
 
 div.sidebar::after,
+aside.sidebar::after,
 div.topic::after,
 div.admonition::after,
 blockquote::after {
@@ -455,20 +458,22 @@ td > :last-child {
 
 /* -- figures --------------------------------------------------------------- */
 
-div.figure {
+div.figure, figure {
     margin: 0.5em;
     padding: 0.5em;
 }
 
-div.figure p.caption {
+div.figure p.caption, figcaption {
     padding: 0.3em;
 }
 
-div.figure p.caption span.caption-number {
+div.figure p.caption span.caption-number,
+figcaption span.caption-number {
     font-style: italic;
 }
 
-div.figure p.caption span.caption-text {
+div.figure p.caption span.caption-text,
+figcaption span.caption-text {
 }
 
 /* -- field list styles ----------------------------------------------------- */
@@ -503,6 +508,63 @@ table.hlist td {
     vertical-align: top;
 }
 
+/* -- object description styles --------------------------------------------- */
+
+.sig {
+	font-family: 'Consolas', 'Menlo', 'DejaVu Sans Mono', 'Bitstream Vera Sans Mono', monospace;
+}
+
+.sig-name, code.descname {
+    background-color: transparent;
+    font-weight: bold;
+}
+
+.sig-name {
+	font-size: 1.1em;
+}
+
+code.descname {
+    font-size: 1.2em;
+}
+
+.sig-prename, code.descclassname {
+    background-color: transparent;
+}
+
+.optional {
+    font-size: 1.3em;
+}
+
+.sig-paren {
+    font-size: larger;
+}
+
+.sig-param.n {
+	font-style: italic;
+}
+
+/* C++ specific styling */
+
+.sig-inline.c-texpr,
+.sig-inline.cpp-texpr {
+	font-family: unset;
+}
+
+.sig.c   .k, .sig.c   .kt,
+.sig.cpp .k, .sig.cpp .kt {
+	color: #0033B3;
+}
+
+.sig.c   .m,
+.sig.cpp .m {
+	color: #1750EB;
+}
+
+.sig.c   .s, .sig.c   .sc,
+.sig.cpp .s, .sig.cpp .sc {
+	color: #067D17;
+}
+
 
 /* -- other body styles ----------------------------------------------------- */
 
@@ -629,14 +691,6 @@ dl.glossary dt {
     font-size: 1.1em;
 }
 
-.optional {
-    font-size: 1.3em;
-}
-
-.sig-paren {
-    font-size: larger;
-}
-
 .versionmodified {
     font-style: italic;
 }
@@ -677,8 +731,9 @@ dl.glossary dt {
 
 .classifier:before {
     font-style: normal;
-    margin: 0.5em;
+    margin: 0 0.5em;
     content: ":";
+    display: inline-block;
 }
 
 abbr, acronym {
@@ -702,6 +757,7 @@ span.pre {
     -ms-hyphens: none;
     -webkit-hyphens: none;
     hyphens: none;
+    white-space: nowrap;
 }
 
 div[class*="highlight-"] {
@@ -765,8 +821,12 @@ div.code-block-caption code {
 
 table.highlighttable td.linenos,
 span.linenos,
-div.doctest > div.highlight span.gp {  /* gp: Generic.Prompt */
-    user-select: none;
+div.highlight span.gp {  /* gp: Generic.Prompt */
+  user-select: none;
+  -webkit-user-select: text; /* Safari fallback only */
+  -webkit-user-select: none; /* Chrome/Safari */
+  -moz-user-select: none; /* Firefox */
+  -ms-user-select: none; /* IE10+ */
 }
 
 div.code-block-caption span.caption-number {
@@ -781,16 +841,6 @@ div.literal-block-wrapper {
     margin: 1em 0;
 }
 
-code.descname {
-    background-color: transparent;
-    font-weight: bold;
-    font-size: 1.2em;
-}
-
-code.descclassname {
-    background-color: transparent;
-}
-
 code.xref, a code {
     background-color: transparent;
     font-weight: bold;
diff --git a/docs/html/_static/doctools.js b/docs/html/_static/doctools.js
index 61ac9d26..e509e483 100644
--- a/docs/html/_static/doctools.js
+++ b/docs/html/_static/doctools.js
@@ -4,7 +4,7 @@
  *
  * Sphinx JavaScript utilities for all documentation.
  *
- * :copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS.
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
  * :license: BSD, see LICENSE for details.
  *
  */
@@ -264,6 +264,9 @@ var Documentation = {
   hideSearchWords : function() {
     $('#searchbox .highlight-link').fadeOut(300);
     $('span.highlighted').removeClass('highlighted');
+    var url = new URL(window.location);
+    url.searchParams.delete('highlight');
+    window.history.replaceState({}, '', url);
   },
 
   /**
@@ -301,12 +304,14 @@ var Documentation = {
               window.location.href = prevHref;
               return false;
             }
+            break;
           case 39: // right
             var nextHref = $('link[rel="next"]').prop('href');
             if (nextHref) {
               window.location.href = nextHref;
               return false;
             }
+            break;
         }
       }
     });
diff --git a/docs/html/_static/documentation_options.js b/docs/html/_static/documentation_options.js
index 120e3eec..8b0a86c6 100644
--- a/docs/html/_static/documentation_options.js
+++ b/docs/html/_static/documentation_options.js
@@ -1,6 +1,6 @@
 var DOCUMENTATION_OPTIONS = {
     URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'),
-    VERSION: '1.2.1',
+    VERSION: '1.3',
     LANGUAGE: 'None',
     COLLAPSE_INDEX: false,
     BUILDER: 'html',
diff --git a/docs/html/_static/epub.css b/docs/html/_static/epub.css
index 6e5722fd..165f41d5 100644
--- a/docs/html/_static/epub.css
+++ b/docs/html/_static/epub.css
@@ -4,7 +4,7 @@
  *
  * Sphinx stylesheet -- default theme.
  *
- * :copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS.
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
  * :license: BSD, see LICENSE for details.
  *
  */
diff --git a/docs/html/_static/language_data.js b/docs/html/_static/language_data.js
index 863704b3..ebe2f03b 100644
--- a/docs/html/_static/language_data.js
+++ b/docs/html/_static/language_data.js
@@ -5,7 +5,7 @@
  * This script contains the language-specific data used by searchtools.js,
  * namely the list of stopwords, stemmer, scorer and splitter.
  *
- * :copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS.
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
  * :license: BSD, see LICENSE for details.
  *
  */
diff --git a/docs/html/_static/pygments.css b/docs/html/_static/pygments.css
index 20c4814d..691aeb82 100644
--- a/docs/html/_static/pygments.css
+++ b/docs/html/_static/pygments.css
@@ -1,5 +1,10 @@
+pre { line-height: 125%; }
+td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
 .highlight .hll { background-color: #ffffcc }
-.highlight  { background: #eeffcc; }
+.highlight { background: #eeffcc; }
 .highlight .c { color: #408090; font-style: italic } /* Comment */
 .highlight .err { border: 1px solid #FF0000 } /* Error */
 .highlight .k { color: #007020; font-weight: bold } /* Keyword */
diff --git a/docs/html/_static/pyramid.css b/docs/html/_static/pyramid.css
index 1329388f..e0b1cfac 100644
--- a/docs/html/_static/pyramid.css
+++ b/docs/html/_static/pyramid.css
@@ -4,7 +4,7 @@
  *
  * Sphinx stylesheet -- pylons theme.
  *
- * :copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS.
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
  * :license: BSD, see LICENSE for details.
  *
  */
diff --git a/docs/html/_static/searchtools.js b/docs/html/_static/searchtools.js
index 1a90152e..2d778593 100644
--- a/docs/html/_static/searchtools.js
+++ b/docs/html/_static/searchtools.js
@@ -4,7 +4,7 @@
  *
  * Sphinx JavaScript utilities for the full-text search.
  *
- * :copyright: Copyright 2007-2021 by the Sphinx team, see AUTHORS.
+ * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
  * :license: BSD, see LICENSE for details.
  *
  */
@@ -282,7 +282,10 @@ var Search = {
                   complete: function(jqxhr, textstatus) {
                     var data = jqxhr.responseText;
                     if (data !== '' && data !== undefined) {
-                      listItem.append(Search.makeSearchSummary(data, searchterms, hlterms));
+                      var summary = Search.makeSearchSummary(data, searchterms, hlterms);
+                      if (summary) {
+                        listItem.append(summary);
+                      }
                     }
                     Search.output.append(listItem);
                     setTimeout(function() {
@@ -325,7 +328,9 @@ var Search = {
     var results = [];
 
     for (var prefix in objects) {
-      for (var name in objects[prefix]) {
+      for (var iMatch = 0; iMatch != objects[prefix].length; ++iMatch) {
+        var match = objects[prefix][iMatch];
+        var name = match[4];
         var fullname = (prefix ? prefix + '.' : '') + name;
         var fullnameLower = fullname.toLowerCase()
         if (fullnameLower.indexOf(object) > -1) {
@@ -339,7 +344,6 @@ var Search = {
           } else if (parts[parts.length - 1].indexOf(object) > -1) {
             score += Scorer.objPartialMatch;
           }
-          var match = objects[prefix][name];
           var objname = objnames[match[1]][2];
           var title = titles[match[0]];
           // If more than one term searched for, we require other words to be
@@ -498,6 +502,9 @@ var Search = {
    */
   makeSearchSummary : function(htmlText, keywords, hlwords) {
     var text = Search.htmlToText(htmlText);
+    if (text == "") {
+      return null;
+    }
     var textLower = text.toLowerCase();
     var start = 0;
     $.each(keywords, function() {
@@ -509,7 +516,7 @@ var Search = {
     var excerpt = ((start > 0) ? '...' : '') +
       $.trim(text.substr(start, 240)) +
       ((start + 240 - text.length) ? '...' : '');
-    var rv = $('
    ').text(excerpt);
+    var rv = $('
    ').text(excerpt);
     $.each(hlwords, function() {
       rv = rv.highlightText(this, 'highlighted');
     });
diff --git a/docs/html/_static/underscore-1.13.1.js b/docs/html/_static/underscore-1.13.1.js
new file mode 100644
index 00000000..ffd77af9
--- /dev/null
+++ b/docs/html/_static/underscore-1.13.1.js
@@ -0,0 +1,2042 @@
+(function (global, factory) {
+  typeof exports === 'object' && typeof module !== 'undefined' ? module.exports = factory() :
+  typeof define === 'function' && define.amd ? define('underscore', factory) :
+  (global = typeof globalThis !== 'undefined' ? globalThis : global || self, (function () {
+    var current = global._;
+    var exports = global._ = factory();
+    exports.noConflict = function () { global._ = current; return exports; };
+  }()));
+}(this, (function () {
+  //     Underscore.js 1.13.1
+  //     https://underscorejs.org
+  //     (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors
+  //     Underscore may be freely distributed under the MIT license.
+
+  // Current version.
+  var VERSION = '1.13.1';
+
+  // Establish the root object, `window` (`self`) in the browser, `global`
+  // on the server, or `this` in some virtual machines. We use `self`
+  // instead of `window` for `WebWorker` support.
+  var root = typeof self == 'object' && self.self === self && self ||
+            typeof global == 'object' && global.global === global && global ||
+            Function('return this')() ||
+            {};
+
+  // Save bytes in the minified (but not gzipped) version:
+  var ArrayProto = Array.prototype, ObjProto = Object.prototype;
+  var SymbolProto = typeof Symbol !== 'undefined' ? Symbol.prototype : null;
+
+  // Create quick reference variables for speed access to core prototypes.
+  var push = ArrayProto.push,
+      slice = ArrayProto.slice,
+      toString = ObjProto.toString,
+      hasOwnProperty = ObjProto.hasOwnProperty;
+
+  // Modern feature detection.
+  var supportsArrayBuffer = typeof ArrayBuffer !== 'undefined',
+      supportsDataView = typeof DataView !== 'undefined';
+
+  // All **ECMAScript 5+** native function implementations that we hope to use
+  // are declared here.
+  var nativeIsArray = Array.isArray,
+      nativeKeys = Object.keys,
+      nativeCreate = Object.create,
+      nativeIsView = supportsArrayBuffer && ArrayBuffer.isView;
+
+  // Create references to these builtin functions because we override them.
+  var _isNaN = isNaN,
+      _isFinite = isFinite;
+
+  // Keys in IE < 9 that won't be iterated by `for key in ...` and thus missed.
+  var hasEnumBug = !{toString: null}.propertyIsEnumerable('toString');
+  var nonEnumerableProps = ['valueOf', 'isPrototypeOf', 'toString',
+    'propertyIsEnumerable', 'hasOwnProperty', 'toLocaleString'];
+
+  // The largest integer that can be represented exactly.
+  var MAX_ARRAY_INDEX = Math.pow(2, 53) - 1;
+
+  // Some functions take a variable number of arguments, or a few expected
+  // arguments at the beginning and then a variable number of values to operate
+  // on. This helper accumulates all remaining arguments past the function’s
+  // argument length (or an explicit `startIndex`), into an array that becomes
+  // the last argument. Similar to ES6’s "rest parameter".
+  function restArguments(func, startIndex) {
+    startIndex = startIndex == null ? func.length - 1 : +startIndex;
+    return function() {
+      var length = Math.max(arguments.length - startIndex, 0),
+          rest = Array(length),
+          index = 0;
+      for (; index < length; index++) {
+        rest[index] = arguments[index + startIndex];
+      }
+      switch (startIndex) {
+        case 0: return func.call(this, rest);
+        case 1: return func.call(this, arguments[0], rest);
+        case 2: return func.call(this, arguments[0], arguments[1], rest);
+      }
+      var args = Array(startIndex + 1);
+      for (index = 0; index < startIndex; index++) {
+        args[index] = arguments[index];
+      }
+      args[startIndex] = rest;
+      return func.apply(this, args);
+    };
+  }
+
+  // Is a given variable an object?
+  function isObject(obj) {
+    var type = typeof obj;
+    return type === 'function' || type === 'object' && !!obj;
+  }
+
+  // Is a given value equal to null?
+  function isNull(obj) {
+    return obj === null;
+  }
+
+  // Is a given variable undefined?
+  function isUndefined(obj) {
+    return obj === void 0;
+  }
+
+  // Is a given value a boolean?
+  function isBoolean(obj) {
+    return obj === true || obj === false || toString.call(obj) === '[object Boolean]';
+  }
+
+  // Is a given value a DOM element?
+  function isElement(obj) {
+    return !!(obj && obj.nodeType === 1);
+  }
+
+  // Internal function for creating a `toString`-based type tester.
+  function tagTester(name) {
+    var tag = '[object ' + name + ']';
+    return function(obj) {
+      return toString.call(obj) === tag;
+    };
+  }
+
+  var isString = tagTester('String');
+
+  var isNumber = tagTester('Number');
+
+  var isDate = tagTester('Date');
+
+  var isRegExp = tagTester('RegExp');
+
+  var isError = tagTester('Error');
+
+  var isSymbol = tagTester('Symbol');
+
+  var isArrayBuffer = tagTester('ArrayBuffer');
+
+  var isFunction = tagTester('Function');
+
+  // Optimize `isFunction` if appropriate. Work around some `typeof` bugs in old
+  // v8, IE 11 (#1621), Safari 8 (#1929), and PhantomJS (#2236).
+  var nodelist = root.document && root.document.childNodes;
+  if (typeof /./ != 'function' && typeof Int8Array != 'object' && typeof nodelist != 'function') {
+    isFunction = function(obj) {
+      return typeof obj == 'function' || false;
+    };
+  }
+
+  var isFunction$1 = isFunction;
+
+  var hasObjectTag = tagTester('Object');
+
+  // In IE 10 - Edge 13, `DataView` has string tag `'[object Object]'`.
+  // In IE 11, the most common among them, this problem also applies to
+  // `Map`, `WeakMap` and `Set`.
+  var hasStringTagBug = (
+        supportsDataView && hasObjectTag(new DataView(new ArrayBuffer(8)))
+      ),
+      isIE11 = (typeof Map !== 'undefined' && hasObjectTag(new Map));
+
+  var isDataView = tagTester('DataView');
+
+  // In IE 10 - Edge 13, we need a different heuristic
+  // to determine whether an object is a `DataView`.
+  function ie10IsDataView(obj) {
+    return obj != null && isFunction$1(obj.getInt8) && isArrayBuffer(obj.buffer);
+  }
+
+  var isDataView$1 = (hasStringTagBug ? ie10IsDataView : isDataView);
+
+  // Is a given value an array?
+  // Delegates to ECMA5's native `Array.isArray`.
+  var isArray = nativeIsArray || tagTester('Array');
+
+  // Internal function to check whether `key` is an own property name of `obj`.
+  function has$1(obj, key) {
+    return obj != null && hasOwnProperty.call(obj, key);
+  }
+
+  var isArguments = tagTester('Arguments');
+
+  // Define a fallback version of the method in browsers (ahem, IE < 9), where
+  // there isn't any inspectable "Arguments" type.
+  (function() {
+    if (!isArguments(arguments)) {
+      isArguments = function(obj) {
+        return has$1(obj, 'callee');
+      };
+    }
+  }());
+
+  var isArguments$1 = isArguments;
+
+  // Is a given object a finite number?
+  function isFinite$1(obj) {
+    return !isSymbol(obj) && _isFinite(obj) && !isNaN(parseFloat(obj));
+  }
+
+  // Is the given value `NaN`?
+  function isNaN$1(obj) {
+    return isNumber(obj) && _isNaN(obj);
+  }
+
+  // Predicate-generating function. Often useful outside of Underscore.
+  function constant(value) {
+    return function() {
+      return value;
+    };
+  }
+
+  // Common internal logic for `isArrayLike` and `isBufferLike`.
+  function createSizePropertyCheck(getSizeProperty) {
+    return function(collection) {
+      var sizeProperty = getSizeProperty(collection);
+      return typeof sizeProperty == 'number' && sizeProperty >= 0 && sizeProperty <= MAX_ARRAY_INDEX;
+    }
+  }
+
+  // Internal helper to generate a function to obtain property `key` from `obj`.
+  function shallowProperty(key) {
+    return function(obj) {
+      return obj == null ? void 0 : obj[key];
+    };
+  }
+
+  // Internal helper to obtain the `byteLength` property of an object.
+  var getByteLength = shallowProperty('byteLength');
+
+  // Internal helper to determine whether we should spend extensive checks against
+  // `ArrayBuffer` et al.
+  var isBufferLike = createSizePropertyCheck(getByteLength);
+
+  // Is a given value a typed array?
+  var typedArrayPattern = /\[object ((I|Ui)nt(8|16|32)|Float(32|64)|Uint8Clamped|Big(I|Ui)nt64)Array\]/;
+  function isTypedArray(obj) {
+    // `ArrayBuffer.isView` is the most future-proof, so use it when available.
+    // Otherwise, fall back on the above regular expression.
+    return nativeIsView ? (nativeIsView(obj) && !isDataView$1(obj)) :
+                  isBufferLike(obj) && typedArrayPattern.test(toString.call(obj));
+  }
+
+  var isTypedArray$1 = supportsArrayBuffer ? isTypedArray : constant(false);
+
+  // Internal helper to obtain the `length` property of an object.
+  var getLength = shallowProperty('length');
+
+  // Internal helper to create a simple lookup structure.
+  // `collectNonEnumProps` used to depend on `_.contains`, but this led to
+  // circular imports. `emulatedSet` is a one-off solution that only works for
+  // arrays of strings.
+  function emulatedSet(keys) {
+    var hash = {};
+    for (var l = keys.length, i = 0; i < l; ++i) hash[keys[i]] = true;
+    return {
+      contains: function(key) { return hash[key]; },
+      push: function(key) {
+        hash[key] = true;
+        return keys.push(key);
+      }
+    };
+  }
+
+  // Internal helper. Checks `keys` for the presence of keys in IE < 9 that won't
+  // be iterated by `for key in ...` and thus missed. Extends `keys` in place if
+  // needed.
+  function collectNonEnumProps(obj, keys) {
+    keys = emulatedSet(keys);
+    var nonEnumIdx = nonEnumerableProps.length;
+    var constructor = obj.constructor;
+    var proto = isFunction$1(constructor) && constructor.prototype || ObjProto;
+
+    // Constructor is a special case.
+    var prop = 'constructor';
+    if (has$1(obj, prop) && !keys.contains(prop)) keys.push(prop);
+
+    while (nonEnumIdx--) {
+      prop = nonEnumerableProps[nonEnumIdx];
+      if (prop in obj && obj[prop] !== proto[prop] && !keys.contains(prop)) {
+        keys.push(prop);
+      }
+    }
+  }
+
+  // Retrieve the names of an object's own properties.
+  // Delegates to **ECMAScript 5**'s native `Object.keys`.
+  function keys(obj) {
+    if (!isObject(obj)) return [];
+    if (nativeKeys) return nativeKeys(obj);
+    var keys = [];
+    for (var key in obj) if (has$1(obj, key)) keys.push(key);
+    // Ahem, IE < 9.
+    if (hasEnumBug) collectNonEnumProps(obj, keys);
+    return keys;
+  }
+
+  // Is a given array, string, or object empty?
+  // An "empty" object has no enumerable own-properties.
+  function isEmpty(obj) {
+    if (obj == null) return true;
+    // Skip the more expensive `toString`-based type checks if `obj` has no
+    // `.length`.
+    var length = getLength(obj);
+    if (typeof length == 'number' && (
+      isArray(obj) || isString(obj) || isArguments$1(obj)
+    )) return length === 0;
+    return getLength(keys(obj)) === 0;
+  }
+
+  // Returns whether an object has a given set of `key:value` pairs.
+  function isMatch(object, attrs) {
+    var _keys = keys(attrs), length = _keys.length;
+    if (object == null) return !length;
+    var obj = Object(object);
+    for (var i = 0; i < length; i++) {
+      var key = _keys[i];
+      if (attrs[key] !== obj[key] || !(key in obj)) return false;
+    }
+    return true;
+  }
+
+  // If Underscore is called as a function, it returns a wrapped object that can
+  // be used OO-style. This wrapper holds altered versions of all functions added
+  // through `_.mixin`. Wrapped objects may be chained.
+  function _$1(obj) {
+    if (obj instanceof _$1) return obj;
+    if (!(this instanceof _$1)) return new _$1(obj);
+    this._wrapped = obj;
+  }
+
+  _$1.VERSION = VERSION;
+
+  // Extracts the result from a wrapped and chained object.
+  _$1.prototype.value = function() {
+    return this._wrapped;
+  };
+
+  // Provide unwrapping proxies for some methods used in engine operations
+  // such as arithmetic and JSON stringification.
+  _$1.prototype.valueOf = _$1.prototype.toJSON = _$1.prototype.value;
+
+  _$1.prototype.toString = function() {
+    return String(this._wrapped);
+  };
+
+  // Internal function to wrap or shallow-copy an ArrayBuffer,
+  // typed array or DataView to a new view, reusing the buffer.
+  function toBufferView(bufferSource) {
+    return new Uint8Array(
+      bufferSource.buffer || bufferSource,
+      bufferSource.byteOffset || 0,
+      getByteLength(bufferSource)
+    );
+  }
+
+  // We use this string twice, so give it a name for minification.
+  var tagDataView = '[object DataView]';
+
+  // Internal recursive comparison function for `_.isEqual`.
+  function eq(a, b, aStack, bStack) {
+    // Identical objects are equal. `0 === -0`, but they aren't identical.
+    // See the [Harmony `egal` proposal](https://wiki.ecmascript.org/doku.php?id=harmony:egal).
+    if (a === b) return a !== 0 || 1 / a === 1 / b;
+    // `null` or `undefined` only equal to itself (strict comparison).
+    if (a == null || b == null) return false;
+    // `NaN`s are equivalent, but non-reflexive.
+    if (a !== a) return b !== b;
+    // Exhaust primitive checks
+    var type = typeof a;
+    if (type !== 'function' && type !== 'object' && typeof b != 'object') return false;
+    return deepEq(a, b, aStack, bStack);
+  }
+
+  // Internal recursive comparison function for `_.isEqual`.
+  function deepEq(a, b, aStack, bStack) {
+    // Unwrap any wrapped objects.
+    if (a instanceof _$1) a = a._wrapped;
+    if (b instanceof _$1) b = b._wrapped;
+    // Compare `[[Class]]` names.
+    var className = toString.call(a);
+    if (className !== toString.call(b)) return false;
+    // Work around a bug in IE 10 - Edge 13.
+    if (hasStringTagBug && className == '[object Object]' && isDataView$1(a)) {
+      if (!isDataView$1(b)) return false;
+      className = tagDataView;
+    }
+    switch (className) {
+      // These types are compared by value.
+      case '[object RegExp]':
+        // RegExps are coerced to strings for comparison (Note: '' + /a/i === '/a/i')
+      case '[object String]':
+        // Primitives and their corresponding object wrappers are equivalent; thus, `"5"` is
+        // equivalent to `new String("5")`.
+        return '' + a === '' + b;
+      case '[object Number]':
+        // `NaN`s are equivalent, but non-reflexive.
+        // Object(NaN) is equivalent to NaN.
+        if (+a !== +a) return +b !== +b;
+        // An `egal` comparison is performed for other numeric values.
+        return +a === 0 ? 1 / +a === 1 / b : +a === +b;
+      case '[object Date]':
+      case '[object Boolean]':
+        // Coerce dates and booleans to numeric primitive values. Dates are compared by their
+        // millisecond representations. Note that invalid dates with millisecond representations
+        // of `NaN` are not equivalent.
+        return +a === +b;
+      case '[object Symbol]':
+        return SymbolProto.valueOf.call(a) === SymbolProto.valueOf.call(b);
+      case '[object ArrayBuffer]':
+      case tagDataView:
+        // Coerce to typed array so we can fall through.
+        return deepEq(toBufferView(a), toBufferView(b), aStack, bStack);
+    }
+
+    var areArrays = className === '[object Array]';
+    if (!areArrays && isTypedArray$1(a)) {
+        var byteLength = getByteLength(a);
+        if (byteLength !== getByteLength(b)) return false;
+        if (a.buffer === b.buffer && a.byteOffset === b.byteOffset) return true;
+        areArrays = true;
+    }
+    if (!areArrays) {
+      if (typeof a != 'object' || typeof b != 'object') return false;
+
+      // Objects with different constructors are not equivalent, but `Object`s or `Array`s
+      // from different frames are.
+      var aCtor = a.constructor, bCtor = b.constructor;
+      if (aCtor !== bCtor && !(isFunction$1(aCtor) && aCtor instanceof aCtor &&
+                               isFunction$1(bCtor) && bCtor instanceof bCtor)
+                          && ('constructor' in a && 'constructor' in b)) {
+        return false;
+      }
+    }
+    // Assume equality for cyclic structures. The algorithm for detecting cyclic
+    // structures is adapted from ES 5.1 section 15.12.3, abstract operation `JO`.
+
+    // Initializing stack of traversed objects.
+    // It's done here since we only need them for objects and arrays comparison.
+    aStack = aStack || [];
+    bStack = bStack || [];
+    var length = aStack.length;
+    while (length--) {
+      // Linear search. Performance is inversely proportional to the number of
+      // unique nested structures.
+      if (aStack[length] === a) return bStack[length] === b;
+    }
+
+    // Add the first object to the stack of traversed objects.
+    aStack.push(a);
+    bStack.push(b);
+
+    // Recursively compare objects and arrays.
+    if (areArrays) {
+      // Compare array lengths to determine if a deep comparison is necessary.
+      length = a.length;
+      if (length !== b.length) return false;
+      // Deep compare the contents, ignoring non-numeric properties.
+      while (length--) {
+        if (!eq(a[length], b[length], aStack, bStack)) return false;
+      }
+    } else {
+      // Deep compare objects.
+      var _keys = keys(a), key;
+      length = _keys.length;
+      // Ensure that both objects contain the same number of properties before comparing deep equality.
+      if (keys(b).length !== length) return false;
+      while (length--) {
+        // Deep compare each member
+        key = _keys[length];
+        if (!(has$1(b, key) && eq(a[key], b[key], aStack, bStack))) return false;
+      }
+    }
+    // Remove the first object from the stack of traversed objects.
+    aStack.pop();
+    bStack.pop();
+    return true;
+  }
+
+  // Perform a deep comparison to check if two objects are equal.
+  function isEqual(a, b) {
+    return eq(a, b);
+  }
+
+  // Retrieve all the enumerable property names of an object.
+  function allKeys(obj) {
+    if (!isObject(obj)) return [];
+    var keys = [];
+    for (var key in obj) keys.push(key);
+    // Ahem, IE < 9.
+    if (hasEnumBug) collectNonEnumProps(obj, keys);
+    return keys;
+  }
+
+  // Since the regular `Object.prototype.toString` type tests don't work for
+  // some types in IE 11, we use a fingerprinting heuristic instead, based
+  // on the methods. It's not great, but it's the best we got.
+  // The fingerprint method lists are defined below.
+  function ie11fingerprint(methods) {
+    var length = getLength(methods);
+    return function(obj) {
+      if (obj == null) return false;
+      // `Map`, `WeakMap` and `Set` have no enumerable keys.
+      var keys = allKeys(obj);
+      if (getLength(keys)) return false;
+      for (var i = 0; i < length; i++) {
+        if (!isFunction$1(obj[methods[i]])) return false;
+      }
+      // If we are testing against `WeakMap`, we need to ensure that
+      // `obj` doesn't have a `forEach` method in order to distinguish
+      // it from a regular `Map`.
+      return methods !== weakMapMethods || !isFunction$1(obj[forEachName]);
+    };
+  }
+
+  // In the interest of compact minification, we write
+  // each string in the fingerprints only once.
+  var forEachName = 'forEach',
+      hasName = 'has',
+      commonInit = ['clear', 'delete'],
+      mapTail = ['get', hasName, 'set'];
+
+  // `Map`, `WeakMap` and `Set` each have slightly different
+  // combinations of the above sublists.
+  var mapMethods = commonInit.concat(forEachName, mapTail),
+      weakMapMethods = commonInit.concat(mapTail),
+      setMethods = ['add'].concat(commonInit, forEachName, hasName);
+
+  var isMap = isIE11 ? ie11fingerprint(mapMethods) : tagTester('Map');
+
+  var isWeakMap = isIE11 ? ie11fingerprint(weakMapMethods) : tagTester('WeakMap');
+
+  var isSet = isIE11 ? ie11fingerprint(setMethods) : tagTester('Set');
+
+  var isWeakSet = tagTester('WeakSet');
+
+  // Retrieve the values of an object's properties.
+  function values(obj) {
+    var _keys = keys(obj);
+    var length = _keys.length;
+    var values = Array(length);
+    for (var i = 0; i < length; i++) {
+      values[i] = obj[_keys[i]];
+    }
+    return values;
+  }
+
+  // Convert an object into a list of `[key, value]` pairs.
+  // The opposite of `_.object` with one argument.
+  function pairs(obj) {
+    var _keys = keys(obj);
+    var length = _keys.length;
+    var pairs = Array(length);
+    for (var i = 0; i < length; i++) {
+      pairs[i] = [_keys[i], obj[_keys[i]]];
+    }
+    return pairs;
+  }
+
+  // Invert the keys and values of an object. The values must be serializable.
+  function invert(obj) {
+    var result = {};
+    var _keys = keys(obj);
+    for (var i = 0, length = _keys.length; i < length; i++) {
+      result[obj[_keys[i]]] = _keys[i];
+    }
+    return result;
+  }
+
+  // Return a sorted list of the function names available on the object.
+  function functions(obj) {
+    var names = [];
+    for (var key in obj) {
+      if (isFunction$1(obj[key])) names.push(key);
+    }
+    return names.sort();
+  }
+
+  // An internal function for creating assigner functions.
+  function createAssigner(keysFunc, defaults) {
+    return function(obj) {
+      var length = arguments.length;
+      if (defaults) obj = Object(obj);
+      if (length < 2 || obj == null) return obj;
+      for (var index = 1; index < length; index++) {
+        var source = arguments[index],
+            keys = keysFunc(source),
+            l = keys.length;
+        for (var i = 0; i < l; i++) {
+          var key = keys[i];
+          if (!defaults || obj[key] === void 0) obj[key] = source[key];
+        }
+      }
+      return obj;
+    };
+  }
+
+  // Extend a given object with all the properties in passed-in object(s).
+  var extend = createAssigner(allKeys);
+
+  // Assigns a given object with all the own properties in the passed-in
+  // object(s).
+  // (https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object/assign)
+  var extendOwn = createAssigner(keys);
+
+  // Fill in a given object with default properties.
+  var defaults = createAssigner(allKeys, true);
+
+  // Create a naked function reference for surrogate-prototype-swapping.
+  function ctor() {
+    return function(){};
+  }
+
+  // An internal function for creating a new object that inherits from another.
+  function baseCreate(prototype) {
+    if (!isObject(prototype)) return {};
+    if (nativeCreate) return nativeCreate(prototype);
+    var Ctor = ctor();
+    Ctor.prototype = prototype;
+    var result = new Ctor;
+    Ctor.prototype = null;
+    return result;
+  }
+
+  // Creates an object that inherits from the given prototype object.
+  // If additional properties are provided then they will be added to the
+  // created object.
+  function create(prototype, props) {
+    var result = baseCreate(prototype);
+    if (props) extendOwn(result, props);
+    return result;
+  }
+
+  // Create a (shallow-cloned) duplicate of an object.
+  function clone(obj) {
+    if (!isObject(obj)) return obj;
+    return isArray(obj) ? obj.slice() : extend({}, obj);
+  }
+
+  // Invokes `interceptor` with the `obj` and then returns `obj`.
+  // The primary purpose of this method is to "tap into" a method chain, in
+  // order to perform operations on intermediate results within the chain.
+  function tap(obj, interceptor) {
+    interceptor(obj);
+    return obj;
+  }
+
+  // Normalize a (deep) property `path` to array.
+  // Like `_.iteratee`, this function can be customized.
+  function toPath$1(path) {
+    return isArray(path) ? path : [path];
+  }
+  _$1.toPath = toPath$1;
+
+  // Internal wrapper for `_.toPath` to enable minification.
+  // Similar to `cb` for `_.iteratee`.
+  function toPath(path) {
+    return _$1.toPath(path);
+  }
+
+  // Internal function to obtain a nested property in `obj` along `path`.
+  function deepGet(obj, path) {
+    var length = path.length;
+    for (var i = 0; i < length; i++) {
+      if (obj == null) return void 0;
+      obj = obj[path[i]];
+    }
+    return length ? obj : void 0;
+  }
+
+  // Get the value of the (deep) property on `path` from `object`.
+  // If any property in `path` does not exist or if the value is
+  // `undefined`, return `defaultValue` instead.
+  // The `path` is normalized through `_.toPath`.
+  function get(object, path, defaultValue) {
+    var value = deepGet(object, toPath(path));
+    return isUndefined(value) ? defaultValue : value;
+  }
+
+  // Shortcut function for checking if an object has a given property directly on
+  // itself (in other words, not on a prototype). Unlike the internal `has`
+  // function, this public version can also traverse nested properties.
+  function has(obj, path) {
+    path = toPath(path);
+    var length = path.length;
+    for (var i = 0; i < length; i++) {
+      var key = path[i];
+      if (!has$1(obj, key)) return false;
+      obj = obj[key];
+    }
+    return !!length;
+  }
+
+  // Keep the identity function around for default iteratees.
+  function identity(value) {
+    return value;
+  }
+
+  // Returns a predicate for checking whether an object has a given set of
+  // `key:value` pairs.
+  function matcher(attrs) {
+    attrs = extendOwn({}, attrs);
+    return function(obj) {
+      return isMatch(obj, attrs);
+    };
+  }
+
+  // Creates a function that, when passed an object, will traverse that object’s
+  // properties down the given `path`, specified as an array of keys or indices.
+  function property(path) {
+    path = toPath(path);
+    return function(obj) {
+      return deepGet(obj, path);
+    };
+  }
+
+  // Internal function that returns an efficient (for current engines) version
+  // of the passed-in callback, to be repeatedly applied in other Underscore
+  // functions.
+  function optimizeCb(func, context, argCount) {
+    if (context === void 0) return func;
+    switch (argCount == null ? 3 : argCount) {
+      case 1: return function(value) {
+        return func.call(context, value);
+      };
+      // The 2-argument case is omitted because we’re not using it.
+      case 3: return function(value, index, collection) {
+        return func.call(context, value, index, collection);
+      };
+      case 4: return function(accumulator, value, index, collection) {
+        return func.call(context, accumulator, value, index, collection);
+      };
+    }
+    return function() {
+      return func.apply(context, arguments);
+    };
+  }
+
+  // An internal function to generate callbacks that can be applied to each
+  // element in a collection, returning the desired result — either `_.identity`,
+  // an arbitrary callback, a property matcher, or a property accessor.
+  function baseIteratee(value, context, argCount) {
+    if (value == null) return identity;
+    if (isFunction$1(value)) return optimizeCb(value, context, argCount);
+    if (isObject(value) && !isArray(value)) return matcher(value);
+    return property(value);
+  }
+
+  // External wrapper for our callback generator. Users may customize
+  // `_.iteratee` if they want additional predicate/iteratee shorthand styles.
+  // This abstraction hides the internal-only `argCount` argument.
+  function iteratee(value, context) {
+    return baseIteratee(value, context, Infinity);
+  }
+  _$1.iteratee = iteratee;
+
+  // The function we call internally to generate a callback. It invokes
+  // `_.iteratee` if overridden, otherwise `baseIteratee`.
+  function cb(value, context, argCount) {
+    if (_$1.iteratee !== iteratee) return _$1.iteratee(value, context);
+    return baseIteratee(value, context, argCount);
+  }
+
+  // Returns the results of applying the `iteratee` to each element of `obj`.
+  // In contrast to `_.map` it returns an object.
+  function mapObject(obj, iteratee, context) {
+    iteratee = cb(iteratee, context);
+    var _keys = keys(obj),
+        length = _keys.length,
+        results = {};
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys[index];
+      results[currentKey] = iteratee(obj[currentKey], currentKey, obj);
+    }
+    return results;
+  }
+
+  // Predicate-generating function. Often useful outside of Underscore.
+  function noop(){}
+
+  // Generates a function for a given object that returns a given property.
+  function propertyOf(obj) {
+    if (obj == null) return noop;
+    return function(path) {
+      return get(obj, path);
+    };
+  }
+
+  // Run a function **n** times.
+  function times(n, iteratee, context) {
+    var accum = Array(Math.max(0, n));
+    iteratee = optimizeCb(iteratee, context, 1);
+    for (var i = 0; i < n; i++) accum[i] = iteratee(i);
+    return accum;
+  }
+
+  // Return a random integer between `min` and `max` (inclusive).
+  function random(min, max) {
+    if (max == null) {
+      max = min;
+      min = 0;
+    }
+    return min + Math.floor(Math.random() * (max - min + 1));
+  }
+
+  // A (possibly faster) way to get the current timestamp as an integer.
+  var now = Date.now || function() {
+    return new Date().getTime();
+  };
+
+  // Internal helper to generate functions for escaping and unescaping strings
+  // to/from HTML interpolation.
+  function createEscaper(map) {
+    var escaper = function(match) {
+      return map[match];
+    };
+    // Regexes for identifying a key that needs to be escaped.
+    var source = '(?:' + keys(map).join('|') + ')';
+    var testRegexp = RegExp(source);
+    var replaceRegexp = RegExp(source, 'g');
+    return function(string) {
+      string = string == null ? '' : '' + string;
+      return testRegexp.test(string) ? string.replace(replaceRegexp, escaper) : string;
+    };
+  }
+
+  // Internal list of HTML entities for escaping.
+  var escapeMap = {
+    '&': '&',
+    '<': '<',
+    '>': '>',
+    '"': '"',
+    "'": ''',
+    '`': '`'
+  };
+
+  // Function for escaping strings to HTML interpolation.
+  var _escape = createEscaper(escapeMap);
+
+  // Internal list of HTML entities for unescaping.
+  var unescapeMap = invert(escapeMap);
+
+  // Function for unescaping strings from HTML interpolation.
+  var _unescape = createEscaper(unescapeMap);
+
+  // By default, Underscore uses ERB-style template delimiters. Change the
+  // following template settings to use alternative delimiters.
+  var templateSettings = _$1.templateSettings = {
+    evaluate: /<%([\s\S]+?)%>/g,
+    interpolate: /<%=([\s\S]+?)%>/g,
+    escape: /<%-([\s\S]+?)%>/g
+  };
+
+  // When customizing `_.templateSettings`, if you don't want to define an
+  // interpolation, evaluation or escaping regex, we need one that is
+  // guaranteed not to match.
+  var noMatch = /(.)^/;
+
+  // Certain characters need to be escaped so that they can be put into a
+  // string literal.
+  var escapes = {
+    "'": "'",
+    '\\': '\\',
+    '\r': 'r',
+    '\n': 'n',
+    '\u2028': 'u2028',
+    '\u2029': 'u2029'
+  };
+
+  var escapeRegExp = /\\|'|\r|\n|\u2028|\u2029/g;
+
+  function escapeChar(match) {
+    return '\\' + escapes[match];
+  }
+
+  // In order to prevent third-party code injection through
+  // `_.templateSettings.variable`, we test it against the following regular
+  // expression. It is intentionally a bit more liberal than just matching valid
+  // identifiers, but still prevents possible loopholes through defaults or
+  // destructuring assignment.
+  var bareIdentifier = /^\s*(\w|\$)+\s*$/;
+
+  // JavaScript micro-templating, similar to John Resig's implementation.
+  // Underscore templating handles arbitrary delimiters, preserves whitespace,
+  // and correctly escapes quotes within interpolated code.
+  // NB: `oldSettings` only exists for backwards compatibility.
+  function template(text, settings, oldSettings) {
+    if (!settings && oldSettings) settings = oldSettings;
+    settings = defaults({}, settings, _$1.templateSettings);
+
+    // Combine delimiters into one regular expression via alternation.
+    var matcher = RegExp([
+      (settings.escape || noMatch).source,
+      (settings.interpolate || noMatch).source,
+      (settings.evaluate || noMatch).source
+    ].join('|') + '|$', 'g');
+
+    // Compile the template source, escaping string literals appropriately.
+    var index = 0;
+    var source = "__p+='";
+    text.replace(matcher, function(match, escape, interpolate, evaluate, offset) {
+      source += text.slice(index, offset).replace(escapeRegExp, escapeChar);
+      index = offset + match.length;
+
+      if (escape) {
+        source += "'+\n((__t=(" + escape + "))==null?'':_.escape(__t))+\n'";
+      } else if (interpolate) {
+        source += "'+\n((__t=(" + interpolate + "))==null?'':__t)+\n'";
+      } else if (evaluate) {
+        source += "';\n" + evaluate + "\n__p+='";
+      }
+
+      // Adobe VMs need the match returned to produce the correct offset.
+      return match;
+    });
+    source += "';\n";
+
+    var argument = settings.variable;
+    if (argument) {
+      // Insure against third-party code injection. (CVE-2021-23358)
+      if (!bareIdentifier.test(argument)) throw new Error(
+        'variable is not a bare identifier: ' + argument
+      );
+    } else {
+      // If a variable is not specified, place data values in local scope.
+      source = 'with(obj||{}){\n' + source + '}\n';
+      argument = 'obj';
+    }
+
+    source = "var __t,__p='',__j=Array.prototype.join," +
+      "print=function(){__p+=__j.call(arguments,'');};\n" +
+      source + 'return __p;\n';
+
+    var render;
+    try {
+      render = new Function(argument, '_', source);
+    } catch (e) {
+      e.source = source;
+      throw e;
+    }
+
+    var template = function(data) {
+      return render.call(this, data, _$1);
+    };
+
+    // Provide the compiled source as a convenience for precompilation.
+    template.source = 'function(' + argument + '){\n' + source + '}';
+
+    return template;
+  }
+
+  // Traverses the children of `obj` along `path`. If a child is a function, it
+  // is invoked with its parent as context. Returns the value of the final
+  // child, or `fallback` if any child is undefined.
+  function result(obj, path, fallback) {
+    path = toPath(path);
+    var length = path.length;
+    if (!length) {
+      return isFunction$1(fallback) ? fallback.call(obj) : fallback;
+    }
+    for (var i = 0; i < length; i++) {
+      var prop = obj == null ? void 0 : obj[path[i]];
+      if (prop === void 0) {
+        prop = fallback;
+        i = length; // Ensure we don't continue iterating.
+      }
+      obj = isFunction$1(prop) ? prop.call(obj) : prop;
+    }
+    return obj;
+  }
+
+  // Generate a unique integer id (unique within the entire client session).
+  // Useful for temporary DOM ids.
+  var idCounter = 0;
+  function uniqueId(prefix) {
+    var id = ++idCounter + '';
+    return prefix ? prefix + id : id;
+  }
+
+  // Start chaining a wrapped Underscore object.
+  function chain(obj) {
+    var instance = _$1(obj);
+    instance._chain = true;
+    return instance;
+  }
+
+  // Internal function to execute `sourceFunc` bound to `context` with optional
+  // `args`. Determines whether to execute a function as a constructor or as a
+  // normal function.
+  function executeBound(sourceFunc, boundFunc, context, callingContext, args) {
+    if (!(callingContext instanceof boundFunc)) return sourceFunc.apply(context, args);
+    var self = baseCreate(sourceFunc.prototype);
+    var result = sourceFunc.apply(self, args);
+    if (isObject(result)) return result;
+    return self;
+  }
+
+  // Partially apply a function by creating a version that has had some of its
+  // arguments pre-filled, without changing its dynamic `this` context. `_` acts
+  // as a placeholder by default, allowing any combination of arguments to be
+  // pre-filled. Set `_.partial.placeholder` for a custom placeholder argument.
+  var partial = restArguments(function(func, boundArgs) {
+    var placeholder = partial.placeholder;
+    var bound = function() {
+      var position = 0, length = boundArgs.length;
+      var args = Array(length);
+      for (var i = 0; i < length; i++) {
+        args[i] = boundArgs[i] === placeholder ? arguments[position++] : boundArgs[i];
+      }
+      while (position < arguments.length) args.push(arguments[position++]);
+      return executeBound(func, bound, this, this, args);
+    };
+    return bound;
+  });
+
+  partial.placeholder = _$1;
+
+  // Create a function bound to a given object (assigning `this`, and arguments,
+  // optionally).
+  var bind = restArguments(function(func, context, args) {
+    if (!isFunction$1(func)) throw new TypeError('Bind must be called on a function');
+    var bound = restArguments(function(callArgs) {
+      return executeBound(func, bound, context, this, args.concat(callArgs));
+    });
+    return bound;
+  });
+
+  // Internal helper for collection methods to determine whether a collection
+  // should be iterated as an array or as an object.
+  // Related: https://people.mozilla.org/~jorendorff/es6-draft.html#sec-tolength
+  // Avoids a very nasty iOS 8 JIT bug on ARM-64. #2094
+  var isArrayLike = createSizePropertyCheck(getLength);
+
+  // Internal implementation of a recursive `flatten` function.
+  function flatten$1(input, depth, strict, output) {
+    output = output || [];
+    if (!depth && depth !== 0) {
+      depth = Infinity;
+    } else if (depth <= 0) {
+      return output.concat(input);
+    }
+    var idx = output.length;
+    for (var i = 0, length = getLength(input); i < length; i++) {
+      var value = input[i];
+      if (isArrayLike(value) && (isArray(value) || isArguments$1(value))) {
+        // Flatten current level of array or arguments object.
+        if (depth > 1) {
+          flatten$1(value, depth - 1, strict, output);
+          idx = output.length;
+        } else {
+          var j = 0, len = value.length;
+          while (j < len) output[idx++] = value[j++];
+        }
+      } else if (!strict) {
+        output[idx++] = value;
+      }
+    }
+    return output;
+  }
+
+  // Bind a number of an object's methods to that object. Remaining arguments
+  // are the method names to be bound. Useful for ensuring that all callbacks
+  // defined on an object belong to it.
+  var bindAll = restArguments(function(obj, keys) {
+    keys = flatten$1(keys, false, false);
+    var index = keys.length;
+    if (index < 1) throw new Error('bindAll must be passed function names');
+    while (index--) {
+      var key = keys[index];
+      obj[key] = bind(obj[key], obj);
+    }
+    return obj;
+  });
+
+  // Memoize an expensive function by storing its results.
+  function memoize(func, hasher) {
+    var memoize = function(key) {
+      var cache = memoize.cache;
+      var address = '' + (hasher ? hasher.apply(this, arguments) : key);
+      if (!has$1(cache, address)) cache[address] = func.apply(this, arguments);
+      return cache[address];
+    };
+    memoize.cache = {};
+    return memoize;
+  }
+
+  // Delays a function for the given number of milliseconds, and then calls
+  // it with the arguments supplied.
+  var delay = restArguments(function(func, wait, args) {
+    return setTimeout(function() {
+      return func.apply(null, args);
+    }, wait);
+  });
+
+  // Defers a function, scheduling it to run after the current call stack has
+  // cleared.
+  var defer = partial(delay, _$1, 1);
+
+  // Returns a function, that, when invoked, will only be triggered at most once
+  // during a given window of time. Normally, the throttled function will run
+  // as much as it can, without ever going more than once per `wait` duration;
+  // but if you'd like to disable the execution on the leading edge, pass
+  // `{leading: false}`. To disable execution on the trailing edge, ditto.
+  function throttle(func, wait, options) {
+    var timeout, context, args, result;
+    var previous = 0;
+    if (!options) options = {};
+
+    var later = function() {
+      previous = options.leading === false ? 0 : now();
+      timeout = null;
+      result = func.apply(context, args);
+      if (!timeout) context = args = null;
+    };
+
+    var throttled = function() {
+      var _now = now();
+      if (!previous && options.leading === false) previous = _now;
+      var remaining = wait - (_now - previous);
+      context = this;
+      args = arguments;
+      if (remaining <= 0 || remaining > wait) {
+        if (timeout) {
+          clearTimeout(timeout);
+          timeout = null;
+        }
+        previous = _now;
+        result = func.apply(context, args);
+        if (!timeout) context = args = null;
+      } else if (!timeout && options.trailing !== false) {
+        timeout = setTimeout(later, remaining);
+      }
+      return result;
+    };
+
+    throttled.cancel = function() {
+      clearTimeout(timeout);
+      previous = 0;
+      timeout = context = args = null;
+    };
+
+    return throttled;
+  }
+
+  // When a sequence of calls of the returned function ends, the argument
+  // function is triggered. The end of a sequence is defined by the `wait`
+  // parameter. If `immediate` is passed, the argument function will be
+  // triggered at the beginning of the sequence instead of at the end.
+  function debounce(func, wait, immediate) {
+    var timeout, previous, args, result, context;
+
+    var later = function() {
+      var passed = now() - previous;
+      if (wait > passed) {
+        timeout = setTimeout(later, wait - passed);
+      } else {
+        timeout = null;
+        if (!immediate) result = func.apply(context, args);
+        // This check is needed because `func` can recursively invoke `debounced`.
+        if (!timeout) args = context = null;
+      }
+    };
+
+    var debounced = restArguments(function(_args) {
+      context = this;
+      args = _args;
+      previous = now();
+      if (!timeout) {
+        timeout = setTimeout(later, wait);
+        if (immediate) result = func.apply(context, args);
+      }
+      return result;
+    });
+
+    debounced.cancel = function() {
+      clearTimeout(timeout);
+      timeout = args = context = null;
+    };
+
+    return debounced;
+  }
+
+  // Returns the first function passed as an argument to the second,
+  // allowing you to adjust arguments, run code before and after, and
+  // conditionally execute the original function.
+  function wrap(func, wrapper) {
+    return partial(wrapper, func);
+  }
+
+  // Returns a negated version of the passed-in predicate.
+  function negate(predicate) {
+    return function() {
+      return !predicate.apply(this, arguments);
+    };
+  }
+
+  // Returns a function that is the composition of a list of functions, each
+  // consuming the return value of the function that follows.
+  function compose() {
+    var args = arguments;
+    var start = args.length - 1;
+    return function() {
+      var i = start;
+      var result = args[start].apply(this, arguments);
+      while (i--) result = args[i].call(this, result);
+      return result;
+    };
+  }
+
+  // Returns a function that will only be executed on and after the Nth call.
+  function after(times, func) {
+    return function() {
+      if (--times < 1) {
+        return func.apply(this, arguments);
+      }
+    };
+  }
+
+  // Returns a function that will only be executed up to (but not including) the
+  // Nth call.
+  function before(times, func) {
+    var memo;
+    return function() {
+      if (--times > 0) {
+        memo = func.apply(this, arguments);
+      }
+      if (times <= 1) func = null;
+      return memo;
+    };
+  }
+
+  // Returns a function that will be executed at most one time, no matter how
+  // often you call it. Useful for lazy initialization.
+  var once = partial(before, 2);
+
+  // Returns the first key on an object that passes a truth test.
+  function findKey(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = keys(obj), key;
+    for (var i = 0, length = _keys.length; i < length; i++) {
+      key = _keys[i];
+      if (predicate(obj[key], key, obj)) return key;
+    }
+  }
+
+  // Internal function to generate `_.findIndex` and `_.findLastIndex`.
+  function createPredicateIndexFinder(dir) {
+    return function(array, predicate, context) {
+      predicate = cb(predicate, context);
+      var length = getLength(array);
+      var index = dir > 0 ? 0 : length - 1;
+      for (; index >= 0 && index < length; index += dir) {
+        if (predicate(array[index], index, array)) return index;
+      }
+      return -1;
+    };
+  }
+
+  // Returns the first index on an array-like that passes a truth test.
+  var findIndex = createPredicateIndexFinder(1);
+
+  // Returns the last index on an array-like that passes a truth test.
+  var findLastIndex = createPredicateIndexFinder(-1);
+
+  // Use a comparator function to figure out the smallest index at which
+  // an object should be inserted so as to maintain order. Uses binary search.
+  function sortedIndex(array, obj, iteratee, context) {
+    iteratee = cb(iteratee, context, 1);
+    var value = iteratee(obj);
+    var low = 0, high = getLength(array);
+    while (low < high) {
+      var mid = Math.floor((low + high) / 2);
+      if (iteratee(array[mid]) < value) low = mid + 1; else high = mid;
+    }
+    return low;
+  }
+
+  // Internal function to generate the `_.indexOf` and `_.lastIndexOf` functions.
+  function createIndexFinder(dir, predicateFind, sortedIndex) {
+    return function(array, item, idx) {
+      var i = 0, length = getLength(array);
+      if (typeof idx == 'number') {
+        if (dir > 0) {
+          i = idx >= 0 ? idx : Math.max(idx + length, i);
+        } else {
+          length = idx >= 0 ? Math.min(idx + 1, length) : idx + length + 1;
+        }
+      } else if (sortedIndex && idx && length) {
+        idx = sortedIndex(array, item);
+        return array[idx] === item ? idx : -1;
+      }
+      if (item !== item) {
+        idx = predicateFind(slice.call(array, i, length), isNaN$1);
+        return idx >= 0 ? idx + i : -1;
+      }
+      for (idx = dir > 0 ? i : length - 1; idx >= 0 && idx < length; idx += dir) {
+        if (array[idx] === item) return idx;
+      }
+      return -1;
+    };
+  }
+
+  // Return the position of the first occurrence of an item in an array,
+  // or -1 if the item is not included in the array.
+  // If the array is large and already in sort order, pass `true`
+  // for **isSorted** to use binary search.
+  var indexOf = createIndexFinder(1, findIndex, sortedIndex);
+
+  // Return the position of the last occurrence of an item in an array,
+  // or -1 if the item is not included in the array.
+  var lastIndexOf = createIndexFinder(-1, findLastIndex);
+
+  // Return the first value which passes a truth test.
+  function find(obj, predicate, context) {
+    var keyFinder = isArrayLike(obj) ? findIndex : findKey;
+    var key = keyFinder(obj, predicate, context);
+    if (key !== void 0 && key !== -1) return obj[key];
+  }
+
+  // Convenience version of a common use case of `_.find`: getting the first
+  // object containing specific `key:value` pairs.
+  function findWhere(obj, attrs) {
+    return find(obj, matcher(attrs));
+  }
+
+  // The cornerstone for collection functions, an `each`
+  // implementation, aka `forEach`.
+  // Handles raw objects in addition to array-likes. Treats all
+  // sparse array-likes as if they were dense.
+  function each(obj, iteratee, context) {
+    iteratee = optimizeCb(iteratee, context);
+    var i, length;
+    if (isArrayLike(obj)) {
+      for (i = 0, length = obj.length; i < length; i++) {
+        iteratee(obj[i], i, obj);
+      }
+    } else {
+      var _keys = keys(obj);
+      for (i = 0, length = _keys.length; i < length; i++) {
+        iteratee(obj[_keys[i]], _keys[i], obj);
+      }
+    }
+    return obj;
+  }
+
+  // Return the results of applying the iteratee to each element.
+  function map(obj, iteratee, context) {
+    iteratee = cb(iteratee, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length,
+        results = Array(length);
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      results[index] = iteratee(obj[currentKey], currentKey, obj);
+    }
+    return results;
+  }
+
+  // Internal helper to create a reducing function, iterating left or right.
+  function createReduce(dir) {
+    // Wrap code that reassigns argument variables in a separate function than
+    // the one that accesses `arguments.length` to avoid a perf hit. (#1991)
+    var reducer = function(obj, iteratee, memo, initial) {
+      var _keys = !isArrayLike(obj) && keys(obj),
+          length = (_keys || obj).length,
+          index = dir > 0 ? 0 : length - 1;
+      if (!initial) {
+        memo = obj[_keys ? _keys[index] : index];
+        index += dir;
+      }
+      for (; index >= 0 && index < length; index += dir) {
+        var currentKey = _keys ? _keys[index] : index;
+        memo = iteratee(memo, obj[currentKey], currentKey, obj);
+      }
+      return memo;
+    };
+
+    return function(obj, iteratee, memo, context) {
+      var initial = arguments.length >= 3;
+      return reducer(obj, optimizeCb(iteratee, context, 4), memo, initial);
+    };
+  }
+
+  // **Reduce** builds up a single result from a list of values, aka `inject`,
+  // or `foldl`.
+  var reduce = createReduce(1);
+
+  // The right-associative version of reduce, also known as `foldr`.
+  var reduceRight = createReduce(-1);
+
+  // Return all the elements that pass a truth test.
+  function filter(obj, predicate, context) {
+    var results = [];
+    predicate = cb(predicate, context);
+    each(obj, function(value, index, list) {
+      if (predicate(value, index, list)) results.push(value);
+    });
+    return results;
+  }
+
+  // Return all the elements for which a truth test fails.
+  function reject(obj, predicate, context) {
+    return filter(obj, negate(cb(predicate)), context);
+  }
+
+  // Determine whether all of the elements pass a truth test.
+  function every(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length;
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      if (!predicate(obj[currentKey], currentKey, obj)) return false;
+    }
+    return true;
+  }
+
+  // Determine if at least one element in the object passes a truth test.
+  function some(obj, predicate, context) {
+    predicate = cb(predicate, context);
+    var _keys = !isArrayLike(obj) && keys(obj),
+        length = (_keys || obj).length;
+    for (var index = 0; index < length; index++) {
+      var currentKey = _keys ? _keys[index] : index;
+      if (predicate(obj[currentKey], currentKey, obj)) return true;
+    }
+    return false;
+  }
+
+  // Determine if the array or object contains a given item (using `===`).
+  function contains(obj, item, fromIndex, guard) {
+    if (!isArrayLike(obj)) obj = values(obj);
+    if (typeof fromIndex != 'number' || guard) fromIndex = 0;
+    return indexOf(obj, item, fromIndex) >= 0;
+  }
+
+  // Invoke a method (with arguments) on every item in a collection.
+  var invoke = restArguments(function(obj, path, args) {
+    var contextPath, func;
+    if (isFunction$1(path)) {
+      func = path;
+    } else {
+      path = toPath(path);
+      contextPath = path.slice(0, -1);
+      path = path[path.length - 1];
+    }
+    return map(obj, function(context) {
+      var method = func;
+      if (!method) {
+        if (contextPath && contextPath.length) {
+          context = deepGet(context, contextPath);
+        }
+        if (context == null) return void 0;
+        method = context[path];
+      }
+      return method == null ? method : method.apply(context, args);
+    });
+  });
+
+  // Convenience version of a common use case of `_.map`: fetching a property.
+  function pluck(obj, key) {
+    return map(obj, property(key));
+  }
+
+  // Convenience version of a common use case of `_.filter`: selecting only
+  // objects containing specific `key:value` pairs.
+  function where(obj, attrs) {
+    return filter(obj, matcher(attrs));
+  }
+
+  // Return the maximum element (or element-based computation).
+  function max(obj, iteratee, context) {
+    var result = -Infinity, lastComputed = -Infinity,
+        value, computed;
+    if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) {
+      obj = isArrayLike(obj) ? obj : values(obj);
+      for (var i = 0, length = obj.length; i < length; i++) {
+        value = obj[i];
+        if (value != null && value > result) {
+          result = value;
+        }
+      }
+    } else {
+      iteratee = cb(iteratee, context);
+      each(obj, function(v, index, list) {
+        computed = iteratee(v, index, list);
+        if (computed > lastComputed || computed === -Infinity && result === -Infinity) {
+          result = v;
+          lastComputed = computed;
+        }
+      });
+    }
+    return result;
+  }
+
+  // Return the minimum element (or element-based computation).
+  function min(obj, iteratee, context) {
+    var result = Infinity, lastComputed = Infinity,
+        value, computed;
+    if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) {
+      obj = isArrayLike(obj) ? obj : values(obj);
+      for (var i = 0, length = obj.length; i < length; i++) {
+        value = obj[i];
+        if (value != null && value < result) {
+          result = value;
+        }
+      }
+    } else {
+      iteratee = cb(iteratee, context);
+      each(obj, function(v, index, list) {
+        computed = iteratee(v, index, list);
+        if (computed < lastComputed || computed === Infinity && result === Infinity) {
+          result = v;
+          lastComputed = computed;
+        }
+      });
+    }
+    return result;
+  }
+
+  // Sample **n** random values from a collection using the modern version of the
+  // [Fisher-Yates shuffle](https://en.wikipedia.org/wiki/Fisher–Yates_shuffle).
+  // If **n** is not specified, returns a single random element.
+  // The internal `guard` argument allows it to work with `_.map`.
+  function sample(obj, n, guard) {
+    if (n == null || guard) {
+      if (!isArrayLike(obj)) obj = values(obj);
+      return obj[random(obj.length - 1)];
+    }
+    var sample = isArrayLike(obj) ? clone(obj) : values(obj);
+    var length = getLength(sample);
+    n = Math.max(Math.min(n, length), 0);
+    var last = length - 1;
+    for (var index = 0; index < n; index++) {
+      var rand = random(index, last);
+      var temp = sample[index];
+      sample[index] = sample[rand];
+      sample[rand] = temp;
+    }
+    return sample.slice(0, n);
+  }
+
+  // Shuffle a collection.
+  function shuffle(obj) {
+    return sample(obj, Infinity);
+  }
+
+  // Sort the object's values by a criterion produced by an iteratee.
+  function sortBy(obj, iteratee, context) {
+    var index = 0;
+    iteratee = cb(iteratee, context);
+    return pluck(map(obj, function(value, key, list) {
+      return {
+        value: value,
+        index: index++,
+        criteria: iteratee(value, key, list)
+      };
+    }).sort(function(left, right) {
+      var a = left.criteria;
+      var b = right.criteria;
+      if (a !== b) {
+        if (a > b || a === void 0) return 1;
+        if (a < b || b === void 0) return -1;
+      }
+      return left.index - right.index;
+    }), 'value');
+  }
+
+  // An internal function used for aggregate "group by" operations.
+  function group(behavior, partition) {
+    return function(obj, iteratee, context) {
+      var result = partition ? [[], []] : {};
+      iteratee = cb(iteratee, context);
+      each(obj, function(value, index) {
+        var key = iteratee(value, index, obj);
+        behavior(result, value, key);
+      });
+      return result;
+    };
+  }
+
+  // Groups the object's values by a criterion. Pass either a string attribute
+  // to group by, or a function that returns the criterion.
+  var groupBy = group(function(result, value, key) {
+    if (has$1(result, key)) result[key].push(value); else result[key] = [value];
+  });
+
+  // Indexes the object's values by a criterion, similar to `_.groupBy`, but for
+  // when you know that your index values will be unique.
+  var indexBy = group(function(result, value, key) {
+    result[key] = value;
+  });
+
+  // Counts instances of an object that group by a certain criterion. Pass
+  // either a string attribute to count by, or a function that returns the
+  // criterion.
+  var countBy = group(function(result, value, key) {
+    if (has$1(result, key)) result[key]++; else result[key] = 1;
+  });
+
+  // Split a collection into two arrays: one whose elements all pass the given
+  // truth test, and one whose elements all do not pass the truth test.
+  var partition = group(function(result, value, pass) {
+    result[pass ? 0 : 1].push(value);
+  }, true);
+
+  // Safely create a real, live array from anything iterable.
+  var reStrSymbol = /[^\ud800-\udfff]|[\ud800-\udbff][\udc00-\udfff]|[\ud800-\udfff]/g;
+  function toArray(obj) {
+    if (!obj) return [];
+    if (isArray(obj)) return slice.call(obj);
+    if (isString(obj)) {
+      // Keep surrogate pair characters together.
+      return obj.match(reStrSymbol);
+    }
+    if (isArrayLike(obj)) return map(obj, identity);
+    return values(obj);
+  }
+
+  // Return the number of elements in a collection.
+  function size(obj) {
+    if (obj == null) return 0;
+    return isArrayLike(obj) ? obj.length : keys(obj).length;
+  }
+
+  // Internal `_.pick` helper function to determine whether `key` is an enumerable
+  // property name of `obj`.
+  function keyInObj(value, key, obj) {
+    return key in obj;
+  }
+
+  // Return a copy of the object only containing the allowed properties.
+  var pick = restArguments(function(obj, keys) {
+    var result = {}, iteratee = keys[0];
+    if (obj == null) return result;
+    if (isFunction$1(iteratee)) {
+      if (keys.length > 1) iteratee = optimizeCb(iteratee, keys[1]);
+      keys = allKeys(obj);
+    } else {
+      iteratee = keyInObj;
+      keys = flatten$1(keys, false, false);
+      obj = Object(obj);
+    }
+    for (var i = 0, length = keys.length; i < length; i++) {
+      var key = keys[i];
+      var value = obj[key];
+      if (iteratee(value, key, obj)) result[key] = value;
+    }
+    return result;
+  });
+
+  // Return a copy of the object without the disallowed properties.
+  var omit = restArguments(function(obj, keys) {
+    var iteratee = keys[0], context;
+    if (isFunction$1(iteratee)) {
+      iteratee = negate(iteratee);
+      if (keys.length > 1) context = keys[1];
+    } else {
+      keys = map(flatten$1(keys, false, false), String);
+      iteratee = function(value, key) {
+        return !contains(keys, key);
+      };
+    }
+    return pick(obj, iteratee, context);
+  });
+
+  // Returns everything but the last entry of the array. Especially useful on
+  // the arguments object. Passing **n** will return all the values in
+  // the array, excluding the last N.
+  function initial(array, n, guard) {
+    return slice.call(array, 0, Math.max(0, array.length - (n == null || guard ? 1 : n)));
+  }
+
+  // Get the first element of an array. Passing **n** will return the first N
+  // values in the array. The **guard** check allows it to work with `_.map`.
+  function first(array, n, guard) {
+    if (array == null || array.length < 1) return n == null || guard ? void 0 : [];
+    if (n == null || guard) return array[0];
+    return initial(array, array.length - n);
+  }
+
+  // Returns everything but the first entry of the `array`. Especially useful on
+  // the `arguments` object. Passing an **n** will return the rest N values in the
+  // `array`.
+  function rest(array, n, guard) {
+    return slice.call(array, n == null || guard ? 1 : n);
+  }
+
+  // Get the last element of an array. Passing **n** will return the last N
+  // values in the array.
+  function last(array, n, guard) {
+    if (array == null || array.length < 1) return n == null || guard ? void 0 : [];
+    if (n == null || guard) return array[array.length - 1];
+    return rest(array, Math.max(0, array.length - n));
+  }
+
+  // Trim out all falsy values from an array.
+  function compact(array) {
+    return filter(array, Boolean);
+  }
+
+  // Flatten out an array, either recursively (by default), or up to `depth`.
+  // Passing `true` or `false` as `depth` means `1` or `Infinity`, respectively.
+  function flatten(array, depth) {
+    return flatten$1(array, depth, false);
+  }
+
+  // Take the difference between one array and a number of other arrays.
+  // Only the elements present in just the first array will remain.
+  var difference = restArguments(function(array, rest) {
+    rest = flatten$1(rest, true, true);
+    return filter(array, function(value){
+      return !contains(rest, value);
+    });
+  });
+
+  // Return a version of the array that does not contain the specified value(s).
+  var without = restArguments(function(array, otherArrays) {
+    return difference(array, otherArrays);
+  });
+
+  // Produce a duplicate-free version of the array. If the array has already
+  // been sorted, you have the option of using a faster algorithm.
+  // The faster algorithm will not work with an iteratee if the iteratee
+  // is not a one-to-one function, so providing an iteratee will disable
+  // the faster algorithm.
+  function uniq(array, isSorted, iteratee, context) {
+    if (!isBoolean(isSorted)) {
+      context = iteratee;
+      iteratee = isSorted;
+      isSorted = false;
+    }
+    if (iteratee != null) iteratee = cb(iteratee, context);
+    var result = [];
+    var seen = [];
+    for (var i = 0, length = getLength(array); i < length; i++) {
+      var value = array[i],
+          computed = iteratee ? iteratee(value, i, array) : value;
+      if (isSorted && !iteratee) {
+        if (!i || seen !== computed) result.push(value);
+        seen = computed;
+      } else if (iteratee) {
+        if (!contains(seen, computed)) {
+          seen.push(computed);
+          result.push(value);
+        }
+      } else if (!contains(result, value)) {
+        result.push(value);
+      }
+    }
+    return result;
+  }
+
+  // Produce an array that contains the union: each distinct element from all of
+  // the passed-in arrays.
+  var union = restArguments(function(arrays) {
+    return uniq(flatten$1(arrays, true, true));
+  });
+
+  // Produce an array that contains every item shared between all the
+  // passed-in arrays.
+  function intersection(array) {
+    var result = [];
+    var argsLength = arguments.length;
+    for (var i = 0, length = getLength(array); i < length; i++) {
+      var item = array[i];
+      if (contains(result, item)) continue;
+      var j;
+      for (j = 1; j < argsLength; j++) {
+        if (!contains(arguments[j], item)) break;
+      }
+      if (j === argsLength) result.push(item);
+    }
+    return result;
+  }
+
+  // Complement of zip. Unzip accepts an array of arrays and groups
+  // each array's elements on shared indices.
+  function unzip(array) {
+    var length = array && max(array, getLength).length || 0;
+    var result = Array(length);
+
+    for (var index = 0; index < length; index++) {
+      result[index] = pluck(array, index);
+    }
+    return result;
+  }
+
+  // Zip together multiple lists into a single array -- elements that share
+  // an index go together.
+  var zip = restArguments(unzip);
+
+  // Converts lists into objects. Pass either a single array of `[key, value]`
+  // pairs, or two parallel arrays of the same length -- one of keys, and one of
+  // the corresponding values. Passing by pairs is the reverse of `_.pairs`.
+  function object(list, values) {
+    var result = {};
+    for (var i = 0, length = getLength(list); i < length; i++) {
+      if (values) {
+        result[list[i]] = values[i];
+      } else {
+        result[list[i][0]] = list[i][1];
+      }
+    }
+    return result;
+  }
+
+  // Generate an integer Array containing an arithmetic progression. A port of
+  // the native Python `range()` function. See
+  // [the Python documentation](https://docs.python.org/library/functions.html#range).
+  function range(start, stop, step) {
+    if (stop == null) {
+      stop = start || 0;
+      start = 0;
+    }
+    if (!step) {
+      step = stop < start ? -1 : 1;
+    }
+
+    var length = Math.max(Math.ceil((stop - start) / step), 0);
+    var range = Array(length);
+
+    for (var idx = 0; idx < length; idx++, start += step) {
+      range[idx] = start;
+    }
+
+    return range;
+  }
+
+  // Chunk a single array into multiple arrays, each containing `count` or fewer
+  // items.
+  function chunk(array, count) {
+    if (count == null || count < 1) return [];
+    var result = [];
+    var i = 0, length = array.length;
+    while (i < length) {
+      result.push(slice.call(array, i, i += count));
+    }
+    return result;
+  }
+
+  // Helper function to continue chaining intermediate results.
+  function chainResult(instance, obj) {
+    return instance._chain ? _$1(obj).chain() : obj;
+  }
+
+  // Add your own custom functions to the Underscore object.
+  function mixin(obj) {
+    each(functions(obj), function(name) {
+      var func = _$1[name] = obj[name];
+      _$1.prototype[name] = function() {
+        var args = [this._wrapped];
+        push.apply(args, arguments);
+        return chainResult(this, func.apply(_$1, args));
+      };
+    });
+    return _$1;
+  }
+
+  // Add all mutator `Array` functions to the wrapper.
+  each(['pop', 'push', 'reverse', 'shift', 'sort', 'splice', 'unshift'], function(name) {
+    var method = ArrayProto[name];
+    _$1.prototype[name] = function() {
+      var obj = this._wrapped;
+      if (obj != null) {
+        method.apply(obj, arguments);
+        if ((name === 'shift' || name === 'splice') && obj.length === 0) {
+          delete obj[0];
+        }
+      }
+      return chainResult(this, obj);
+    };
+  });
+
+  // Add all accessor `Array` functions to the wrapper.
+  each(['concat', 'join', 'slice'], function(name) {
+    var method = ArrayProto[name];
+    _$1.prototype[name] = function() {
+      var obj = this._wrapped;
+      if (obj != null) obj = method.apply(obj, arguments);
+      return chainResult(this, obj);
+    };
+  });
+
+  // Named Exports
+
+  var allExports = {
+    __proto__: null,
+    VERSION: VERSION,
+    restArguments: restArguments,
+    isObject: isObject,
+    isNull: isNull,
+    isUndefined: isUndefined,
+    isBoolean: isBoolean,
+    isElement: isElement,
+    isString: isString,
+    isNumber: isNumber,
+    isDate: isDate,
+    isRegExp: isRegExp,
+    isError: isError,
+    isSymbol: isSymbol,
+    isArrayBuffer: isArrayBuffer,
+    isDataView: isDataView$1,
+    isArray: isArray,
+    isFunction: isFunction$1,
+    isArguments: isArguments$1,
+    isFinite: isFinite$1,
+    isNaN: isNaN$1,
+    isTypedArray: isTypedArray$1,
+    isEmpty: isEmpty,
+    isMatch: isMatch,
+    isEqual: isEqual,
+    isMap: isMap,
+    isWeakMap: isWeakMap,
+    isSet: isSet,
+    isWeakSet: isWeakSet,
+    keys: keys,
+    allKeys: allKeys,
+    values: values,
+    pairs: pairs,
+    invert: invert,
+    functions: functions,
+    methods: functions,
+    extend: extend,
+    extendOwn: extendOwn,
+    assign: extendOwn,
+    defaults: defaults,
+    create: create,
+    clone: clone,
+    tap: tap,
+    get: get,
+    has: has,
+    mapObject: mapObject,
+    identity: identity,
+    constant: constant,
+    noop: noop,
+    toPath: toPath$1,
+    property: property,
+    propertyOf: propertyOf,
+    matcher: matcher,
+    matches: matcher,
+    times: times,
+    random: random,
+    now: now,
+    escape: _escape,
+    unescape: _unescape,
+    templateSettings: templateSettings,
+    template: template,
+    result: result,
+    uniqueId: uniqueId,
+    chain: chain,
+    iteratee: iteratee,
+    partial: partial,
+    bind: bind,
+    bindAll: bindAll,
+    memoize: memoize,
+    delay: delay,
+    defer: defer,
+    throttle: throttle,
+    debounce: debounce,
+    wrap: wrap,
+    negate: negate,
+    compose: compose,
+    after: after,
+    before: before,
+    once: once,
+    findKey: findKey,
+    findIndex: findIndex,
+    findLastIndex: findLastIndex,
+    sortedIndex: sortedIndex,
+    indexOf: indexOf,
+    lastIndexOf: lastIndexOf,
+    find: find,
+    detect: find,
+    findWhere: findWhere,
+    each: each,
+    forEach: each,
+    map: map,
+    collect: map,
+    reduce: reduce,
+    foldl: reduce,
+    inject: reduce,
+    reduceRight: reduceRight,
+    foldr: reduceRight,
+    filter: filter,
+    select: filter,
+    reject: reject,
+    every: every,
+    all: every,
+    some: some,
+    any: some,
+    contains: contains,
+    includes: contains,
+    include: contains,
+    invoke: invoke,
+    pluck: pluck,
+    where: where,
+    max: max,
+    min: min,
+    shuffle: shuffle,
+    sample: sample,
+    sortBy: sortBy,
+    groupBy: groupBy,
+    indexBy: indexBy,
+    countBy: countBy,
+    partition: partition,
+    toArray: toArray,
+    size: size,
+    pick: pick,
+    omit: omit,
+    first: first,
+    head: first,
+    take: first,
+    initial: initial,
+    last: last,
+    rest: rest,
+    tail: rest,
+    drop: rest,
+    compact: compact,
+    flatten: flatten,
+    without: without,
+    uniq: uniq,
+    unique: uniq,
+    union: union,
+    intersection: intersection,
+    difference: difference,
+    unzip: unzip,
+    transpose: unzip,
+    zip: zip,
+    object: object,
+    range: range,
+    chunk: chunk,
+    mixin: mixin,
+    'default': _$1
+  };
+
+  // Default Export
+
+  // Add all of the Underscore functions to the wrapper object.
+  var _ = mixin(allExports);
+  // Legacy Node.js API.
+  _._ = _;
+
+  return _;
+
+})));
+//# sourceMappingURL=underscore-umd.js.map
diff --git a/docs/html/_static/underscore.js b/docs/html/_static/underscore.js
index 166240ef..cf177d42 100644
--- a/docs/html/_static/underscore.js
+++ b/docs/html/_static/underscore.js
@@ -1,6 +1,6 @@
-!function(n,r){"object"==typeof exports&&"undefined"!=typeof module?module.exports=r():"function"==typeof define&&define.amd?define("underscore",r):(n=n||self,function(){var t=n._,e=n._=r();e.noConflict=function(){return n._=t,e}}())}(this,(function(){
-//     Underscore.js 1.12.0
+!function(n,r){"object"==typeof exports&&"undefined"!=typeof module?module.exports=r():"function"==typeof define&&define.amd?define("underscore",r):(n="undefined"!=typeof globalThis?globalThis:n||self,function(){var t=n._,e=n._=r();e.noConflict=function(){return n._=t,e}}())}(this,(function(){
+//     Underscore.js 1.13.1
 //     https://underscorejs.org
-//     (c) 2009-2020 Jeremy Ashkenas, DocumentCloud and Investigative Reporters & Editors
+//     (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors
 //     Underscore may be freely distributed under the MIT license.
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e,u,o=1/0,i=1/0;if(null==r||"number"==typeof r&&"object"!=typeof n[0]&&null!=n)for(var a=0,f=(n=er(n)?n:jn(n)).length;ae||void 0===t)return 1;if(t
     
     
-    Index — IDTxl 1.2.1 documentation
-    
-    
-    
+    Index — IDTxl 1.3 documentation
+    
+    
+    
     
     
     
@@ -31,7 +31,7 @@ 
    Navigation
    
         
  • 
           modules |
    • 
-        
  • IDTxl 1.2.1 documentation »
    • 
+        
  • IDTxl 1.3 documentation »
    • 
         
  • Index
    •  
       
     
      
@@ -51,6 +51,7 @@ 
    Index
    
  | D
  | E
  | G
+ | H
  | I
  | J
  | L
@@ -78,6 +79,10 @@ 
    A
    
       
  • ais_fdr() (in module idtxl.stats)
 
    • 
       
  • AlgorithmExhaustedError
+
    • 
+      
  • alpha_ML() (idtxl.estimators_Rudelt.RudeltAbstractNSBEstimator method)
+
    • 
+      
  • analyse_auto_MI() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
 
    • 
       
  • analyse_network() (idtxl.active_information_storage.ActiveInformationStorage method)
 
@@ -123,12 +128,14 @@ 
    A
    
 
    B
    
 
    
   
   
 
    
@@ -169,12 +180,26 @@ 
    C
    
 
    D
    
 
@@ -273,18 +306,74 @@ 
    G
  • (idtxl.estimators_jidt.JidtGaussianCMI method)
 
    • 
+      
  • get_auto_MI() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
+
    • 
+      
  • get_bbc_term() (idtxl.estimators_Rudelt.RudeltBBCEstimator method)
+
    • 
+      
  • get_beta_MAP() (idtxl.estimators_Rudelt.RudeltAbstractNSBEstimator method)
+
    • 
+      
  • get_bootstrap_history_dependence() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
+
  • get_edge_list() (idtxl.results.AdjacencyMatrix method)
+
    • 
+      
  • get_embeddings() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
+
    • 
+      
  • get_embeddings_that_maximise_R() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
+
    • 
+      
  • get_H0_X_past_cond_X() (idtxl.estimators_Rudelt.RudeltShufflingEstimator method)
+
    • 
+      
  • get_H0_X_past_cond_X_eq_x() (idtxl.estimators_Rudelt.RudeltShufflingEstimator method)
+
    • 
+      
  • get_H_X_past_cond_X() (idtxl.estimators_Rudelt.RudeltShufflingEstimator method)
+
    • 
+      
  • get_H_X_past_uncond() (idtxl.estimators_Rudelt.RudeltShufflingEstimator method)
+
    • 
+      
  • get_history_dependence() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
+
    • 
+      
  • get_information_timescale_tau_R() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
+
    • 
+      
  • get_integration_bounds() (idtxl.estimators_Rudelt.RudeltAbstractNSBEstimator method)
+
    • 
+      
  • get_marginal_frequencies_of_spikes_in_bins() (idtxl.estimators_Rudelt.RudeltShufflingEstimator method)
 
    • 
   
   
   
         
       
       
   
   
+      
  • get_temporal_depth_T_D() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
+
    • 
+      
  • get_window_delimiters() (idtxl.estimators_Rudelt.RudeltAbstractEstimator method)
+
    • 
       
  • getit() (idtxl.multivariate_te.MultivariateTE method)
 
    • 
       		
 
    
 
+
    H
    
+
+  
+
    
+
 
    I
    
 
@@ -576,6 +682,8 @@ 
    L
    
   
      
@@ -353,6 +456,13 @@ 
      I
      
 
       
+      
    • 
+    idtxl.embedding_optimization_ais_Rudelt
+
+      
      • 
       
    • 
@@ -384,17 +494,17 @@ 
      I
      
 
      • 
       
    
       
  • 
-    idtxl.idtxl_exceptions
+    idtxl.estimators_Rudelt
 
       
    • 
       
  • 
-    idtxl.idtxl_import
+    idtxl.idtxl_exceptions
 
       
    • 
       
  • 
@@ -483,18 +593,10 @@ 
    I
    
         
  • module
 
    • 
       
-      
  • import_fieldtrip() (in module idtxl.idtxl_import)
-
-      
    • 
-      
  • import_matarray() (in module idtxl.idtxl_import)
-
-      
    • 
       
  • is_analytic_null_estimator() (idtxl.estimators_jidt.JidtDiscrete method)
 
       
    • 
       
  • is_parallel() (idtxl.estimators_jidt.JidtEstimator method)
@@ -521,6 +625,8 @@ 
    I
    
         
  • (idtxl.estimators_pid.SydneyPID method)
 
    • 
         
  • (idtxl.estimators_pid.TartuPID method)
+
    • 
+        
  • (idtxl.estimators_Rudelt.RudeltAbstractEstimator method)
 
    • 
       
       		
   
   
 
    
@@ -608,6 +716,8 @@ 
    M
    
         
  • idtxl.bivariate_te
 
    • 
         
  • idtxl.data
+
    • 
+        
  • idtxl.embedding_optimization_ais_Rudelt
 
    • 
         
  • idtxl.estimators_jidt
 
    • 
@@ -617,9 +727,9 @@ 
    M
    
 
         
  • idtxl.estimators_pid
 
    • 
-        
  • idtxl.idtxl_exceptions
+        
  • idtxl.estimators_Rudelt
 
    • 
-        
  • idtxl.idtxl_import
+        
  • idtxl.idtxl_exceptions
 
    • 
         
  • idtxl.idtxl_io
 
    • 
@@ -672,10 +782,10 @@ 
    N
    
 
       
  • network_fdr() (in module idtxl.stats)
 
    • 
-  
-  
+  
 
@@ -698,11 +812,17 @@ 
    O
    
 
       
  • OpenCLKraskov (class in idtxl.estimators_opencl)
 
    • 
-  
-  
+  
 
@@ -721,10 +841,14 @@ 
    P
    
       
  • plot_network() (in module idtxl.visualise_graph)
 
    • 
       
  • plot_network_comparison() (in module idtxl.visualise_graph)
+
    • 
+      
  • plot_selected_vars() (in module idtxl.visualise_graph)
 
    • 
   
   
       
  • print_matrix() (idtxl.results.AdjacencyMatrix method)
 
    • 
-      
  • processes_analysed() (idtxl.results.ResultsSingleProcessAnalysis property)
+      
  • processes_analysed (idtxl.results.ResultsSingleProcessAnalysis property)
+
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    • 
   
 
 
@@ -747,6 +875,8 @@ 
    R
    
       
  • remove_column() (in module idtxl.idtxl_utils)
 
    • 
       
  • remove_row() (in module idtxl.idtxl_utils)
+
    • 
+      
  • remove_subresults_single_process() (idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt method)
 
    • 
       
  • Results (class in idtxl.results)
 
    • 
@@ -754,17 +884,33 @@ 
    R
    
 
       
  • ResultsNetworkAnalysis (class in idtxl.results)
 
    • 
-  
-  
+  
 
@@ -776,15 +922,17 @@ 
    S
    
 
       
  • save_pickle() (in module idtxl.idtxl_io)
 
    • 
-      
  • selected_vars_full() (idtxl.network_analysis.NetworkAnalysis property)
+      
  • selected_vars_full (idtxl.network_analysis.NetworkAnalysis property)
 
    • 
-      
  • selected_vars_sources() (idtxl.network_analysis.NetworkAnalysis property)
+      
  • selected_vars_sources (idtxl.network_analysis.NetworkAnalysis property)
 
    • 
-      
  • selected_vars_target() (idtxl.network_analysis.NetworkAnalysis property)
+      
  • selected_vars_target (idtxl.network_analysis.NetworkAnalysis property)
 
    • 
       
  • separate_arrays() (in module idtxl.idtxl_utils)
 
    • 
       
  • set_data() (idtxl.data.Data method)
+
    • 
+      
  • shuffling_MI() (idtxl.estimators_Rudelt.RudeltShufflingEstimator method)
 
    • 
       
  • SingleProcessAnalysis (class in idtxl.single_process_analysis)
 
    • 
@@ -803,6 +951,10 @@ 
    S
    
       
  • SxPID (class in idtxl.estimators_multivariate_pid)
 
    • 
       
  • SydneyPID (class in idtxl.estimators_pid)
+
    • 
+      
  • symbol_array_to_binary() (idtxl.estimators_Rudelt.RudeltAbstractEstimator method)
+
    • 
+      
  • symbol_binary_to_array() (idtxl.estimators_Rudelt.RudeltAbstractEstimator method)
 
    • 
       
  • syn_shd_against_surrogates() (in module idtxl.stats)
 
    • 
@@ -812,7 +964,7 @@ 
    S
    
 
    T
    
 
    
   
   
      
@@ -823,6 +975,10 @@ 
      T
      
 
 
      U
      
 
+  
+     
+       
+       
@@ -189,7 +194,7 @@ 
      Python Module Index
      Quick search
      
     
      
-      
+      
       
     
     
      
@@ -208,13 +213,13 @@ 
      Navigation
    • 
           modules |
      • 
-        
    • IDTxl 1.2.1 documentation »
      • 
+        
    • IDTxl 1.3 documentation »
    • Python Module Index
      • 
\ No newline at end of file
diff --git a/docs/html/search.html b/docs/html/search.html
index dd89e61a..e0858ff7 100644
--- a/docs/html/search.html
+++ b/docs/html/search.html
@@ -5,11 +5,11 @@
   
-    Search — IDTxl 1.2.1 documentation
-    
-    
+    Search — IDTxl 1.3 documentation
+    
+    
     
-    
+    
@@ -37,7 +37,7 @@ 
      Navigation
    • 
           modules |
      • 
-        
    • IDTxl 1.2.1 documentation »
      • 
+        
    • IDTxl 1.3 documentation »
    • Search
      •   
@@ -48,26 +48,35 @@ 
      Navigation
      
             
   
      Search
      
-  
      
-  
+  
+  
+  
+  
   
      
     Searching for multiple words only shows matches that contain
     all words.
   
      
+  
+  
   
      
-    
+    
     
     
   
   
+  
+  
   
      
   
   
      
+  
 
             
      
           
      
@@ -88,13 +97,13 @@ 
      Navigation
      
         
    • 
           modules |
      • 
-        
    • IDTxl 1.2.1 documentation »
      • 
+        
    • IDTxl 1.3 documentation »
      • 
         
    • Search
      •  
       
     
      
\ No newline at end of file
diff --git a/docs/html/searchindex.js b/docs/html/searchindex.js
index 2cd4ba0d..e9fef345 100644
--- a/docs/html/searchindex.js
+++ b/docs/html/searchindex.js
@@ -1 +1 @@
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id:[0,3,7,8],pid_analysi:[0,6],pid_calc_nam:[0,6],pid_estim:[0,6],pipa:[0,5],pl00007990:[0,1],planck:[0,5],platform:[0,2],plo:[0,1,6],plot:[0,6],plot_mute_graph:0,plot_network:0,plot_network_comparison:0,plot_selected_var:0,plug:[0,2,6],plugin:[0,6],plugin_entropi:[0,6],plugin_estim:[0,6],point:[0,1,2,3,6,7],pointless:[0,3],pointwis:0,pone:[0,1],porta:[0,5,6],possibl:[0,2,3,6],posterior:[0,6],posteriori:[0,6],potenti:[0,5],power:[0,3],practic:[0,2],predict:[0,3],prefix:[0,6],preprint2012_25:[0,5],preprint:[0,5],present:[0,3],preserv:[0,6],press:[0,6],pretti:[0,3],previous:[0,5,6],priesemann:[0,6],principl:[0,6],print:[0,2,3,7],print_dict:[0,3],print_edge_list:[0,7],print_matrix:0,printer:[0,3],prior:[0,6],probabl:[0,2,6],problem:8,proc:[0,1],process:[0,1,2,3,4,5,6,7],process_index:[],process_set:[0,6],processes_analys:[0,7],produc:[0,6],product:[0,6],program:[0,2],project:0,prokopenko:[0,2,6],properti:[0,1,2,7],propos:[0,1,2],protocol:0,provid:[0,1,2,3,4,6,7],prune:[0,3,5,6],ps:[0,1,6],psr:[0,1],ptw:[0,7],pval:[0,7],pvalu:[0,5,6,7],pvalue_omnibu:[0,5],pvalues_sign_sourc:[0,5],py:[],pypi:[0,2],python3:0,python:[0,2,3,6],qualit:0,quantifi:[0,2,6],quantiti:[0,6],r:[0,1,2,3,5,6,7],r_max:[0,6],r_nsb:[0,6],r_plugin:[0,6],r_tot:[0,6,7],rais:[0,1,2,3],randint:[0,6],random:[0,1,2,6,7],rang:[0,1,2,3,6],rank:[0,3],rate:[0,3,6,7],rauh:[0,2,6],raw:[0,1,2,3,4,5,6],raw_symbol:[0,6],re:[0,3,6],read:[0,6],realis:[0,1,2,3,5,6,7],realiz:[0,2],rebas:[0,6],recogn:0,recommend:[0,6],record:[0,4,6,7],recording_length:[0,6,7],recurs:0,reduc:[0,6],redund:[0,7],refer:[0,1,2,3,5,6],regard:[0,7],regular:0,rel:[0,6],relat:[0,6],relationship:[0,2],relev:[0,3,5,6],remain:[0,3],reman:[0,3],remanin:[0,3],remov:[0,2,3],remove_column:[0,3],remove_row:[0,3],remove_subresults_single_process:[0,6],repeat:[0,3],repetit:[0,1],repl:[0,1],replic:[0,1,2,3,4,5,6],repo:[],report:[0,8],repres:[0,1,2,3,6,7],represent:[0,6],request:[0,1,3],requir:[0,2,3],res_network:[0,7],res_pid:[0,7],resampl:[0,6],reshap:[0,1],respect:[0,2,5],respons:[0,6],restrict:[0,3],result:[1,2,3,4,5,6,8,9],resultsmultivariatepid:[0,6,7],resultsnetworkanalysi:0,resultsnetworkcomparison:[0,4,7],resultsnetworkinfer:[0,3,5,7],resultspid:[0,6,7],resultssingleprocessanalysi:[0,3,6,7],resultssingleprocessrudelt:[0,6,7],resum:[0,5,6],resume_checkpoint:[0,5,6],retriev:[0,5,6],return_averaged_r:[0,6],return_calc:[0,2],return_count:[0,2],rev:[0,2,5,6],reveal:[0,6],revisit:[0,6],rho:[0,6],rlz:[0,7],robot:[0,2],round:[0,3],routin:0,row:[0,2,3],rtot:[0,6,7],rubinov:[0,5],rudelt:[0,6,7],rudeltabstractestim:[0,6],rudeltabstractnsbestim:[0,6],rudeltbbcestim:[0,6],rudeltnsbestimatorsymbolsmi:[0,6],rudeltpluginestimatorsymbolsmi:[0,6],rudeltshufflingestim:[0,6],run:[0,1,2],s10827:[0,5],s1:[0,2,6],s1_unq:[0,7],s2:[0,2,6],s2_unq:[0,7],s3:[0,6],s:[0,1,2,3,4,5,6],same:[0,1,2,3,4,5,6],sameshima:[0,1],sampl:[0,1,2,3,5,6,7],save:[0,2,3,6],save_json:0,save_pickl:0,scale:[0,6,7],scaling_k:[0,6],scheme:[0,1],schreiber:[0,2,5],sci:[0,2,6],scipi:[0,3],search:[0,2,5,6,8],second:[0,2,3,4,6,7],see:[0,2,3,4,5,6,7],seed:[0,1],select:[0,3,4,5,6],selected_sources_pv:0,selected_sources_t:0,selected_var:[0,7],selected_vars_ful:[0,5,6],selected_vars_sourc:[0,5,7],selected_vars_target:[0,5,7],self:[0,6],separ:[0,3,5,6],separate_arrai:[0,3],sequenti:[0,3],seri:[0,1,2],set:[0,1,2,3,4,5,6,7],set_data:[0,1],setup:[0,3],sever:[0,1],sh:[0,6],shafe:[0,6],shape:[0,3],share:[0,2,3,6,7],shd_s1_s2:[0,7],shift:[0,1],should:[0,2,3,6],show:[0,6],shuffl:[0,1,3,4,5,6],shuffling_mi:[0,6],side:[0,4],sign:[0,6],sign_ominbu:[0,5],sign_sourc:0,signal:0,signific:[0,3,4,5,6,7],significantli:[0,3],similar:[0,2,6,7],simplic:[0,6],simplifi:[0,6],simul:[0,1,6],singl:[0,3,4,5,6,7],single_process_analysi:[8,9],single_process_result:[],singleprocessanalysi:0,singleton:0,size:[0,1,2,6,7],slice:[0,1],slice_permute_repl:[0,1],slice_permute_sampl:[0,1],slid:[0,6],small:[0,1,3],smaller:[0,2,3],smallest:[0,6,7],sn:0,so:[0,2,3,6],soft:[0,2],softwar:0,solver:[0,2],solver_arg:[0,2],some:[0,2,6],sort:[0,3],sort_descend:[0,3],sortind:[0,3],sourc:[0,1,2,3,4,5,6,7],source_1:[0,7],source_2:[0,7],source_i:[0,7],source_set:[0,5],source_target_delai:[0,2],sources_test:0,space:[0,2,5,6],specif:[0,7],specifi:[0,1,5,6,7],spike:[0,7,8],spike_tim:[0,6],spikes_in_window:[0,6],spr:[0,1],squar:[0,1],squeez:0,stage:[0,2],stai:[0,1],stamp:0,stand:0,standard:[0,1,3,6],standardis:[0,1,2,3,7],start:[0,1],stat:[5,6,7,8,9],state:[0,1,2,5,6,7],statis:[0,3],statist:[0,1,3,4,5,6,7],statistic_omnibu:[0,5],statistic_sign_sourc:[0,5],stats_typ:[0,4],step:[0,5,6],stochast:[0,1,7],stoegbauer:[0,2],stop:[0,3],storag:[0,2,3,7,8],store:[0,1,7],str:[0,1,2,4,5,6,7],strategi:[0,1],string:[0,1,3,5,6,7],structur:[0,1],studi:[0,2],style:0,subject:[0,4],submodul:[8,9],subplot:0,subset:[0,3,6],suffici:[0,1,6],sum:[0,6],sum_i:[0,3],summar:[0,6],support:[0,2,6],surrog:[0,1,2,3,4,5,6,7],surrogate_distribut:[0,7],swap:[0,1,2,3],swap_char:[0,3],sxpid:[0,6],sydneypid:[0,2,6],symbol:[0,6],symbol_arrai:[0,6],symbol_array_to_binari:[0,6],symbol_binari:[0,6],symbol_binary_to_arrai:[0,6],symbol_block_length:[0,6],symbol_count:[0,6],symmetr:[0,6],syn_s1_s2:[0,7],syn_shd_against_surrog:[0,3],synergist:[0,2,3,6,7],system:[0,2,6],t:[0,2,3,5,6,7],t_0:[0,6],t_d:[0,6,7],t_max:[0,6],tabl:[0,3],tail:[0,4],tail_comp:[0,4],take:[0,1,6],taken:[0,2,3],target1:[0,6],target:[0,2,3,4,5,6,7],target_r:[0,6],targets_analys:[0,7],tartupid:[0,2],task:[0,4,6],tau:[0,2,6],tau_1:[0,6],tau_r:[0,6,7],tau_sourc:[0,2,5],tau_target:[0,2,5],td:[0,6,7],te:[0,2,3,4,5,6,7],te_max_candid:[0,3],te_min_candid:[0,3],technic:[0,2],techniqu:[0,5,6],tempor:[0,1,2,5,6,7],term:[0,6],test:[0,1,3,4,5,6,7],text:0,than:[0,2,3,6],thei:[0,2,6],theiler:[0,2],theiler_t:[0,2],them:0,theoret:[0,3,7,8],theori:0,therefor:[0,6],thereof:[0,6],thi:[0,1,2,3,4,5,6,7],third:[0,2,6],those:0,thread:[0,2],three:[0,1,2,5,6],threshold:[0,3],through:[0,5,6],thu:[0,1,6],time:[0,1,2,3,4,5,6,7],timescal:[0,6,7],timescale_minimum_past_rang:[0,6],tmax:[0,6],todo:[0,6],togeth:[0,6],toggl:[0,4,5,6],toler:[0,6],too:[0,1,2],tool:[0,6],toolbox:[0,1],toolkit:[0,2],tot:[0,6,7],total:[0,1,2,6,7],toward:[0,6],tracker:8,train:[0,6,7],tranfer:[0,7],transfer:[0,1,2,3,4,7,8],treat:[0,5],trial:[0,1,2],tupl:[0,1,3,5,6,7],tutori:8,two:[0,1,2,3,4,5,6,7],type:[0,1,2,4,6],typeerror:[0,1],typic:[0,6],u:[0,1],unbias:[0,6],uncorrect:[0,7],uncorrel:[0,1],under:[0,2,4,6],understand:[0,6],unequ:[0,3],uniform:[0,2,5,6],union:[0,7],uniqu:[0,2,3,6,7],unit:[0,4],univari:[0,3],unnorm:[0,6],unnormalized_posterior:[0,6],unq_against_surrog:[0,3],unsuccess:[0,2],until:[0,2],unweight:[0,7],up:[0,4,6],upon:0,upper:[0,2,6],us:[0,1,2,3,4,5,6,7],usag:[0,2],use_al:[0,2],user:[0,8],util:[0,8],utilis:[0,2],utl:[0,6],v4:0,v6:0,v7:0,v:[0,6],valid:0,valu:[0,1,2,3,4,5,6,7],var1:[0,2],var2:[0,2],vari:[0,2],variabl:[0,1,2,3,4,5,6,7],variou:[0,1],vars_count:[0,7],vector:[0,1,3],verbos:[0,2,4,5,6],version:[0,2],via:[0,1,2,5,6,7],vicent:[0,2,5],viewer:0,virtualis:[0,2],visual:[0,6],visualis:0,visualise_graph:[8,9],visul:[0,6],vitrualis:[0,2],vs:[0,4],vstack:[0,6],w:[0,6],wa:[0,1,2,4,6,7],wai:[0,7],wang:0,want:[0,8],warn:0,we:[0,1,6],weight:[0,7],weight_typ:0,well:[0,2,3,6],were:[0,5,6,7],what:[0,6],when:[0,1,2,3,5,6],where:[0,1,2,3,5,6,7],whether:[0,6,7],which:[0,1,2,3,4,6],whole:[0,3,4,5,6],whose:[0,6],wibral:[0,5,6],wiki:[0,3,8],william:[0,6],window:[0,2,6],wise:[0,1],within:[0,1,4],without:[0,5,6],workaround:[0,2],write:[0,5,6],write_ckp:[0,5,6],written:0,wrt:[0,5,6,7],www:[0,5],x:[0,1,2,3,6],x_past:[0,6],xi:[0,6],xia:0,xl:0,y:[0,2,6],yield:[0,6,7],you:[0,8],your:[0,2],z:[0,1,2,3,6,7],zero:[0,6],zomaya:[0,2,6]},titles:["idtxl 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\ No newline at end of file
diff --git a/docs/idtxl.rst b/docs/idtxl.rst
index e9d46f28..6dc6d645 100644
--- a/docs/idtxl.rst
+++ b/docs/idtxl.rst
@@ -69,6 +69,22 @@ idtxl.active_information_storage module
     :undoc-members:
     :show-inheritance:
 
+idtxl.embedding_optimization_ais_Rudelt module
+----------------------------------------------
+
+.. automodule:: idtxl.embedding_optimization_ais_Rudelt
+    :members:
+    :undoc-members:
+    :show-inheritance:
+
+idtxl.estimators_Rudelt module
+------------------------------
+
+.. automodule:: idtxl.estimators_Rudelt
+    :members:
+    :undoc-members:
+    :show-inheritance:
+
 idtxl.estimators_jidt module
 ----------------------------
 
@@ -102,14 +118,6 @@ idtxl.estimators_pid module
     :show-inheritance:
 
 
-idtxl.idtxl_import module
--------------------------
-
-.. automodule:: idtxl.idtxl_import
-    :members:
-    :undoc-members:
-    :show-inheritance:
-
 idtxl.idtxl_exceptions module
 -----------------------------
 
diff --git a/docs/idtxl_process_analysis.rst b/docs/idtxl_process_analysis.rst
index 4d58d0cd..684e1e84 100644
--- a/docs/idtxl_process_analysis.rst
+++ b/docs/idtxl_process_analysis.rst
@@ -18,3 +18,16 @@ Bivariate Partial Information Decomposition (PID)
 .. autoclass:: idtxl.bivariate_pid.BivariatePID
     :members:
     :noindex:
+
+History-dependence estimator for neural spiking data
+----------------------------------------------------
+Embedding optimization
+^^^^^^^^^^^^^^^^^^^^^^
+.. autoclass:: idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt
+    :members:
+    :noindex:
+Estimators
+^^^^^^^^^^
+.. automodule:: idtxl.estimators_Rudelt
+    :members:
+    :noindex:
diff --git a/docs/idtxl_results_class.rst b/docs/idtxl_results_class.rst
index dd16bdd0..31a48a37 100644
--- a/docs/idtxl_results_class.rst
+++ b/docs/idtxl_results_class.rst
@@ -24,3 +24,6 @@ Results node dynamics
 .. autoclass:: idtxl.results.ResultsMultivariatePID
     :members:
     :noindex:
+.. autoclass:: idtxl.results.ResultsSingleProcessRudelt
+    :members:
+    :noindex:
diff --git a/idtxl/embedding_optimization_ais_Rudelt.py b/idtxl/embedding_optimization_ais_Rudelt.py
index 2e1ca2d2..4f106519 100644
--- a/idtxl/embedding_optimization_ais_Rudelt.py
+++ b/idtxl/embedding_optimization_ais_Rudelt.py
@@ -27,7 +27,7 @@ class OptimizationRudelt():
 
     Args:
         settings : dict
-            . estimation_method : string
+            - estimation_method : string
                 The method to be used to estimate the history dependence 'bbc' or 'shuffling'.
             - embedding_step_size : float
                 Step size delta t (in seconds) with which the window is slid through the data.
@@ -37,21 +37,20 @@ class OptimizationRudelt():
                 (default: [1, 2, 3, 4, 5])
             - embedding_past_range_set : list of floating-point values
                 Set of values for T, the past range (in seconds) to be used for embeddings.
-                (default: [0.005, 0.00561, 0.00629, 0.00706, 0.00792, 0.00889, 0.00998, 0.01119, 0.01256,
-                           0.01409, 0.01581, 0.01774, 0.01991, 0.02233, 0.02506, 0.02812, 0.03155, 0.0354,
-                           0.03972, 0.04456, 0.05, 0.0561, 0.06295, 0.07063, 0.07924, 0.08891, 0.09976, 0.11194,
-                           0.12559, 0.14092, 0.15811, 0.17741, 0.19905, 0.22334, 0.25059, 0.28117, 0.31548,
-                           0.35397, 0.39716, 0.44563, 0.5, 0.56101, 0.62946, 0.70627, 0.79245, 0.88914, 0.99763,
-                           1.11936, 1.25594, 1.40919, 1.58114, 1.77407, 1.99054, 2.23342, 2.50594, 2.81171, 3.15479,
-                           3.53973, 3.97164, 4.45625, 5.0])
+                (default: [0.005, 0.00561, 0.00629, 0.00706, 0.00792, 0.00889,
+                0.00998, 0.01119, 0.01256, 0.01409, 0.01581, 0.01774, 0.01991,
+                0.02233, 0.02506, 0.02812, 0.03155, 0.0354, 0.03972, 0.04456,
+                0.05, 0.0561, 0.06295, 0.07063, 0.07924, 0.08891, 0.09976,
+                0.11194, 0.12559, 0.14092, 0.15811, 0.17741, 0.19905, 0.22334,
+                0.25059, 0.28117, 0.31548, 0.35397, 0.39716, 0.44563, 0.5,
+                0.56101, 0.62946, 0.70627, 0.79245, 0.88914, 0.99763, 1.11936,
+                1.25594, 1.40919, 1.58114, 1.77407, 1.99054, 2.23342, 2.50594,
+                2.81171, 3.15479, 3.53973, 3.97164, 4.45625, 5.0])
             - embedding_scaling_exponent_set : dict
                 Set of values for kappa, the scaling exponent for the bins in the embedding.
                 Should be a python-dictionary with the three entries 'number_of_scalings', 'min_first_bin_size' and
                 'min_step_for_scaling'.
-                defaults: {'number_of_scalings': 10,
-                           'min_first_bin_size': 0.005,
-                           'min_step_for_scaling': 0.01})
-
+                defaults: {'number_of_scalings': 10, 'min_first_bin_size': 0.005, 'min_step_for_scaling': 0.01})
             - bbc_tolerance : float
                 The tolerance for the Bayesian Bias Criterion. Influences which embeddings are
                 discarded from the analysis.
@@ -65,7 +64,6 @@ class OptimizationRudelt():
                 Minimum past range T_0 (in seconds) to take into consideration for the estimation of the
                 information timescale tau_R.
                 (default: 0.01)
-
             - number_of_bootstraps_R_max : int
                 The number of bootstrap re-shuffles that should be used to determine the optimal
                 embedding. (Bootstrap the estimates of R_max to determine R_tot.)
@@ -103,11 +101,11 @@ class OptimizationRudelt():
                 The upper percentiles for the confidence interval.
                 This has no effect if bootstrap_CI_use_sd is set to True
                 (default: 97.5)
-
             - analyse_auto_MI : bool
                 perform calculation of auto mutual information of the spike train
                 (default: True)
                 If set to True:
+
                 - auto_MI_bin_size_set : list of floating-point values
                     Set of values for the sizes of the bins (in seconds).
                     (default: [0.005, 0.01, 0.025, 0.05, 0.25, 0.5])
@@ -120,6 +118,7 @@ class OptimizationRudelt():
                 the history dependence and the auto mutual information
                 (default: False)
                 if set to True:
+
                 - output_path : String
                     Path where the .eps images should be saved
                 - output_prefix : String
@@ -652,6 +651,7 @@ def compute_CIs(self,
 
         Args:
             data : data_spiketime object
+                Input data
             target_R : String
                 One of 'R_max', 'R_tot' or 'nonessential'.
                 If set to R_max, replications of R are produced for the T at which
@@ -879,40 +879,42 @@ def optimize_single_run(self, data, process):
             optimizes a single realisation of spike time data given the process number
 
             Args:
-            data : Data_spiketime instance
-                raw data for analysis
-            process : int
-                index of process;
+                data : Data_spiketime instance
+                    raw data for analysis
+                process : int
+                    index of process;
 
             Returns:
-            returns a DotDict with the following keys
-                Process : int
-                    Process that was optimized
-                estimation_method : String
-                    Estimation method that was used for optimization
-                T_D : float
-                    Estimated optimal value for the temporal depth TD
-                tau_R :
-                    Information timescale tau_R, a characteristic timescale of history
-                    dependence similar to an autocorrelation time.
-                R_tot : float
-                    Estimated value for the total history dependence Rtot,
-                AIS_tot : float
-                    Estimated value for the total active information storage
-                opt_number_of_bins_d : int
-                    Number of bins d for the embedding that yields (R̂tot ,T̂D)
-                opt_scaling_k : int
-                    Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
-                opt_first_bin_size : int
-                    Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
-                history_dependence : array with floating-point values
-                    Estimated history dependence for each embedding
-                firing_rate : float
-                    Firing rate of the neuron/ spike train
-                recording_length : float
-                    Length of the recording (in seconds)
-                H_spiking : float
-                    Entropy of the spike times
+                DotDict
+                    with the following keys
+
+                    Process : int
+                        Process that was optimized
+                    estimation_method : String
+                        Estimation method that was used for optimization
+                    T_D : float
+                        Estimated optimal value for the temporal depth TD
+                    tau_R :
+                        Information timescale tau_R, a characteristic timescale of history
+                        dependence similar to an autocorrelation time.
+                    R_tot : float
+                        Estimated value for the total history dependence Rtot,
+                    AIS_tot : float
+                        Estimated value for the total active information storage
+                    opt_number_of_bins_d : int
+                        Number of bins d for the embedding that yields (R̂tot ,T̂D)
+                    opt_scaling_k : int
+                        Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
+                    opt_first_bin_size : int
+                        Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
+                    history_dependence : array with floating-point values
+                        Estimated history dependence for each embedding
+                    firing_rate : float
+                        Firing rate of the neuron/ spike train
+                    recording_length : float
+                        Length of the recording (in seconds)
+                    H_spiking : float
+                        Entropy of the spike times
 
             if analyse_auto_MI was set to True additionally:
                 auto_MI : dict
diff --git a/idtxl/estimators_Rudelt.py b/idtxl/estimators_Rudelt.py
index ba0f7795..5f74f4fa 100644
--- a/idtxl/estimators_Rudelt.py
+++ b/idtxl/estimators_Rudelt.py
@@ -1020,48 +1020,48 @@ def shuffling_MI(self, symbol_counts, number_of_bins_d):
         a correction term to reduce its bias.
 
         For the plug-in estimate:
+
         - Extract the past_symbol_counts from the symbol_counts.
         - I_plugin = H(X_past) - H(X_past | X)
 
         Notation:
-        X: current activity, aka response
-        X_past: past activity
-
-        P_X_uncond: P(X)
-        P_X_past_uncond: P(X_past)
-        P_X_past_cond_X: P(X_past | X)
-
-        H_X_past_uncond: H(X_past)
-        H_X_past_cond_X: H(X_past | X)
-
-        I_plugin: plugin estimate of I(X_past; X)
 
+        - X: current activity, aka response
+        - X_past: past activity
+        - P_X_uncond: P(X)
+        - P_X_past_uncond: P(X_past)
+        - P_X_past_cond_X: P(X_past | X)
+        - H_X_past_uncond: H(X_past)
+        - H_X_past_cond_X: H(X_past | X)
+        - I_plugin: plugin estimate of I(X_past; X)
 
         For the correction term:
+
         - Simulate additional data under the assumption that activity
-        in the past contributes independently towards the current activity.
+            in the past contributes independently towards the current activity.
         - Compute the entropy under the assumptions of the model, which
-        due to its simplicity is easy to sample and the estimate unbiased
+            due to its simplicity is easy to sample and the estimate unbiased
         - Compute the entropy using the plug-in estimate, whose bias is
-        similar to that of the plug-in estimate on the original data
+            similar to that of the plug-in estimate on the original data
         - Compute the correction term as the difference between the
-        unbiased and biased terms
+            unbiased and biased terms
 
         Notation:
-        P0_sh_X_past_cond_X: P_0,sh(X_past | X), equiv. to P(X_past | X)
-                             on the shuffled data
-
-        H0_X_past_cond_X: H_0(X_past | X), based on the model of independent
-        contributions
-        H0_sh_X_past_cond_X: H_0,sh(X_past | X), based on
-        P0_sh_X_past_cond_X, ie the plug-in estimate
 
-        I_corr: the correction term to reduce the bias of I_plugin
+        - P0_sh_X_past_cond_X: P_0,sh(X_past | X), equiv. to P(X_past | X)
+          on the shuffled data
+        - H0_X_past_cond_X: H_0(X_past | X), based on the model of independent
+          contributions
+        - H0_sh_X_past_cond_X: H_0,sh(X_past | X), based on
+        - P0_sh_X_past_cond_X, ie the plug-in estimate
+        - I_corr: the correction term to reduce the bias of I_plugin
 
-
-        :param symbol_counts: the activity of a spike train is embedded into symbols,
-        whose occurences are counted (cf emb.get_symbol_counts)
-        :param number_of_bins_d: the number of bins of the embedding
+        Args:
+            symbol_counts : iterable
+                the activity of a spike train is embedded into symbols,
+                whose occurrences are counted (cf emb.get_symbol_counts)
+            number_of_bins_d : int
+                the number of bins of the embedding
         """
 
         # plug-in estimate
diff --git a/idtxl/results.py b/idtxl/results.py
index aaa6685d..22dea4be 100644
--- a/idtxl/results.py
+++ b/idtxl/results.py
@@ -1041,7 +1041,9 @@ def get_target_sources(self, target):
 class ResultsSingleProcessRudelt():
     """Store results of single process analysis.
 
-    Provides a container for the results Rudelt optimization algorithm
+    Provides a container for the results Rudelt optimization algorithm. To
+    obtain results for individual processes, call the .get_single_process()
+    method (see docstring for details).
 
     Note that for convenience all dictionaries in this class can additionally
     be accessed using dot-notation:
@@ -1060,45 +1062,6 @@ class ResultsSingleProcessRudelt():
                 - n_processes : int - total number of processes analysed
         processes_analysed : list
             list of analysed processes
-        _single_processes : dict
-            containing the results of all analysed processes
-            The results of each process can be extracted using:
-                >>>  = .get_single_process()
-                The extracted single process results contains a DotDict with the following entries:
-                    Process : int
-                        Process that was optimized
-                    estimation_method : String
-                        Estimation method that was used for optimization
-                    T_D : float
-                        Estimated optimal value for the temporal depth TD
-                    tau_R :
-                        Information timescale tau_R, a characteristic timescale of history
-                        dependence similar to an autocorrelation time.
-                    R_tot : float
-                        Estimated value for the total history dependence Rtot,
-                    AIS_tot : float
-                        Estimated value for the total active information storage
-                    opt_number_of_bins_d : int
-                        Number of bins d for the embedding that yields (R̂tot ,T̂D)
-                    opt_scaling_k : int
-                        Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
-                    opt_first_bin_size : int
-                        Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
-                    history_dependence : array with floating-point values
-                        Estimated history dependence for each embedding
-                    firing_rate : float
-                        Firing rate of the neuron/ spike train
-                    recording_length : float
-                        Length of the recording (in seconds)
-                    H_spiking : float
-                        Entropy of the spike times
-
-                    if analyse_auto_MI was set to True additionally:
-                    auto_MI : dict
-                        numpy array of MI values for each delay
-                    auto_MI_delays : list of int
-                        list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay
-
     """
 
     def __init__(self, processes):
@@ -1132,50 +1095,51 @@ def get_single_process(self, process):
 
         Return results for individual processes, contains for each process
 
-            Process : int
-                Process that was optimized
-            estimation_method : String
-                Estimation method that was used for optimization
-            T_D : float
-                Estimated optimal value for the temporal depth TD
-            tau_R :
-                Information timescale tau_R, a characteristic timescale of history
-                dependence similar to an autocorrelation time.
-            R_tot : float
-                Estimated value for the total history dependence Rtot,
-            AIS_tot : float
-                Estimated value for the total active information storage
-            opt_number_of_bins_d : int
-                Number of bins d for the embedding that yields (R̂tot ,T̂D)
-            opt_scaling_k : int
-                Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
-            opt_first_bin_size : int
-                Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
-            history_dependence : array with floating-point values
-                Estimated history dependence for each embedding
-            firing_rate : float
-                Firing rate of the neuron/ spike train
-            recording_length : float
-                Length of the recording (in seconds)
-            H_spiking : float
-                Entropy of the spike times
-
-            if analyse_auto_MI was set to True additionally:
-            auto_MI : dict
-                numpy array of MI values for each delay
-            auto_MI_delays : list of int
-                list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay
-
         Args:
             process : int
                 process id
-          
+
         Returns:
             dict
                 results for single process. Note that for convenience
                 dictionary entries can either be accessed via keywords
                 (result['selected_vars']) or via dot-notation
-                (result.selected_vars).
+                (result.selected_vars). Contains keys
+
+                Process : int
+                    Process that was optimized
+                estimation_method : String
+                    Estimation method that was used for optimization
+                T_D : float
+                    Estimated optimal value for the temporal depth TD
+                tau_R :
+                    Information timescale tau_R, a characteristic timescale of history
+                    dependence similar to an autocorrelation time.
+                R_tot : float
+                    Estimated value for the total history dependence Rtot,
+                AIS_tot : float
+                    Estimated value for the total active information storage
+                opt_number_of_bins_d : int
+                    Number of bins d for the embedding that yields (R̂tot ,T̂D)
+                opt_scaling_k : int
+                    Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
+                opt_first_bin_size : int
+                    Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
+                history_dependence : array with floating-point values
+                    Estimated history dependence for each embedding
+                firing_rate : float
+                    Firing rate of the neuron/ spike train
+                recording_length : float
+                    Length of the recording (in seconds)
+                H_spiking : float
+                    Entropy of the spike times
+
+                if analyse_auto_MI was set to True additionally:
+                auto_MI : dict
+                    numpy array of MI values for each delay
+                auto_MI_delays : list of int
+                    list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay
+
         """
         # Return required key from required _single_process dictionary
         if process not in self.processes_analysed:

      
   
     
     
   
 
\ No newline at end of file
diff --git a/docs/html/idtxl.html b/docs/html/idtxl.html
index d75987f5..187e432b 100644
--- a/docs/html/idtxl.html
+++ b/docs/html/idtxl.html
@@ -4,11 +4,12 @@
 
   
     
-    
-    idtxl package — IDTxl 1.2.1 documentation
-    
-    
-    
+    
+
+    idtxl package — IDTxl 1.3 documentation
+    
+    
+    
     
     
     
@@ -35,7 +36,7 @@ 
      Navigation
      
         
    • 
           previous |
      • 
-        
    • IDTxl 1.2.1 documentation »
      • 
+        
    • IDTxl 1.3 documentation »
      • 
         
    • idtxl package
      •  
       
       
@@ -45,17 +46,17 @@ 
      Navigation
      
         
      
           
      
             
-  
      
+  
      
 
      idtxl package
      
-
      
+
      
 
      Submodules
      
-
      
-
      
+
      
+
      
 
      idtxl.data module
      
 
      Provide data structures for IDTxl analysis.
      
 
      
-
      
-class idtxl.data.Data(data=None, dim_order='psr', normalise=True, seed=None)[source]
      
+
      
+class idtxl.data.Data(data=None, dim_order='psr', normalise=True, seed=None)[source]
      
 
      Bases: object
      
 
      Store data for information dynamics estimation.
      
 
      Data takes a 1- to 3-dimensional array representing realisations of random
@@ -119,15 +120,15 @@ 
      Submodules
-
      
-property data
      
+
      
+
      
+property data
      
 
      Return data array.
      
 
      
 
 
      
-
      
-generate_logistic_maps_data(n_samples=1000, n_replications=10, coefficient_matrices=array([[[0.5, 0.0], [0.4, 0.5]]]), noise_std=0.1)[source]
      
+
      
+generate_logistic_maps_data(n_samples=1000, n_replications=10, coefficient_matrices=array([[[0.5, 0.], [0.4, 0.5]]]), noise_std=0.1)[source]
      
 
      Generate discrete-time coupled-logistic-maps time series.
      
 
      Generate data and overwrite the instance’s current data.
      
 
      The implemented logistic map function is f(x) = 4 * x * (1 - x).
      
@@ -153,8 +154,8 @@ 
      Submodules
-
      
-generate_mute_data(n_samples=1000, n_replications=10)[source]
      
+
      
+generate_mute_data(n_samples=1000, n_replications=10)[source]
      
 
      Generate example data for a 5-process network.
      
 
      Generate example data and overwrite the instance’s current data. The
 network is used as an example the paper on the MuTE toolbox (Montalto,
@@ -188,8 +189,8 @@ 
      Submodules
-
      
-generate_var_data(n_samples=1000, n_replications=10, coefficient_matrices=array([[[0.5, 0.0], [0.4, 0.5]]]), noise_std=0.1)[source]
      
+
      
+generate_var_data(n_samples=1000, n_replications=10, coefficient_matrices=array([[[0.5, 0.], [0.4, 0.5]]]), noise_std=0.1)[source]
      
 
      Generate discrete-time VAR (vector autoregressive) time series.
      
 
      Generate data and overwrite the instance’s current data.
      
 
      
@@ -214,8 +215,8 @@ 
      Submodules
-
      
-get_realisations(current_value, idx_list, shuffle=False)[source]
      
+
      
+get_realisations(current_value, idx_list, shuffle=False)[source]
      
 
      Return realisations for a list of indices.
      
 
      Return realisations for indices in list. Optionally, realisations can
 be shuffled to create surrogate data for statistical testing. For
@@ -306,20 +307,20 @@ 
      Submodules
-
      
-get_seed()[source]
      
+
      
+get_seed()[source]
      
 
      return the initial seed of the data
      
 
      
 
 
      
-
      
-get_state()[source]
      
+
      
+get_state()[source]
      
 
      return the current state of the random seed
      
 
      
 
 
      
-
      
-n_realisations(current_value=None)[source]
      
+
      
+n_realisations(current_value=None)[source]
      
 
      Number of realisations over samples and replications.
      
 
      
 
      Args:
      
@@ -335,14 +336,14 @@ 
      Submodules
-
      
-n_realisations_repl()[source]
      
+
      
+n_realisations_repl()[source]
      
 
      Number of realisations over replications.
      
 
      
 
 
      
-
      
-n_realisations_samples(current_value=None)[source]
      
+
      
+n_realisations_samples(current_value=None)[source]
      
 
      Number of realisations over samples.
      
 
      
 
      Args:
      
@@ -357,8 +358,8 @@ 
      Submodules
-
      
-permute_replications(current_value, idx_list)[source]
      
+
      
+permute_replications(current_value, idx_list)[source]
      
 
      Return realisations with permuted replications (time stays intact).
      
 
      Create surrogate data by permuting realisations over replications while
 keeping the temporal structure (order of samples) intact. Return
@@ -447,8 +448,8 @@ 
      Submodules
-
      
-permute_samples(current_value, idx_list, perm_settings)[source]
      
+
      
+permute_samples(current_value, idx_list, perm_settings)[source]
      
 
      Return realisations with permuted samples (repl. stays intact).
      
 
      Create surrogate data by permuting realisations over samples (time)
 while keeping the order of replications intact. Surrogates can be
@@ -658,8 +659,8 @@ 
      Submodules
-
      
-set_data(data, dim_order)[source]
      
+
      
+set_data(data, dim_order)[source]
      
 
      Overwrite data in an existing Data object.
      
 
      
 
      Args:
      
@@ -675,8 +676,8 @@ 
      Submodules
-
      
-slice_permute_replications(process)[source]
      
+
      
+slice_permute_replications(process)[source]
      
 
      Return data slice with permuted replications (time stays intact).
      
 
      Create surrogate data by permuting realisations over replications while
 keeping the temporal structure (order of samples) intact. Return
@@ -686,8 +687,8 @@ 
      Submodules
-
      
-slice_permute_samples(process, perm_settings)[source]
      
+
      
+slice_permute_samples(process, perm_settings)[source]
      
 
      Return slice of data with permuted samples (repl. stays intact).
      
 
      Create surrogate data by permuting data in a slice over samples (time)
 while keeping the order of replications intact. Return slice for the
@@ -886,8 +887,8 @@ 
      Submodules
+
      
+
      
 
      idtxl.bivariate_te module
      
 
      Perform network inference using multivarate transfer entropy.
      
 
      Estimate multivariate transfer entropy (TE) for network inference using a
@@ -898,8 +899,8 @@ 
      Submodules
-
      
-class idtxl.bivariate_te.BivariateTE[source]
      
+
      
+class idtxl.bivariate_te.BivariateTE[source]
      
 
      Bases: idtxl.network_inference.NetworkInferenceTE, idtxl.network_inference.NetworkInferenceBivariate
      
 
      Perform network inference using bivariate transfer entropy.
      
 
      Perform network inference using bivariate transfer entropy (TE). To
@@ -955,8 +956,8 @@ 
      Submodules
-
      
-analyse_network(settings, data, targets='all', sources='all')[source]
      
+
      
+analyse_network(settings, data, targets='all', sources='all')[source]
      
 
      Find bivariate transfer entropy between all nodes in the network.
      
 
      Estimate bivariate transfer entropy (TE) between all nodes in the
 network or between selected sources and targets.
      
@@ -1017,8 +1018,8 @@ 
      Submodules
-
      
-analyse_single_target(settings, data, target, sources='all')[source]
      
+
      
+analyse_single_target(settings, data, target, sources='all')[source]
      
 
      Find bivariate transfer entropy between sources and a target.
      
 
      Find bivariate transfer entropy (TE) between all potential source
 processes and the target process. Uses bivariate, non-uniform embedding
@@ -1131,8 +1132,8 @@ 
      Submodules
+
      
+
      
 
      idtxl.bivariate_mi module
      
 
      Perform network inference using bivarate mutual information.
      
 
      Estimate bivariate mutual information (MI) for network inference using a
@@ -1143,8 +1144,8 @@ 
      Submodules
-
      
-class idtxl.bivariate_mi.BivariateMI[source]
      
+
      
+class idtxl.bivariate_mi.BivariateMI[source]
      
 
      Bases: idtxl.network_inference.NetworkInferenceMI, idtxl.network_inference.NetworkInferenceBivariate
      
 
      Perform network inference using bivariate mutual information.
      
 
      Perform network inference using bivariate mutual information (MI). To
@@ -1194,8 +1195,8 @@ 
      Submodules
-
      
-analyse_network(settings, data, targets='all', sources='all')[source]
      
+
      
+analyse_network(settings, data, targets='all', sources='all')[source]
      
 
      Find bivariate mutual information between all nodes in the network.
      
 
      Estimate bivariate mutual information (MI) between all nodes in the
 network or between selected sources and targets.
      
@@ -1259,8 +1260,8 @@ 
      Submodules
-
      
-analyse_single_target(settings, data, target, sources='all')[source]
      
+
      
+analyse_single_target(settings, data, target, sources='all')[source]
      
 
      Find bivariate mutual information between sources and a target.
      
 
      Find bivariate mutual information (MI) between all potential source
 processes and the target process. Uses bivariate, non-uniform embedding
@@ -1374,8 +1375,8 @@ 
      Submodules
+
      
+
      
 
      idtxl.bivariate_pid module
      
 
      Estimate partial information decomposition (PID).
      
 
      Estimate PID for two source and one target process using different estimators.
      
@@ -1384,8 +1385,8 @@ 
      Submodules
-
      
-class idtxl.bivariate_pid.BivariatePID[source]
      
+
      
+class idtxl.bivariate_pid.BivariatePID[source]
      
 
      Bases: idtxl.single_process_analysis.SingleProcessAnalysis
      
 
      Perform partial information decomposition for individual processes.
      
 
      Perform partial information decomposition (PID) for two source processes
@@ -1417,8 +1418,8 @@ 
      Submodules
-
      
-analyse_network(settings, data, targets, sources)[source]
      
+
      
+analyse_network(settings, data, targets, sources)[source]
      
 
      Estimate partial information decomposition for network nodes.
      
 
      Estimate partial information decomposition (PID) for multiple nodes in
 the network.
      
@@ -1481,8 +1482,8 @@ 
      Submodules
-
      
-analyse_single_target(settings, data, target, sources)[source]
      
+
      
+analyse_single_target(settings, data, target, sources)[source]
      
 
      Estimate partial information decomposition for a network node.
      
 
      Estimate partial information decomposition (PID) for a target node in
 the network.
      
@@ -1546,8 +1547,8 @@ 
      Submodules
+
      
+
      
 
      idtxl.multivariate_te module
      
 
      Perform network inference using multivarate transfer entropy.
      
 
      Estimate multivariate transfer entropy (TE) for network inference using a
@@ -1558,8 +1559,8 @@ 
      Submodules
-
      
-class idtxl.multivariate_te.MultivariateTE[source]
      
+
      
+class idtxl.multivariate_te.MultivariateTE[source]
      
 
      Bases: idtxl.network_inference.NetworkInferenceTE, idtxl.network_inference.NetworkInferenceMultivariate
      
 
      Perform network inference using multivariate transfer entropy.
      
 
      Perform network inference using multivariate transfer entropy (TE). To
@@ -1583,8 +1584,9 @@ 
      Submoduleshttp://doi.org/10.1103/PhysRevE.83.051112
      
 
-
      
-
      Attributes:
      
+
      Attributes:
      
+
      
+
      
 
      source_setlist
      indices of source processes tested for their influence on the
 target
      
 
      
@@ -1612,11 +1614,10 @@ 
      Submodulesbool
      statistical significance of the over-all TE
      
 
      
 
      
-
      
-
      
+
      
 
      
-
      
-analyse_network(settings, data, targets='all', sources='all')[source]
      
+
      
+analyse_network(settings, data, targets='all', sources='all')[source]
      
 
      Find multivariate transfer entropy between all nodes in the network.
      
 
      Estimate multivariate transfer entropy (TE) between all nodes in the
 network or between selected sources and targets.
      
@@ -1678,8 +1679,8 @@ 
      Submodules
-
      
-analyse_single_target(settings, data, target, sources='all')[source]
      
+
      
+analyse_single_target(settings, data, target, sources='all')[source]
      
 
      Find multivariate transfer entropy between sources and a target.
      
 
      Find multivariate transfer entropy (TE) between all source processes
 and the target process. Uses multivariate, non-uniform embedding found
@@ -1787,14 +1788,14 @@ 
      Submodules
-
      
-getit()[source]
      
+
      
+getit()[source]
      
 
      
 
 
 
-
      
-
      
+
+
      
 
      idtxl.multivariate_mi module
      
 
      Perform network inference using multivarate mutual information.
      
 
      Estimate multivariate mutual information (MI) for network inference using a
@@ -1805,8 +1806,8 @@ 
      Submodules
-
      
-class idtxl.multivariate_mi.MultivariateMI[source]
      
+
      
+class idtxl.multivariate_mi.MultivariateMI[source]
      
 
      Bases: idtxl.network_inference.NetworkInferenceMI, idtxl.network_inference.NetworkInferenceMultivariate
      
 
      Perform network inference using multivariate mutual information.
      
 
      Perform network inference using multivariate mutual information (MI). To
@@ -1854,8 +1855,8 @@ 
      Submodules
-
      
-analyse_network(settings, data, targets='all', sources='all')[source]
      
+
      
+analyse_network(settings, data, targets='all', sources='all')[source]
      
 
      Find multivariate mutual information between nodes in the network.
      
 
      Estimate multivariate mutual information (MI) between all nodes in the
 network or between selected sources and targets.
      
@@ -1923,8 +1924,8 @@ 
      Submodules
-
      
-analyse_single_target(settings, data, target, sources='all')[source]
      
+
      
+analyse_single_target(settings, data, target, sources='all')[source]
      
 
      Find multivariate mutual information between sources and a target.
      
 
      Find multivariate mutual information (MI) between all source processes
 and the target process. Uses multivariate, non-uniform embedding found
@@ -2035,8 +2036,8 @@ 
      Submodules
+
      
+
      
 
      idtxl.multivariate_pid module
      
 
      Estimate partial information decomposition (PID).
      
 
      Estimate PID for multiple sources (up to 4 sources) and one target process
@@ -2046,8 +2047,8 @@ 
      Submodules
-
      
-class idtxl.multivariate_pid.MultivariatePID[source]
      
+
      
+class idtxl.multivariate_pid.MultivariatePID[source]
      
 
      Bases: idtxl.single_process_analysis.SingleProcessAnalysis
      
 
      Perform partial information decomposition for individual processes.
      
 
      Perform partial information decomposition (PID) for multiple source
@@ -2079,8 +2080,8 @@ 
      Submodules
-
      
-analyse_network(settings, data, targets, sources)[source]
      
+
      
+analyse_network(settings, data, targets, sources)[source]
      
 
      Estimate partial information decomposition for network nodes.
      
 
      Estimate, for multiple nodes (target processes), the partial
 information decomposition (PID) for multiple source processes
@@ -2144,8 +2145,8 @@ 
      Submodules
-
      
-analyse_single_target(settings, data, target, sources)[source]
      
+
      
+analyse_single_target(settings, data, target, sources)[source]
      
 
      Estimate partial information decomposition for a network node.
      
 
      Estimate partial information decomposition (PID) for multiple source
 processes (up to 4 sources) and a target process in the network.
      
@@ -2206,8 +2207,8 @@ 
      Submodules
+
      
+
      
 
      idtxl.active_information_storage module
      
 
      Analysis of AIS in a network of processes.
      
 
      Analysis of active information storage (AIS) in individual processes of a
@@ -2217,8 +2218,8 @@ 
      Submodules
-
      
-class idtxl.active_information_storage.ActiveInformationStorage[source]
      
+
      
+class idtxl.active_information_storage.ActiveInformationStorage[source]
      
 
      Bases: idtxl.single_process_analysis.SingleProcessAnalysis
      
 
      Estimate active information storage in individual processes.
      
 
      Estimate active information storage (AIS) in individual processes of the
@@ -2261,8 +2262,8 @@ 
      Submodules
-
      
-analyse_network(settings, data, processes='all')[source]
      
+
      
+analyse_network(settings, data, processes='all')[source]
      
 
      Estimate active information storage for multiple network processes.
      
 
      Estimate active information storage for all or a subset of processes in
 the network.
      
@@ -2317,74 +2318,1206 @@ 
      Submodules
-
      
-analyse_single_process(settings, data, process)[source]
      
-
      Estimate active information storage for a single process.
      
-
      Estimate active information storage for one process in the network.
-Uses non-uniform embedding found through information maximisation. This
-is done in three steps (see Lizier and Faes for details):
      
-
        
-
      1. Find all relevant samples in the processes’ own past, by
-iteratively adding candidate samples that have significant
-conditional mutual information (CMI) with the current value
-(conditional on all samples that were added previously)
        2. 
-
      2. Prune the final conditional set by testing the CMI between each
-sample in the final set and the current value, conditional on all
-other samples in the final set
        3. 
-
      3. Calculate AIS using the final set of candidates as the past state
-(calculate MI between samples in the past and the current value);
-test for statistical significance using a permutation test
        4. 
-
      
-
      
-
      Note:
      For a further description of the algorithm see references in the
-class docstring.
      
+
      
+
+
      
+
      
+analyse_single_process(settings, data, process)[source]
      
+
      Estimate active information storage for a single process.
      
+
      Estimate active information storage for one process in the network.
+Uses non-uniform embedding found through information maximisation. This
+is done in three steps (see Lizier and Faes for details):
      
+
        
+
      1. Find all relevant samples in the processes’ own past, by
+iteratively adding candidate samples that have significant
+conditional mutual information (CMI) with the current value
+(conditional on all samples that were added previously)
        2. 
+
      2. Prune the final conditional set by testing the CMI between each
+sample in the final set and the current value, conditional on all
+other samples in the final set
        3. 
+
      3. Calculate AIS using the final set of candidates as the past state
+(calculate MI between samples in the past and the current value);
+test for statistical significance using a permutation test
        4. 
+
      
+
      
+
      Note:
      For a further description of the algorithm see references in the
+class docstring.
      
+
      
+
      Args:
      
+
      settingsdict
      parameters for estimator use and statistics:
      
+
        
+
      • cmi_estimator : str - estimator to be used for CMI and MI
+calculation (for estimator settings see the documentation in
+the estimators_* modules)
        • 
+
      • max_lag : int - maximum temporal search depth for candidates
+in the processes’ past in samples
        • 
+
      • tau : int [optional] - spacing between candidates in the
+sources’ past in samples (default=1)
        • 
+
      • n_perm_* : int [optional] - number of permutations, where *
+can be ‘max_stat’, ‘min_stat’, ‘mi’ (default=500)
        • 
+
      • alpha_* : float [optional] - critical alpha level for
+statistical significance, where * can be ‘max_stat’,
+‘min_stat’, ‘mi’ (default=0.05)
        • 
+
      • add_conditionals : list of tuples | str [optional] - force
+the estimator to add these conditionals when estimating TE;
+can either be a list of variables, where each variable is
+described as (idx process, lag wrt to current value) or can
+be a string: ‘faes’ for Faes-Method (see references)
        • 
+
      • permute_in_time : bool [optional] - force surrogate creation
+by shuffling realisations in time instead of shuffling
+replications; see documentation of Data.permute_samples() for
+further settings (default=False)
        • 
+
      • verbose : bool [optional] - toggle console output
+(default=True)
        • 
+
      • write_ckp : bool [optional] - enable checkpointing, writes
+analysis state to disk every time a variable is selected;
+resume crashed analysis using
+network_analysis.resume_checkpoint() (default=False)
        • 
+
      • filename_ckp : str [optional] - checkpoint file name (without
+extension) (default=’./idtxl_checkpoint’)
        • 
+
      
+
      
+
      dataData instance
      raw data for analysis
      
+
      
+
      processint
      index of process
      
+
      
+
      
+
      
+
      Returns:
      
+
      ResultsSingleProcessAnalysis instance
      results of AIS estimation, see documentation of
+ResultsSingleProcessAnalysis()
      
+
      
+
      
+
      
+
      
+
      
+
+
      
+
+
      
+
      
+
      idtxl.embedding_optimization_ais_Rudelt module
      
+
      Optimization of embedding parameters of spike times using the history dependence estimators
      
+
      
+
      
+class idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt(settings=None)[source]
      
+
      Bases: object
      
+
      Optimization of embedding parameters of spike times using the history dependence estimators
      
+
      References:
      
+
      
+
      
+
      [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
      optimization reveals long-lasting history dependence in
+neural spiking activity, 2021, PLOS Computational Biology, 17(6)
      
+
      
+
      
+
      [2]: https://github.com/Priesemann-Group/hdestimator
      
+
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      
+
      Args:
      
+
      settingsdict
        
+
      • 
+
        estimation_methodstring
        The method to be used to estimate the history dependence ‘bbc’ or ‘shuffling’.
        
+
        
+
        
+
        • 
+
      • 
+
        embedding_step_sizefloat
        Step size delta t (in seconds) with which the window is slid through the data.
+(default: 0.005)
        
+
        
+
        
+
        • 
+
      • 
+
        embedding_number_of_bins_setlist of integer values
        Set of values for d, the number of bins in the embedding.
+(default: [1, 2, 3, 4, 5])
        
+
        
+
        
+
        • 
+
      • 
+
        embedding_past_range_setlist of floating-point values
        Set of values for T, the past range (in seconds) to be used for embeddings.
+(default: [0.005, 0.00561, 0.00629, 0.00706, 0.00792, 0.00889,
+0.00998, 0.01119, 0.01256, 0.01409, 0.01581, 0.01774, 0.01991,
+0.02233, 0.02506, 0.02812, 0.03155, 0.0354, 0.03972, 0.04456,
+0.05, 0.0561, 0.06295, 0.07063, 0.07924, 0.08891, 0.09976,
+0.11194, 0.12559, 0.14092, 0.15811, 0.17741, 0.19905, 0.22334,
+0.25059, 0.28117, 0.31548, 0.35397, 0.39716, 0.44563, 0.5,
+0.56101, 0.62946, 0.70627, 0.79245, 0.88914, 0.99763, 1.11936,
+1.25594, 1.40919, 1.58114, 1.77407, 1.99054, 2.23342, 2.50594,
+2.81171, 3.15479, 3.53973, 3.97164, 4.45625, 5.0])
        
+
        
+
        
+
        • 
+
      • 
+
        embedding_scaling_exponent_setdict
        Set of values for kappa, the scaling exponent for the bins in the embedding.
+Should be a python-dictionary with the three entries ‘number_of_scalings’, ‘min_first_bin_size’ and
+‘min_step_for_scaling’.
+defaults: {‘number_of_scalings’: 10, ‘min_first_bin_size’: 0.005, ‘min_step_for_scaling’: 0.01})
        
+
        
+
        
+
        • 
+
      • 
+
        bbc_tolerancefloat
        The tolerance for the Bayesian Bias Criterion. Influences which embeddings are
+discarded from the analysis.
+(default: 0.05)
        
+
        
+
        
+
        • 
+
      • 
+
        return_averaged_Rbool
        Return R_tot as the average over R(T) for T in [T_D, T_max], instead of R_tot = R(T_D).
+If set to True, the setting for number_of_bootstraps_R_tot (see below) is ignored and set to 0
+and CI bounds are not calculated.
+(default: True)
        
+
        
+
        
+
        • 
+
      • 
+
        timescale_minimum_past_rangefloat
        Minimum past range T_0 (in seconds) to take into consideration for the estimation of the
+information timescale tau_R.
+(default: 0.01)
        
+
        
+
        
+
        • 
+
      • 
+
        number_of_bootstraps_R_maxint
        The number of bootstrap re-shuffles that should be used to determine the optimal
+embedding. (Bootstrap the estimates of R_max to determine R_tot.)
+These are computed during the ‘history-dependence’ task because they are essential
+to obtain R_tot.
+(default: 250)
        
+
        
+
        
+
        • 
+
      • 
+
        number_of_bootstraps_R_totint
        The number of bootstrap re-shuffles that should be used to estimate the confidence
+interval of the optimal embedding. (Bootstrap the estimates of R_tot = R(T_D) to
+obtain a confidence interval for R_tot.).
+These are computed during the ‘confidence-intervals’ task.
+The setting return_averaged_R (see above) needs to be set to False for this setting
+to take effect.
+(default: 250)
        
+
        
+
        
+
        • 
+
      • 
+
        number_of_bootstraps_nonessentialint
        The number of bootstrap re-shuffles that should be used to estimate the confidence
+intervals for embeddings other than the optimal one. (Bootstrap the estimates of
+R(T) for all other T.)
+(These are not necessary for the main analysis and therefore default to 0.)
        
+
        
+
        
+
        • 
+
      • 
+
        symbol_block_lengthint
        The number of symbols that should be drawn in each block for bootstrap resampling
+If it is set to None (recommended), the length is automatically chosen, based
+on heuristics
+(default: None)
        
+
        
+
        
+
        • 
+
      • 
+
        bootstrap_CI_use_sdbool
        Most of the time we observed normally-distributed bootstrap replications,
+so it is sufficient (and more efficient) to compute confidence intervals
+based on the standard deviation
+(default: True)
        
+
        
+
        
+
        • 
+
      • 
+
        bootstrap_CI_percentile_lofloat
        The lower percentile for the confidence interval.
+This has no effect if bootstrap_CI_use_sd is set to True
+(default: 2.5)
        
+
        
+
        
+
        • 
+
      • 
+
        bootstrap_CI_percentile_hifloat
        The upper percentiles for the confidence interval.
+This has no effect if bootstrap_CI_use_sd is set to True
+(default: 97.5)
        
+
        
+
        
+
        • 
+
      • 
+
        analyse_auto_MIbool
        perform calculation of auto mutual information of the spike train
+(default: True)
+If set to True:
        
+
          
+
        • 
+
          auto_MI_bin_size_setlist of floating-point values
          Set of values for the sizes of the bins (in seconds).
+(default: [0.005, 0.01, 0.025, 0.05, 0.25, 0.5])
          
+
          
+
          
+
          • 
+
        • 
+
          auto_MI_max_delayint
          The maximum delay (in seconds) between the past bin and the response.
+(default: 5)
          
+
          
+
          
+
          • 
+
        
+
        
+
        
+
        • 
+
      • 
+
        visualizationbool
        create .eps output image showing the optimization values and graphs for
+the history dependence and the auto mutual information
+(default: False)
+if set to True:
        
+
          
+
        • 
+
          output_pathString
          Path where the .eps images should be saved
          
+
          
+
          
+
          • 
+
        • 
+
          output_prefixString
          Prefix of the output images
+e.g. <output_prefix>_process0.eps
          
+
          
+
          
+
          • 
+
        
+
        
+
        
+
        • 
+
      • 
+
        debug: bool
        show values while calculating
+(default: False)
        
+
        
+
        
+
        • 
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+analyse_auto_MI(spike_times)[source]
      
+
      Get the auto MI for the spike times.  If it is available from file, load
+it, else compute it.
      
+
      
+
+
      
+
      
+check_inputs()[source]
      
+
      Check input settings for completeness
      
+
      
+
+
      
+
      
+compute_CIs(data, target_R='R_max', symbol_block_length=None)[source]
      
+
      Compute bootstrap replications of the history dependence estimate
+which can be used to obtain confidence intervals.
      
+
      
+
      Args:
      
+
      datadata_spiketime object
      Input data
      
+
      
+
      target_RString
      One of ‘R_max’, ‘R_tot’ or ‘nonessential’.
+If set to R_max, replications of R are produced for the T at which
+R is maximised.
+If set to R_tot, replications of R are produced for T = T_D (cf
+get_temporal_depth_T_D).
+If set to nonessential, replications of R are produced for each T
+(one embedding per T, cf get_embeddings_that_maximise_R).  These
+are not otherwise used in the analysis and are probably only useful
+if the resulting plot is visually inspected, so in most cases it can
+be set to zero.
      
+
      
+
      symbol_block_lengthint
      The number of symbols that should be drawn in each block for bootstrap resampling
+If it is set to None (recommended), the length is automatically chosen, based
+on heuristics
      
+
      
+
      
+
      
+
      
+
      
+
+
      
+
      
+get_R_tot(return_averaged_R=False, **kwargs)[source]
      
+
      
+
+
      
+
      
+get_auto_MI(spike_times, bin_size, number_of_delays)[source]
      
+
      Compute the auto mutual information in the neuron’s activity, a
+measure closely related to history dependence.
      
+
      
+
+
      
+
      
+get_bootstrap_history_dependence(data, embedding, number_of_bootstraps, symbol_block_length=None)[source]
      
+
      For a given embedding, return bootstrap replications for R.
      
+
      
+
+
      
+
      
+get_embeddings(embedding_past_range_set, embedding_number_of_bins_set, embedding_scaling_exponent_set)[source]
      
+
      Get all combinations of parameters T, d, k, based on the
+sets of selected parameters.
      
+
      
+
+
      
+
      
+get_embeddings_that_maximise_R(bbc_tolerance=None, dependent_var='T', get_as_list=False)[source]
      
+
      For each T (or d), get the embedding for which R is maximised.
      
+
      For the bbc estimator, here the bbc_tolerance is applied, ie
+get the unbiased embeddings that maximise R.
      
+
      
+
+
      
+
      
+get_history_dependence(data, process)[source]
      
+
      Estimate the history dependence for each embedding to all given processes.
      
+
      
+
+
      
+
      
+get_information_timescale_tau_R()[source]
      
+
      Get the information timescale tau_R, a characteristic
+timescale of history dependence similar to an autocorrelation
+time.
      
+
      
+
+
      
+
      
+get_past_range(number_of_bins_d, first_bin_size, scaling_k)[source]
      
+
      Get the past range T of the embedding, based on the parameters d, tau_1 and k.
      
+
      
+
+
      
+
      
+get_set_of_scalings(past_range_T, number_of_bins_d, number_of_scalings, min_first_bin_size, min_step_for_scaling)[source]
      
+
      Get scaling exponents such that the uniform embedding as well as
+the embedding for which the first bin has a length of
+min_first_bin_size (in seconds), as well as linearly spaced
+scaling factors in between, such that in total
+number_of_scalings scalings are obtained.
      
+
      
+
+
      
+
      
+get_temporal_depth_T_D(get_R_thresh=False)[source]
      
+
      Get the temporal depth T_D, the past range for the
+‘optimal’ embedding parameters.
      
+
      Given the maximal history dependence R at each past range T,
+(cf get_embeddings_that_maximise_R), first find the smallest T at
+which R is maximised (cf get_max_R_T).  If bootstrap replications
+for this R are available, get the smallest T at which this R minus
+one standard deviation of the bootstrap estimates is attained.
      
+
      
+
+
      
+
      
+optimize(data, processes='all')[source]
      
+
      Optimize the embedding parameters of spike time data using the Rudelt history dependence estimator.
      
+
      References:
      
+
      
+
      
+
      [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
      optimization reveals long-lasting history dependence in
+neural spiking activity, 2021, PLOS Computational Biology, 17(6)
      
+
      
+
      
+
      [2]: https://github.com/Priesemann-Group/hdestimator
      
+
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      
+
      Args:
      
+
      dataData_spiketime instance
      raw data for analysis
      
+
      
+
      processeslist of int
      index of processes;
+spike times are optimized all processes specified in the list separately.
      
+
      
+
      
+
      
+
      Returns: # ——————————————————————————————————– TODO
      
+
      ResultsSingleProcessRudelt instance
      results of Rudelt optimization, see documentation of
+ResultsSingleProcessRudelt()
      
+
      
+
      if visulization in settings was set True (see class OptimizationRudelt):
      
+
      .eps images are created for each optimized process containing:
        
+
      • optimized values for the process
        • 
+
      • graph for the history dependence
        • 
+
      • graph for auto mutual information (if calculated)
        • 
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+
+
      
+
      
+optimize_single_run(data, process)[source]
      
+
      optimizes a single realisation of spike time data given the process number
      
+
      
+
      Args:
      
+
      dataData_spiketime instance
      raw data for analysis
      
+
      
+
      processint
      index of process;
      
+
      
+
      
+
      
+
      Returns:
      
+
      DotDict
      with the following keys
      
+
      
+
      Processint
      Process that was optimized
      
+
      
+
      estimation_methodString
      Estimation method that was used for optimization
      
+
      
+
      T_Dfloat
      Estimated optimal value for the temporal depth TD
      
+
      
+
      tau_R :
      Information timescale tau_R, a characteristic timescale of history
+dependence similar to an autocorrelation time.
      
+
      
+
      R_totfloat
      Estimated value for the total history dependence Rtot,
      
+
      
+
      AIS_totfloat
      Estimated value for the total active information storage
      
+
      
+
      opt_number_of_bins_dint
      Number of bins d for the embedding that yields (R̂tot ,T̂D)
      
+
      
+
      opt_scaling_kint
      Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
      
+
      
+
      opt_first_bin_sizeint
      Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
      
+
      
+
      history_dependencearray with floating-point values
      Estimated history dependence for each embedding
      
+
      
+
      firing_ratefloat
      Firing rate of the neuron/ spike train
      
+
      
+
      recording_lengthfloat
      Length of the recording (in seconds)
      
+
      
+
      H_spikingfloat
      Entropy of the spike times
      
+
      
+
      
+
      
+
      
+
      
+
      if analyse_auto_MI was set to True additionally:
      
+
      auto_MIdict
      numpy array of MI values for each delay
      
+
      
+
      auto_MI_delayslist of int
      list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay
      
+
      
+
      
+
      
+
      
+
      
+
+
      
+
      
+remove_subresults_single_process()[source]
      
+
      delete results from self from single process
      
+
      
+
+
      
+
+
      
+
      
+
      idtxl.estimators_Rudelt module
      
+
      Provide HDE estimators.
      
+
      
+
      
+class idtxl.estimators_Rudelt.RudeltAbstractEstimator(settings=None)[source]
      
+
      Bases: idtxl.estimator.Estimator
      
+
      Abstract class for implementation of nsb and plugin estimators from Rudelt.
      
+
      Abstract class for implementation of nsb and plugin estimators, child classes
+implement estimators for mutual information (MI) .
      
+
      References:
      
+
      
+
      
+
      [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
      optimization reveals long-lasting history dependence in
+neural spiking activity, 2021, PLOS Computational Biology, 17(6)
      
+
      
+
      
+
      [2]: https://github.com/Priesemann-Group/hdestimator
      
+
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      
+
      Args:
      
+
      settingsdict
        
+
      • 
+
        embedding_step_sizefloat [optional]
        Step size delta t (in seconds) with which the window is slid through the data
+(default = 0.005).
        
+
        
+
        
+
        • 
+
      • 
+
        normalisebool [optional]
        rebase spike times to zero
+(default=True)
        
+
        
+
        
+
        • 
+
      • 
+
        return_averaged_Rbool [optional]
        If set to True, compute R̂tot as the average over R̂(T ) for T ∈ [T̂D, Tmax ] instead of
+R̂tot = R(T̂D ). If set to True, the setting for number_of_bootstraps_R_tot is ignored and
+set to 0
+(default=True)
        
+
        
+
        
+
        • 
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+get_median_number_of_spikes_per_bin(raw_symbols)[source]
      
+
      Given raw symbols (in which the number of spikes per bin are counted,
+ie not necessarily binary quantity), get the median number of spikes
+for each bin, among all symbols obtained by the embedding.
      
+
      
+
+
      
+
      
+get_multiplicities(symbol_counts, alphabet_size)[source]
      
+
      Get the multiplicities of some given symbol counts.
      
+
      To estimate the entropy of a system, it is only important how
+often a symbol/ event occurs (the probability that it occurs), not
+what it represents. Therefore, computations can be simplified by
+summarizing symbols by their frequency, as represented by the
+multiplicities.
      
+
      
+
+
      
+
      
+get_past_range(number_of_bins_d, first_bin_size, scaling_k)[source]
      
+
      Get the past range T of the embedding, based on the parameters d, tau_1 and k.
      
+
      
+
+
      
+
      
+get_raw_symbols(spike_times, embedding, first_bin_size)[source]
      
+
      Get the raw symbols (in which the number of spikes per bin are counted,
+ie not necessarily binary quantity), as obtained by applying the
+embedding.
      
+
      
+
+
      
+
      
+get_symbol_counts(symbol_array)[source]
      
+
      Count how often symbols occur
      
+
      
+
+
      
+
      
+get_window_delimiters(number_of_bins_d, scaling_k, first_bin_size)[source]
      
+
      Get delimiters of the window, used to describe the embedding. The
+window includes both the past embedding and the response.
      
+
      The delimiters are times, relative to the first bin, that separate
+two consequent bins.
      
+
      
+
+
      
+
      
+is_analytic_null_estimator()[source]
      
+
      Indicate if estimator supports analytic surrogates.
      
+
      Return true if the estimator implements estimate_surrogates_analytic()
+where data is formatted as per the estimate method for this estimator.
      
+
      
+
      Returns:
      bool
      
+
      
+
      
+
      
+
+
      
+
      
+is_parallel()[source]
      
+
      Indicate if estimator supports parallel estimation over chunks.
      
+
      Return true if the supports parallel estimation over chunks, where a
+chunk is one independent data set.
      
+
      
+
      Returns:
      bool
      
+
      
+
      
+
      
+
+
      
+
      
+symbol_array_to_binary(spikes_in_window, number_of_bins_d)[source]
      
+
      Given an array of 1s and 0s, representing spikes and the absence
+thereof, read the array as a binary number to obtain a
+(base 10) integer.
      
+
      
+
+
      
+
      
+symbol_binary_to_array(symbol_binary, number_of_bins_d)[source]
      
+
      Given a binary representation of a symbol (cf symbol_array_to_binary),
+convert it back into its array-representation.
      
+
      
+
+
      
+
+
      
+
      
+class idtxl.estimators_Rudelt.RudeltAbstractNSBEstimator(settings=None)[source]
      
+
      Bases: idtxl.estimators_Rudelt.RudeltAbstractEstimator
      
+
      Abstract class for implementation of NSB estimators from Rudelt.
      
+
      Abstract class for implementation of Nemenman-Shafee-Bialek (NSB)
+estimators, child classes implement nsb estimators for mutual information
+(MI).
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      References:
      
+
      
+
      
+
      [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
      optimization reveals long-lasting history dependence in
+neural spiking activity, 2021, PLOS Computational Biology, 17(6)
      
+
      
+
      [2]: I. Nemenman, F. Shafee, W. Bialek: Entropy and inference,
      revisited. In T.G. Dietterich, S. Becker, and Z. Ghahramani,
+editors, Advances in Neural Information Processing Systems 14,
+Cambridge, MA, 2002. MIT Press.
      
+
      
+
      
+
      
+
      
+
      Args:
      
+
      settingsdict
        
+
      • 
+
        embedding_step_sizefloat [optional]
        Step size delta t (in seconds) with which the window is slid through the data
+(default = 0.005).
        
+
        
+
        
+
        • 
+
      • 
+
        normalisebool [optional]
        rebase spike times to zero
+(default=True)
        
+
        
+
        
+
        • 
+
      • 
+
        return_averaged_Rbool [optional]
        If set to True, compute R̂tot as the average over R̂(T ) for T ∈ [T̂D, Tmax ] instead of
+R̂tot = R(T̂D ). If set to True, the setting for number_of_bootstraps_R_tot is ignored and
+set to 0
+(default=True)
        
+
        
+
        
+
        • 
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+H1(beta, mk, K, N)[source]
      
+
      Compute the first moment (expectation value) of the entropy H.
      
+
      H is the entropy one obtains with a symmetric Dirichlet prior
+with concentration parameter beta and a multinomial likelihood.
      
+
      
+
+
      
+
      
+alpha_ML(mk, K1, N)[source]
      
+
      Compute first guess for the beta_MAP (cf get_beta_MAP) parameter
+via the posterior of a Dirichlet process.
      
+
      
+
+
      
+
      
+d2_log_rho(beta, mk, K, N)[source]
      
+
      Second derivate of the logarithm of the Dirichlet multinomial likelihood.
      
+
      
+
+
      
+
      
+d2_log_rho_xi(beta, mk, K, N)[source]
      
+
      Second derivative of the logarithm of the nsb (unnormalized) posterior.
      
+
      
+
+
      
+
      
+d2_xi(beta, K)[source]
      
+
      Second derivative of xi(beta) (cf d_xi).
      
+
      
+
+
      
+
      
+d3_xi(beta, K)[source]
      
+
      Third derivative of xi(beta) (cf d_xi).
      
+
      
+
+
      
+
      
+d_log_rho(beta, mk, K, N)[source]
      
+
      First derivate of the logarithm of the Dirichlet multinomial likelihood.
      
+
      
+
+
      
+
      
+d_log_rho_xi(beta, mk, K, N)[source]
      
+
      First derivative of the logarithm of the nsb (unnormalized) posterior.
      
+
      
+
+
      
+
      
+d_xi(beta, K)[source]
      
+
      First derivative of xi(beta).
      
+
      xi(beta) is the entropy of the system when no data has been observed.
+d_xi is the prior for the nsb estimator
      
+
      
+
+
      
+
      
+get_beta_MAP(mk, K, N)[source]
      
+
      Get the maximum a posteriori (MAP) value for beta.
      
+
      Provides the location of the peak, around which we integrate.
      
+
      beta_MAP is the value for beta for which the posterior of the
+NSB estimator is maximised (or, equivalently, of the logarithm
+thereof, as computed here).
      
+
      
+
+
      
+
      
+get_integration_bounds(mk, K, N)[source]
      
+
      Find the integration bounds for the estimator.
      
+
      Typically it is a delta-like distribution so it is sufficient
+to integrate around this peak. (If not this function is not
+called.)
      
+
      
+
+
      
+
      
+log_likelihood_DP_alpha(a, K1, N)[source]
      
+
      Alpha-dependent terms of the log-likelihood of a Dirichlet Process.
      
+
      
+
+
      
+
      
+nsb_entropy(mk, K, N)[source]
      
+
      Estimate the entropy of a system using the NSB estimator.
      
+
      
+
      Parameters
      
+
        
+
      • mk – multiplicities
        • 
+
      • K – number of possible symbols/ state space of the system
        • 
+
      • N – total number of observed symbols
        • 
+
      
+
      
+
      
+
      
+
+
      
+
      
+rho(beta, mk, K, N)[source]
      
+
      rho(beta, data) is the Dirichlet multinomial likelihood.
      
+
      rho(beta, data) together with the d_xi(beta) make up
+the posterior for the nsb estimator
      
+
      
+
+
      
+
      
+unnormalized_posterior(beta, mk, K, N)[source]
      
+
      The (unnormalized) posterior in the nsb estimator.
      
+
      Product of the likelihood rho and the prior d_xi;
+the normalizing factor is given by the marginal likelihood
      
+
      
+
+
      
+
+
      
+
      
+class idtxl.estimators_Rudelt.RudeltBBCEstimator(settings=None)[source]
      
+
      Bases: idtxl.estimators_Rudelt.RudeltAbstractEstimator
      
+
      Bayesian bias criterion (BBC) Estimator using NSB and Plugin estimator
      
+
      Calculate the mutual information (MI) of one variable depending on its past
+using nsb and plugin estimator and check if bias criterion is passed.
+See parent class for references.
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      
+
      Args:
      
+
      settingsdict
        
+
      • 
+
        embedding_step_sizefloat [optional]
        Step size delta t (in seconds) with which the window is slid through the data
+(default = 0.005).
        
+
        
+
        
+
        • 
+
      • 
+
        normalisebool [optional]
        rebase spike times to zero
+(default=True)
        
+
        
+
        
+
        • 
+
      • 
+
        return_averaged_Rbool [optional]
        If set to True, compute R̂tot as the average over R̂(T ) for T ∈ [T̂D, Tmax ] instead of
+R̂tot = R(T̂D ). If set to True, the setting for number_of_bootstraps_R_tot is ignored and
+set to 0
+(default=True)
        
+
        
+
        
+
        • 
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+bayesian_bias_criterion(R_nsb, R_plugin, bbc_tolerance)[source]
      
+
      Get whether the Bayesian bias criterion (bbc) is passed.
      
+
      
+
      Parameters
      
+
        
+
      • R_nsb – history dependence computed with NSB estimator
        • 
+
      • R_plugin – history dependence computed with plugin estimator
        • 
+
      • bbc_tolerance – tolerance for the Bayesian bias criterion
        • 
+
      
+
      
+
      
+
      
+
+
      
+
      
+estimate(symbol_array, past_symbol_array, current_symbol_array, bbc_tolerance=None)[source]
      
+
      Calculate the mutual information (MI) of one variable depending on its past
+using nsb and plugin estimator and check if bias criterion is passed/
      
+
      
+
      Args:
      
+
      symbol_array1D numpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      past_symbol_arraynumpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      current_symbol_arraynumpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      
+
      
+
      Returns:
      
+
      I (float)
      MI (AIS)
      
+
      
+
      R (float)
      MI / H_uncond (History dependence)
      
+
      
+
      bbc_term (float)
      bbc tolerance-independent term of the Bayesian bias
+criterion (bbc)
      
+
      
+
      
+
      
+
      
+
      
+
+
      
+
      
+get_bbc_term(R_nsb, R_plugin)[source]
      
+
      Get the bbc tolerance-independent term of the Bayesian bias
+criterion (bbc).
      
+
      
+
      Parameters
      
+
        
+
      • R_nsb – history dependence computed with NSB estimator
        • 
+
      • R_plugin – history dependence computed with plugin estimator
        • 
+
      
+
      
+
      
+
      
+
+
      
+
+
      
+
      
+class idtxl.estimators_Rudelt.RudeltNSBEstimatorSymbolsMI(settings=None)[source]
      
+
      Bases: idtxl.estimators_Rudelt.RudeltAbstractNSBEstimator
      
+
      History dependence NSB estimator
      
+
      Calculate the mutual information (MI) of one variable depending on its past
+using NSB estimator. See parent class for references.
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      
+
      Args:
      
+
      settingsdict
        
+
      • 
+
        embedding_step_sizefloat [optional]
        Step size delta t (in seconds) with which the window is slid through the data
+(default = 0.005).
        
+
        
+
        
+
        • 
+
      • 
+
        normalisebool [optional]
        rebase spike times to zero
+(default=True)
        
+
        
+
        
+
        • 
+
      • 
+
        return_averaged_Rbool [optional]
        If set to True, compute R̂tot as the average over R̂(T ) for T ∈ [T̂D, Tmax ] instead of
+R̂tot = R(T̂D ). If set to True, the setting for number_of_bootstraps_R_tot is ignored and
+set to 0
+(default=True)
        
+
        
+
        
+
        • 
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+estimate(symbol_array, past_symbol_array, current_symbol_array)[source]
      
+
      Estimate mutual information using NSB estimator.
      
+
      
+
      Args:
      
+
      symbol_array1D numpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      past_symbol_arraynumpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      current_symbol_arraynumpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      
+
      
+
      Returns:
      
+
      I (float)
      MI (AIS)
      
+
      
+
      R (float)
      MI / H_uncond (History dependence)
      
+
      
+
      
+
      
+
      
+
      
+
+
      
+
      
+nsb_estimator(symbol_counts, past_symbol_counts, alphabet_size, alphabet_size_past, H_uncond)[source]
      
+
      Estimate the entropy of a system using the NSB estimator.
      
+
      
+
+
      
+
+
      
+
      
+class idtxl.estimators_Rudelt.RudeltPluginEstimatorSymbolsMI(settings=None)[source]
      
+
      Bases: idtxl.estimators_Rudelt.RudeltAbstractEstimator
      
+
      Plugin History dependence estimator
      
+
      Calculate the mutual information (MI) of one variable depending on its past
+using plugin estimator. See parent class for references.
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      
+
      Args:
      
+
      settingsdict
        
+
      • 
+
        embedding_step_sizefloat [optional] - Step size delta t (in seconds) with which the window is slid
        through the data (default = 0.005).
        
+
        
+
        
+
        • 
+
      • normalise : bool [optional] - rebase spike times to zero (default=True)
        • 
+
      • return_averaged_R : bool [optional] - rebase spike times to zero (default=True)
        • 
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+estimate(symbol_array, past_symbol_array, current_symbol_array)[source]
      
+
      Estimate mutual information using plugin estimator.
      
+
      
+
      Args:
      
+
      symbol_array1D numpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      past_symbol_arraynumpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      current_symbol_arraynumpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      
+
      
+
      Returns:
      
+
      I (float)
      MI (AIS)
      
+
      
+
      R (float)
      MI / H_uncond (History dependence)
      
+
      
+
      
+
      
+
      
+
      
+
+
      
+
      
+plugin_entropy(mk, N)[source]
      
+
      Estimate the entropy of a system using the Plugin estimator.
      
+
      (In principle this is the same function as utl.get_shannon_entropy,
+only here it is a function of the multiplicities, not the probabilities.)
      
+
      
+
      Parameters
      
+
        
+
      • mk – multiplicities
        • 
+
      • N – total number of observed symbols
        • 
+
      
+
      
+
      
+
      
+
+
      
+
      
+plugin_estimator(symbol_counts, past_symbol_counts, alphabet_size, alphabet_size_past, H_uncond)[source]
      
+
      Estimate the entropy of a system using the BBC estimator.
      
+
      
+
+
      
+
+
      
+
      
+class idtxl.estimators_Rudelt.RudeltShufflingEstimator(settings=None)[source]
      
+
      Bases: idtxl.estimators_Rudelt.RudeltAbstractEstimator
      
+
      Estimate the history dependence in a spike train using the shuffling estimator.
      
+
      See parent class for references.
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      
+
      
+estimate(symbol_array)[source]
      
+
      Estimate the history dependence in a spike train using the shuffling estimator.
      
+
      
+
      
+
      Args:
      
+
      symbol_array1D numpy array
      realisations of symbols based on current and past states.
+(first output of get_realisations_symbol from data_spiketimes object)
      
+
      
+
      
+
      
+
      
+
      
+
      
+
      Returns:
      
+
      I (float)
      MI (AIS)
      
+
      
+
      R (float)
      MI / H_uncond (History dependence)
      
+
      
+
      
+
      
+
      
+
      
+
+
      
+
      
+get_H0_X_past_cond_X(marginal_probabilities, number_of_bins_d, P_X_uncond)[source]
      
+
      Compute H_0(X_past | X), the estimate of the entropy for the past
+symbols given a response, under the assumption that activity in
+the past contributes independently towards the response.
      
+
      
+
+
      
+
      
+get_H0_X_past_cond_X_eq_x(marginal_probabilities, number_of_bins_d)[source]
      
+
      Compute H_0(X_past | X = x), cf get_H0_X_past_cond_X.
      
+
      
+
+
      
+
      
+get_H_X_past_cond_X(P_X_uncond, P_X_past_cond_X)[source]
      
+
      Compute H(X_past | X), the plug-in estimate of the conditional entropy for the past
+symbols, conditioned on the response X,  given their probabilities.
      
+
      
+
+
      
+
      
+get_H_X_past_uncond(P_X_past_uncond)[source]
      
+
      Compute H(X_past), the plug-in estimate of the entropy for the past symbols, given
+their probabilities.
      
+
      
+
+
      
+
      
+get_P_X_past_cond_X(past_symbol_counts, number_of_symbols)[source]
      
+
      Compute P(X_past | X), the probability of the past activity conditioned
+on the response X using the plug-in estimator.
      
+
      
+
+
      
+
      
+get_P_X_past_uncond(past_symbol_counts, number_of_symbols)[source]
      
+
      Compute P(X_past), the probability of the past activity using
+the plug-in estimator.
      
+
      
+
+
      
+
      
+get_P_X_uncond(number_of_symbols)[source]
      
+
      Compute P(X), the probability of the current activity using
+the plug-in estimator.
      
+
      
+
+
      
+
      
+get_marginal_frequencies_of_spikes_in_bins(symbol_counts, number_of_bins_d)[source]
      
+
      Compute for each past bin 1…d the sum of spikes found in that bin across all
+observed symbols.
      
+
      
+
+
      
+
      
+get_shuffled_symbol_counts(symbol_counts, past_symbol_counts, number_of_bins_d, number_of_symbols)[source]
      
+
      Simulate new data by, for each past bin 1…d, permutating the activity
+across all observed past_symbols (for a given response X). The marginal
+probability of observing a spike given the response is thus preserved for
+each past bin.
      
+
      
+
+
      
+
      
+shuffling_MI(symbol_counts, number_of_bins_d)[source]
      
+
      Estimate the mutual information between current and past activity
+in a spike train using the shuffling estimator.
      
+
      To obtain the shuffling estimate, compute the plug-in estimate and
+a correction term to reduce its bias.
      
+
      For the plug-in estimate:
      
+
        
+
      • Extract the past_symbol_counts from the symbol_counts.
        • 
+
      • I_plugin = H(X_past) - H(X_past | X)
        • 
+
      
+
      Notation:
      
+
        
+
      • X: current activity, aka response
        • 
+
      • X_past: past activity
        • 
+
      • P_X_uncond: P(X)
        • 
+
      • P_X_past_uncond: P(X_past)
        • 
+
      • P_X_past_cond_X: P(X_past | X)
        • 
+
      • H_X_past_uncond: H(X_past)
        • 
+
      • H_X_past_cond_X: H(X_past | X)
        • 
+
      • I_plugin: plugin estimate of I(X_past; X)
        • 
+
      
+
      For the correction term:
      
+
        
+
      • 
+
        Simulate additional data under the assumption that activity
        in the past contributes independently towards the current activity.
        
+
        
+
        
+
        • 
+
      • 
+
        Compute the entropy under the assumptions of the model, which
        due to its simplicity is easy to sample and the estimate unbiased
        
+
        
+
        
+
        • 
+
      • 
+
        Compute the entropy using the plug-in estimate, whose bias is
        similar to that of the plug-in estimate on the original data
        
+
        
+
        
+
        • 
+
      • 
+
        Compute the correction term as the difference between the
        unbiased and biased terms
        
 
        
-
        Args:
        
-
        settingsdict
        parameters for estimator use and statistics:
        
+
        
+
        • 
+
      
+
      Notation:
      
 
        
-
      • cmi_estimator : str - estimator to be used for CMI and MI
-calculation (for estimator settings see the documentation in
-the estimators_* modules)
        • 
-
      • max_lag : int - maximum temporal search depth for candidates
-in the processes’ past in samples
        • 
-
      • tau : int [optional] - spacing between candidates in the
-sources’ past in samples (default=1)
        • 
-
      • n_perm_* : int [optional] - number of permutations, where *
-can be ‘max_stat’, ‘min_stat’, ‘mi’ (default=500)
        • 
-
      • alpha_* : float [optional] - critical alpha level for
-statistical significance, where * can be ‘max_stat’,
-‘min_stat’, ‘mi’ (default=0.05)
        • 
-
      • add_conditionals : list of tuples | str [optional] - force
-the estimator to add these conditionals when estimating TE;
-can either be a list of variables, where each variable is
-described as (idx process, lag wrt to current value) or can
-be a string: ‘faes’ for Faes-Method (see references)
        • 
-
      • permute_in_time : bool [optional] - force surrogate creation
-by shuffling realisations in time instead of shuffling
-replications; see documentation of Data.permute_samples() for
-further settings (default=False)
        • 
-
      • verbose : bool [optional] - toggle console output
-(default=True)
        • 
-
      • write_ckp : bool [optional] - enable checkpointing, writes
-analysis state to disk every time a variable is selected;
-resume crashed analysis using
-network_analysis.resume_checkpoint() (default=False)
        • 
-
      • filename_ckp : str [optional] - checkpoint file name (without
-extension) (default=’./idtxl_checkpoint’)
        • 
+
      • P0_sh_X_past_cond_X: P_0,sh(X_past | X), equiv. to P(X_past | X)
+on the shuffled data
        • 
+
      • H0_X_past_cond_X: H_0(X_past | X), based on the model of independent
+contributions
        • 
+
      • H0_sh_X_past_cond_X: H_0,sh(X_past | X), based on
        • 
+
      • P0_sh_X_past_cond_X, ie the plug-in estimate
        • 
+
      • I_corr: the correction term to reduce the bias of I_plugin
        • 
 
      
+
      
+
      Args:
      
+
      symbol_countsiterable
      the activity of a spike train is embedded into symbols,
+whose occurrences are counted (cf emb.get_symbol_counts)
      
 
      
-
      dataData instance
      raw data for analysis
      
-
      
-
      processint
      index of process
      
-
      
-
      
-
      
-
      Returns:
      
-
      ResultsSingleProcessAnalysis instance
      results of AIS estimation, see documentation of
-ResultsSingleProcessAnalysis()
      
+
      number_of_bins_dint
      the number of bins of the embedding
      
 
      
 
      
 
      
@@ -2393,13 +3526,13 @@ 
      Submodules
+
      
+
      
 
      idtxl.estimators_jidt module
      
 
      Provide JIDT estimators.
      
 
      
-
      
-class idtxl.estimators_jidt.JidtDiscrete(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscrete(settings)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtEstimator
      
 
      Abstract class for implementation of discrete JIDT-estimators.
      
 
      Abstract class for implementation of plug-in JIDT-estimators for discrete
@@ -2432,8 +3565,8 @@ 
      Submodules
-
      
-estimate_surrogates_analytic(n_perm=200, **data)[source]
      
+
      
+estimate_surrogates_analytic(n_perm=200, **data)[source]
      
 
      Return estimate of the analytical surrogate distribution.
      
 
      This method must be implemented because this class’
 is_analytic_null_estimator() method returns true.
      
@@ -2458,8 +3591,8 @@ 
      Submodules
-
      
-abstract get_analytic_distribution(**data)[source]
      
+
      
+abstract get_analytic_distribution(**data)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -2481,8 +3614,8 @@ 
      Submodules
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -2495,8 +3628,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtDiscreteAIS(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscreteAIS(settings)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtDiscrete
      
 
      Calculate AIS with JIDT’s discrete-variable implementation.
      
 
      Calculate the active information storage (AIS) for one process. Call JIDT
@@ -2527,8 +3660,8 @@ 
      Submodules
-
      
-estimate(process, return_calc=False)[source]
      
+
      
+estimate(process, return_calc=False)[source]
      
 
      Estimate active information storage.
      
 
      
 
      Args:
      
@@ -2560,8 +3693,8 @@ 
      Submodules
-
      
-get_analytic_distribution(process)[source]
      
+
      
+get_analytic_distribution(process)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -2586,8 +3719,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtDiscreteCMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscreteCMI(settings=None)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtDiscrete
      
 
      Calculate CMI with JIDT’s implementation for discrete variables.
      
 
      Calculate the conditional mutual information between two variables given
@@ -2622,8 +3755,8 @@ 
      Submodules
-
      
-estimate(var1, var2, conditional=None, return_calc=False)[source]
      
+
      
+estimate(var1, var2, conditional=None, return_calc=False)[source]
      
 
      Estimate conditional mutual information.
      
 
      
 
      Args:
      
@@ -2660,8 +3793,8 @@ 
      Submodules
-
      
-get_analytic_distribution(var1, var2, conditional=None)[source]
      
+
      
+get_analytic_distribution(var1, var2, conditional=None)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -2691,8 +3824,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtDiscreteMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscreteMI(settings=None)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtDiscrete
      
 
      Calculate MI with JIDT’s discrete-variable implementation.
      
 
      Calculate the mutual information (MI) between two variables. Call JIDT via
@@ -2726,8 +3859,8 @@ 
      Submodules
-
      
-estimate(var1, var2, return_calc=False)[source]
      
+
      
+estimate(var1, var2, return_calc=False)[source]
      
 
      Estimate mutual information.
      
 
      
 
      Args:
      
@@ -2761,8 +3894,8 @@ 
      Submodules
-
      
-get_analytic_distribution(var1, var2)[source]
      
+
      
+get_analytic_distribution(var1, var2)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -2789,8 +3922,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtDiscreteTE(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscreteTE(settings)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtDiscrete
      
 
      Calculate TE with JIDT’s implementation for discrete variables.
      
 
      Calculate the transfer entropy between two time series processes.
@@ -2836,8 +3969,8 @@ 
      Submodules
-
      
-estimate(source, target, return_calc=False)[source]
      
+
      
+estimate(source, target, return_calc=False)[source]
      
 
      Estimate transfer entropy from a source to a target variable.
      
 
      
 
      Args:
      
@@ -2871,8 +4004,8 @@ 
      Submodules
-
      
-get_analytic_distribution(source, target)[source]
      
+
      
+get_analytic_distribution(source, target)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -2899,8 +4032,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtEstimator(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtEstimator(settings=None)[source]
      
 
      Bases: idtxl.estimator.Estimator
      
 
      Abstract class for implementation of JIDT estimators.
      
 
      Abstract class for implementation of JIDT estimators, child classes
@@ -2937,8 +4070,8 @@ 
      Submodules
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -2951,8 +4084,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtGaussian(CalcClass, settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussian(CalcClass, settings)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtEstimator
      
 
      Abstract class for implementation of JIDT Gaussian-estimators.
      
 
      Abstract class for implementation of JIDT Gaussian-estimators, child
@@ -2979,10 +4112,10 @@ 
      Submodules
-
      
-estimate_surrogates_analytic(n_perm=200, **data)[source]
      
-
      Estimate the surrogate distribution analytically.
-This method must be implemented because this class’
+
      
+estimate_surrogates_analytic(n_perm=200, **data)[source]
      
+
      Estimate the surrogate distribution analytically.
      
+
      This method must be implemented because this class’
 is_analytic_null_estimator() method returns true
      
 
      
 
      Args:
      
@@ -3005,8 +4138,8 @@ 
      Submodules
-
      
-get_analytic_distribution(**data)[source]
      
+
      
+get_analytic_distribution(**data)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -3028,8 +4161,8 @@ 
      Submodules
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -3042,8 +4175,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtGaussianAIS(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussianAIS(settings)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtGaussian
      
 
      Calculate active information storage with JIDT’s Gaussian implementation.
      
 
      Calculate active information storage (AIS) for some process using JIDT’s
@@ -3078,8 +4211,8 @@ 
      Submodules
-
      
-estimate(process)[source]
      
+
      
+estimate(process)[source]
      
 
      Estimate active information storage.
      
 
      
 
      Args:
      
@@ -3101,8 +4234,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtGaussianCMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussianCMI(settings=None)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtGaussian
      
 
      Calculate conditional mutual infor with JIDT’s Gaussian implementation.
      
 
      Computes the differential conditional mutual information of two
@@ -3134,8 +4267,8 @@ 
      Submodules
-
      
-estimate(var1, var2, conditional=None)[source]
      
+
      
+estimate(var1, var2, conditional=None)[source]
      
 
      Estimate conditional mutual information.
      
 
      
 
      Args:
      
@@ -3160,8 +4293,8 @@ 
      Submodules
-
      
-get_analytic_distribution(var1, var2, conditional=None)[source]
      
+
      
+get_analytic_distribution(var1, var2, conditional=None)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -3190,8 +4323,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtGaussianMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussianMI(settings=None)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtGaussian
      
 
      Calculate mutual information with JIDT’s Gaussian implementation.
      
 
      Calculate the mutual information between two variables. Call JIDT via jpype
@@ -3219,8 +4352,8 @@ 
      Submodules
-
      
-estimate(var1, var2)[source]
      
+
      
+estimate(var1, var2)[source]
      
 
      Estimate mutual information.
      
 
      
 
      Args:
      
@@ -3244,8 +4377,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtGaussianTE(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussianTE(settings)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtGaussian
      
 
      Calculate transfer entropy with JIDT’s Gaussian implementation.
      
 
      Calculate transfer entropy between a source and a target variable using
@@ -3289,8 +4422,8 @@ 
      Submodules
-
      
-estimate(source, target)[source]
      
+
      
+estimate(source, target)[source]
      
 
      Estimate transfer entropy from a source to a target variable.
      
 
      
 
      Args:
      
@@ -3314,8 +4447,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtKraskov(CalcClass, settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskov(CalcClass, settings=None)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtEstimator
      
 
      Abstract class for implementation of JIDT Kraskov-estimators.
      
 
      Abstract class for implementation of JIDT Kraskov-estimators, child classes
@@ -3356,8 +4489,8 @@ 
      Submodules
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -3370,8 +4503,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtKraskovAIS(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskovAIS(settings)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtKraskov
      
 
      Calculate active information storage with JIDT’s Kraskov implementation.
      
 
      Calculate active information storage (AIS) for some process using JIDT’s
@@ -3418,8 +4551,8 @@ 
      Submodules
-
      
-estimate(process)[source]
      
+
      
+estimate(process)[source]
      
 
      Estimate active information storage.
      
 
      
 
      Args:
      
@@ -3441,8 +4574,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtKraskovCMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskovCMI(settings=None)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtKraskov
      
 
      Calculate conditional mutual inform with JIDT’s Kraskov implementation.
      
 
      Calculate the conditional mutual information (CMI) between three variables.
@@ -3484,8 +4617,8 @@ 
      Submodules
-
      
-estimate(var1, var2, conditional=None)[source]
      
+
      
+estimate(var1, var2, conditional=None)[source]
      
 
      Estimate conditional mutual information.
      
 
      
 
      Args:
      
@@ -3512,8 +4645,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtKraskovMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskovMI(settings=None)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtKraskov
      
 
      Calculate mutual information with JIDT’s Kraskov implementation.
      
 
      Calculate the mutual information between two variables. Call JIDT via jpype
@@ -3553,8 +4686,8 @@ 
      Submodules
-
      
-estimate(var1, var2)[source]
      
+
      
+estimate(var1, var2)[source]
      
 
      Estimate mutual information.
      
 
      
 
      Args:
      
@@ -3578,8 +4711,8 @@ 
      Submodules
-
      
-class idtxl.estimators_jidt.JidtKraskovTE(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskovTE(settings)[source]
      
 
      Bases: idtxl.estimators_jidt.JidtKraskov
      
 
      Calculate transfer entropy with JIDT’s Kraskov implementation.
      
 
      Calculate transfer entropy between a source and a target variable using
@@ -3625,8 +4758,8 @@ 
      Submodules
-
      
-estimate(source, target)[source]
      
+
      
+estimate(source, target)[source]
      
 
      Estimate transfer entropy from a source to a target variable.
      
 
      
 
      Args:
      
@@ -3650,8 +4783,8 @@ 
      Submodules
-
      
-idtxl.estimators_jidt.common_estimate_surrogates_analytic(estimator, n_perm=200, **data)[source]
      
+
      
+idtxl.estimators_jidt.common_estimate_surrogates_analytic(estimator, n_perm=200, **data)[source]
      
 
      Estimate the surrogate distribution analytically for JidtEstimator.
      
 
      Estimate the surrogate distribution analytically for a JidtEstimator
 which is_analytic_null_estimator(), by sampling estimates at random
@@ -3677,12 +4810,12 @@ 
      Submodules
+
      
+
      
 
      idtxl.estimators_opencl module
      
 
      
-
      
-class idtxl.estimators_opencl.OpenCLKraskov(settings=None)[source]
      
+
      
+class idtxl.estimators_opencl.OpenCLKraskov(settings=None)[source]
      
 
      Bases: idtxl.estimator.Estimator
      
 
      Abstract class for implementation of OpenCL estimators.
      
 
      Abstract class for implementation of OpenCL estimators, child classes
@@ -3735,8 +4868,8 @@ 
      Submodules
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -3747,8 +4880,8 @@ 
      Submodules
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -3761,8 +4894,8 @@ 
      Submodules
-
      
-class idtxl.estimators_opencl.OpenCLKraskovCMI(settings=None)[source]
      
+
      
+class idtxl.estimators_opencl.OpenCLKraskovCMI(settings=None)[source]
      
 
      Bases: idtxl.estimators_opencl.OpenCLKraskov
      
 
      Calculate conditional mutual inform with OpenCL Kraskov implementation.
      
 
      Calculate the conditional mutual information (CMI) between three variables
@@ -3795,8 +4928,8 @@ 
      Submodules
-
      
-estimate(var1, var2, conditional=None, n_chunks=1)[source]
      
+
      
+estimate(var1, var2, conditional=None, n_chunks=1)[source]
      
 
      Estimate conditional mutual information.
      
 
      If conditional is None, the mutual information between var1 and var2 is
 calculated.
      
@@ -3831,8 +4964,8 @@ 
      Submodules
-
      
-class idtxl.estimators_opencl.OpenCLKraskovMI(settings=None)[source]
      
+
      
+class idtxl.estimators_opencl.OpenCLKraskovMI(settings=None)[source]
      
 
      Bases: idtxl.estimators_opencl.OpenCLKraskov
      
 
      Calculate mutual information with OpenCL Kraskov implementation.
      
 
      Calculate the mutual information (MI) between two variables using OpenCL
@@ -3865,8 +4998,8 @@ 
      Submodules
-
      
-estimate(var1, var2, n_chunks=1)[source]
      
+
      
+estimate(var1, var2, n_chunks=1)[source]
      
 
      Estimate mutual information.
      
 
      
 
      Args:
      
@@ -3896,8 +5029,8 @@ 
      Submodules
+
      
+
      
 
      idtxl.estimators_multivariate_pid module
      
 
      Multivariate Partical information decomposition for discrete random variables.
      
 
      This module provides an estimator for multivariate partial information
@@ -3908,8 +5041,8 @@ 
      Submoduleshttp://arxiv.org/abs/2002.03356
      
 
 
      
-
      
-class idtxl.estimators_multivariate_pid.SxPID(settings)[source]
      
+
      
+class idtxl.estimators_multivariate_pid.SxPID(settings)[source]
      
 
      Bases: idtxl.estimator.Estimator
      
 
      Estimate partial information decomposition for multiple inputs.
      
 
      Implementation of the multivariate partial information decomposition (PID)
@@ -3939,8 +5072,8 @@ 
      Submodules
-
      
-estimate(s, t)[source]
      
+
      
+estimate(s, t)[source]
      
 
      
 
      Args:
      
 
      slist of numpy arrays
      1D arrays containing realizations of a discrete random variable
      
@@ -3967,8 +5100,8 @@ 
      Submodules
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -3979,8 +5112,8 @@ 
      Submodules
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -3992,8 +5125,8 @@ 
      Submodules
+
      
+
      
 
      idtxl.estimators_pid module
      
 
      Partical information decomposition for discrete random variables.
      
 
      This module provides an estimator for partial information decomposition
@@ -4001,8 +5134,8 @@ 
      Submoduleshttp://doi.org/10.3390/e16042161
      
 
      
-
      
-class idtxl.estimators_pid.SydneyPID(settings)[source]
      
+
      
+class idtxl.estimators_pid.SydneyPID(settings)[source]
      
 
      Bases: idtxl.estimator.Estimator
      
 
      Estimate partial information decomposition of discrete variables.
      
 
      Fast implementation of the BROJA partial information decomposition (PID)
@@ -4062,8 +5195,8 @@ 
      Submodules
-
      
-estimate(s1, s2, t)[source]
      
+
      
+estimate(s1, s2, t)[source]
      
 
      
 
      Args:
      
 
      s1numpy array
      1D array containing realizations of a discrete random variable
      
@@ -4084,8 +5217,8 @@ 
      Submodules
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -4096,8 +5229,8 @@ 
      Submodules
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -4110,8 +5243,8 @@ 
      Submodules
-
      
-class idtxl.estimators_pid.TartuPID(settings)[source]
      
+
      
+class idtxl.estimators_pid.TartuPID(settings)[source]
      
 
      Bases: idtxl.estimator.Estimator
      
 
      Estimate partial information decomposition for two inputs and one output
      
 
      Implementation of the partial information decomposition (PID) estimator for
@@ -4144,8 +5277,8 @@ 
      Submodules
-
      
-estimate(s1, s2, t)[source]
      
+
      
+estimate(s1, s2, t)[source]
      
 
      
 
      Args:
      
 
      s1numpy array
      1D array containing realizations of a discrete random variable
      
@@ -4165,8 +5298,8 @@ 
      Submodules
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -4177,8 +5310,8 @@ 
      Submodules
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -4190,132 +5323,13 @@ 
      Submodules
-
      idtxl.idtxl_import module
      
-
      Import external file formats into IDTxl.
      
-
      Provide functions to import the following into IDTxl:
      
-
      
-
        
-
      • mat-files (version>7.3, hdf5)
        • 
-
      • FieldTrip-style mat-files (version>7.3, hdf5)
        • 
-
      
-
      
-
      Matlab supports hdf5 only for files saved as version 7.3 or higher:
-https://au.mathworks.com/help/matlab/ref/save.html#inputarg_version
      
-
      Creates a numpy array usable as input to IDTxl.
      
-
      
-
      Methods:
      ft_trial_2_numpyarray(file_name, ft_struct_name)
-matarray2idtxlconverter(file_name, array_name, order) =     takes a file_name,
      
-
      
-
      the name of the array variable (array_name) inside,
-and the order of sensor axis,  time axisand (CHECK THIS!!)
-repetition axis (as a list)
      
-
      
-
      
-
      Note:
      Written for Python 3.4+
      
-
      
-
      
-
      Created on Wed Mar 19 12:34:36 2014
      
-
      @author: Michael Wibral
      
-
      
-
      
-idtxl.idtxl_import.import_fieldtrip(file_name, ft_struct_name, file_version, normalise=True)[source]
      
-
      Convert FieldTrip-style MATLAB-file into an IDTxl Data object.
      
-
      Import a MATLAB structure with fields  “trial” (data), “label” (channel
-labels), “time” (time stamps for data samples), and “fsample” (sampling
-rate). This structure is the standard file format in the MATLAB toolbox
-FieldTrip and commonly use to represent neurophysiological data (see also
-http://www.fieldtriptoolbox.org/reference/ft_datatype_raw). The data is
-returned as a IDTxl Data() object.
      
-
      The structure is assumed to be saved as a matlab hdf5 file (“-v7.3’ or
-higher, .mat) with a SINGLE FieldTrip data structure inside.
      
-
      
-
      Args:
      
-
      file_namestring
      full (matlab) file_name on disk
      
-
      
-
      ft_struct_namestring
      variable name of the MATLAB structure that is in FieldTrip format
-(autodetect will hopefully be possible later …)
      
-
      
-
      file_versionstring
      version of the file, e.g. ‘v7.3’ for MATLAB’s 7.3 format
      
-
      
-
      normalisebool [optional]
      normalise data after import (default=True)
      
-
      
-
      
-
      
-
      Returns:
      
-
      Data() instance
      instance of IDTxl Data object, containing data from the ‘trial’
-field
      
-
      
-
      list of strings
      list of channel labels, corresponding to the ‘label’ field
      
-
      
-
      numpy array
      time stamps for samples, corresponding to one entry in the ‘time’
-field
      
-
      
-
      int
      sampling rate, corresponding to the ‘fsample’ field
      
-
      
-
      
-
      
-
      
-
      @author: Michael Wibral
      
-
      
-
-
      
-
      
-idtxl.idtxl_import.import_matarray(file_name, array_name, file_version, dim_order, normalise=True)[source]
      
-
      Read Matlab hdf5 file into IDTxl.
      
-
      reads a matlab hdf5 file (“-v7.3’ or higher, .mat) with a SINGLE
-array inside and returns a numpy array with dimensions that
-are channel x time x trials, using np.swapaxes where necessary
      
-
      
-
      Note:
      The import function squeezes the loaded mat-file, i.e., any singleton
-dimension will be removed. Hence do not enter singleton dimension into
-the ‘dim_order’, e.g., don’t pass dim_order=’ps’ but dim_order=’s‘ if
-you want to load a 1D-array where entries represent samples recorded
-from a single channel.
      
-
      
-
      Args:
      
-
      file_namestring
      full (matlab) file_name on disk
      
-
      
-
      array_namestring
      variable name of the MATLAB structure to be read
      
-
      
-
      file_versionstring
      version of the file, e.g. ‘v7.3’ for MATLAB’s 7.3 format, currently
-versions ‘v4’, ‘v6’, ‘v7’, and ‘v7’ are supported
      
-
      
-
      dim_orderstring
      order of dimensions, accepts any combination of the characters
-‘p’, ‘s’, and ‘r’ for processes, samples, and replications; must
-have the same length as the data dimensionality, e.g., ‘ps’ for a
-two-dimensional array of data from several processes over time
      
-
      
-
      normalisebool [optional]
      normalise data after import (default=True)
      
-
      
-
      
-
      
-
      Returns:
      
-
      Data() instance
      instance of IDTxl Data object, containing data from the ‘trial’
-field
      
-
      
-
      list of strings
      list of channel labels, corresponding to the ‘label’ field
      
-
      
-
      numpy array
      time stamps for samples, corresponding to one entry in the ‘time’
-field
      
-
      
-
      int
      sampling rate, corresponding to the ‘fsample’ field
      
-
      
-
      
-
      
-
      
-
      Created on Wed Mar 19 12:34:36 2014
      
-
      @author: Michael Wibral
      
-
      
-
-
      
-
      
+
+
      
 
      idtxl.idtxl_exceptions module
      
 
      Provide error handling and warnings.
      
 
      
-
      
-exception idtxl.idtxl_exceptions.AlgorithmExhaustedError(message)[source]
      
+
      
+exception idtxl.idtxl_exceptions.AlgorithmExhaustedError(message)[source]
      
 
      Bases: Exception
      
 
      Exception raised to signal that the estimators can no longer be used
 for this particular target (e.g. because of memory errors in high
@@ -4327,14 +5341,14 @@ 
      Submodules
-
      
-exception idtxl.idtxl_exceptions.BROJA_2PID_Exception[source]
      
+
      
+exception idtxl.idtxl_exceptions.BROJA_2PID_Exception[source]
      
 
      Bases: Exception
      
 
      
 
 
      
-
      
-exception idtxl.idtxl_exceptions.JidtOutOfMemoryError(message)[source]
      
+
      
+exception idtxl.idtxl_exceptions.JidtOutOfMemoryError(message)[source]
      
 
      Bases: idtxl.idtxl_exceptions.AlgorithmExhaustedError
      
 
      
 
      Exception raised to signal a Java OutOfMemoryException.
      It is a child class of AlgorithmExhaustedError.
      
@@ -4345,20 +5359,20 @@ 
      Submodules
-
      
-idtxl.idtxl_exceptions.package_missing(err, message)[source]
      
+
      
+idtxl.idtxl_exceptions.package_missing(err, message)[source]
      
 
      Report a missing optional package upon import.
      
 
      
 
-
      
-
      
+
+
      
 
      idtxl.idtxl_io module
      
 
      Provide I/O functionality.
      
 
      Provide functions to load and save IDTxl data, provide import functions (e.g.,
 mat-files, FieldTrip) and export functions (e.g., networkx, BrainNet Viewer).
      
 
      
-
      
-idtxl.idtxl_io.export_brain_net_viewer(adjacency_matrix, mni_coord, file_name, **kwargs)[source]
      
+
      
+idtxl.idtxl_io.export_brain_net_viewer(adjacency_matrix, mni_coord, file_name, **kwargs)[source]
      
 
      Export network to BrainNet Viewer.
      
 
      Export networks to BrainNet Viewer (project home page:
 http://www.nitrc.org/projects/bnv/). BrainNet Viewer is a MATLAB
@@ -4399,8 +5413,8 @@ 
      Submodules
-
      
-idtxl.idtxl_io.export_networkx_graph(adjacency_matrix, weights)[source]
      
+
      
+idtxl.idtxl_io.export_networkx_graph(adjacency_matrix, weights)[source]
      
 
      Export networkx graph object for an inferred network.
      
 
      Export a weighted, directed graph object from the network of inferred
 (multivariate) interactions (e.g., multivariate TE), using the networkx
@@ -4424,8 +5438,8 @@ 
      Submodules
-
      
-idtxl.idtxl_io.export_networkx_source_graph(results, target, sign_sources=True, fdr=True)[source]
      
+
      
+idtxl.idtxl_io.export_networkx_source_graph(results, target, sign_sources=True, fdr=True)[source]
      
 
      Export graph object of source variables for a single target.
      
 
      Export graph object from the network of (multivariate) interactions (e.g.,
 multivariate TE) between single source variables and a target process using
@@ -4455,8 +5469,8 @@ 
      Submodules
-
      
-idtxl.idtxl_io.import_fieldtrip(file_name, ft_struct_name, file_version, normalise=True)[source]
      
+
      
+idtxl.idtxl_io.import_fieldtrip(file_name, ft_struct_name, file_version, normalise=True)[source]
      
 
      Convert FieldTrip-style MATLAB-file into an IDTxl Data object.
      
 
      Import a MATLAB structure with fields  “trial” (data), “label” (channel
 labels), “time” (time stamps for data samples), and “fsample” (sampling
@@ -4496,15 +5510,15 @@ 
      Submodules
-
      
-idtxl.idtxl_io.import_matarray(file_name, array_name, file_version, dim_order, normalise=True)[source]
      
+
      
+idtxl.idtxl_io.import_matarray(file_name, array_name, file_version, dim_order, normalise=True)[source]
      
 
      Read Matlab hdf5 file into IDTxl.
      
 
      reads a matlab hdf5 file (“-v7.3’ or higher, .mat) or non-hdf5 files with a
 SINGLE array inside and returns an IDTxl Data() object.
      
 
      
 
      Note:
      The import function squeezes the loaded mat-file, i.e., any singleton
 dimension will be removed. Hence do not enter singleton dimension into
-the ‘dim_order’, e.g., don’t pass dim_order=’ps’ but dim_order=’s‘ if
+the ‘dim_order’, e.g., don’t pass dim_order=’ps’ but dim_order=’s’ if
 you want to load a 1D-array where entries represent samples recorded
 from a single channel.
      
 
      
@@ -4535,8 +5549,8 @@ 
      Submodules
-
      
-idtxl.idtxl_io.load_json(file_path)[source]
      
+
      
+idtxl.idtxl_io.load_json(file_path)[source]
      
 
      Load dictionary saved as JSON file from disk.
      
 
      
 
      Args:
      
@@ -4554,14 +5568,14 @@ 
      Submodules
-
      
-idtxl.idtxl_io.load_pickle(name)[source]
      
+
      
+idtxl.idtxl_io.load_pickle(name)[source]
      
 
      Load objects that have been saved using Python’s pickle module.
      
 
      
 
 
      
-
      
-idtxl.idtxl_io.save_json(d, file_path)[source]
      
+
      
+idtxl.idtxl_io.save_json(d, file_path)[source]
      
 
      Save dictionary to disk as JSON file.
      
 
      Writes dictionary to disk at the specified file path.
      
 
      
@@ -4578,8 +5592,8 @@ 
      Submodules
-
      
-idtxl.idtxl_io.save_pickle(obj, name)[source]
      
+
      
+idtxl.idtxl_io.save_pickle(obj, name)[source]
      
 
      Save objects using Python’s pickle module.
      
 
      
 
      Note:
      pickle.HIGHEST_PROTOCOL is a binary format, which may be inconvenient,
@@ -4588,31 +5602,31 @@ 
      Submodules
+
      
+
      
 
      idtxl.idtxl_utils module
      
 
      Provide IDTxl utility functions.
      
 
      
-
      
-idtxl.idtxl_utils.argsort_descending(a)[source]
      
+
      
+idtxl.idtxl_utils.argsort_descending(a)[source]
      
 
      Sort array in descending order and return sortind indices.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.autocorrelation(x)[source]
      
+
      
+idtxl.idtxl_utils.autocorrelation(x)[source]
      
 
      Calculate autocorrelation of a vector.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.calculate_mi(corr)[source]
      
+
      
+idtxl.idtxl_utils.calculate_mi(corr)[source]
      
 
      Calculate mutual information from correlation coefficient.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.combine_discrete_dimensions(a, numBins)[source]
      
+
      
+idtxl.idtxl_utils.combine_discrete_dimensions(a, numBins)[source]
      
 
      Combine multi-dimensional discrete variable into a single dimension.
      
 
      Combine all dimensions for a discrete variable down into a single
 dimensional value for each sample. This is done basically by multiplying
@@ -4638,8 +5652,8 @@ 
      Submodules
-
      
-idtxl.idtxl_utils.conflicting_entries(dict_1, dict_2)[source]
      
+
      
+idtxl.idtxl_utils.conflicting_entries(dict_1, dict_2)[source]
      
 
      Test two dictionaries for unequal entries.
      
 
      Note that only keys that are present in both dicts are compared. If one
 dictionary contains an entry not present in the other dictionary, the
@@ -4647,8 +5661,8 @@ 
      Submodules
-
      
-idtxl.idtxl_utils.discretise(a, numBins)[source]
      
+
      
+idtxl.idtxl_utils.discretise(a, numBins)[source]
      
 
      Discretise continuous data.
      
 
      Discretise continuous data into discrete values (with 0 as lowest) by
 evenly partitioning the range of the data, one dimension at a time.
@@ -4672,8 +5686,8 @@ 
      Submodules
-
      
-idtxl.idtxl_utils.discretise_max_ent(a, numBins)[source]
      
+
      
+idtxl.idtxl_utils.discretise_max_ent(a, numBins)[source]
      
 
      Discretise continuous data using maximum entropy partitioning.
      
 
      Discretise continuous data into discrete values (with 0 as lowest) by
 making a maximum entropy partitioning, one dimension at a time. Adapted
@@ -4697,20 +5711,20 @@ 
      Submodules
-
      
-idtxl.idtxl_utils.equal_dicts(dict_1, dict_2)[source]
      
+
      
+idtxl.idtxl_utils.equal_dicts(dict_1, dict_2)[source]
      
 
      Test two dictionaries for equality.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.print_dict(d, indent=4)[source]
      
+
      
+idtxl.idtxl_utils.print_dict(d, indent=4)[source]
      
 
      Use Python’s pretty printer to print dictionaries to the console.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.remove_column(a, j)[source]
      
+
      
+idtxl.idtxl_utils.remove_column(a, j)[source]
      
 
      Remove a column from a numpy array.
      
 
      This is faster than logical indexing (‘25 times faster’), because it does
 not make copies, see
@@ -4727,8 +5741,8 @@ 
      Submodules
-
      
-idtxl.idtxl_utils.remove_row(a, i)[source]
      
+
      
+idtxl.idtxl_utils.remove_row(a, i)[source]
      
 
      Remove a row from a numpy array.
      
 
      This is faster than logical indexing (‘25 times faster’), because it does
 not make copies, see
@@ -4745,8 +5759,8 @@ 
      Submodules
-
      
-idtxl.idtxl_utils.separate_arrays(idx_all, idx_single, a)[source]
      
+
      
+idtxl.idtxl_utils.separate_arrays(idx_all, idx_single, a)[source]
      
 
      Separate a single column from all other columns in a 2D-array.
      
 
      Return the separated single column and the remaining columns of a 2D-
 array.
      
@@ -4773,14 +5787,14 @@ 
      Submodules
-
      
-idtxl.idtxl_utils.sort_descending(a)[source]
      
+
      
+idtxl.idtxl_utils.sort_descending(a)[source]
      
 
      Sort array in descending order.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.standardise(a, dimension=0, df=1)[source]
      
+
      
+idtxl.idtxl_utils.standardise(a, dimension=0, df=1)[source]
      
 
      Z-standardise a numpy array along a given dimension.
      
 
      Standardise array along the axis defined in dimension using the denominator
 (N - df) for the calculation of the standard deviation.
      
@@ -4803,8 +5817,8 @@ 
      Submodules
-
      
-idtxl.idtxl_utils.swap_chars(s, i_1, i_2)[source]
      
+
      
+idtxl.idtxl_utils.swap_chars(s, i_1, i_2)[source]
      
 
      Swap to characters in a string.
      
 
      
 
      Example:
      >>> print(swap_chars('heLlotHere', 2, 6))
@@ -4815,25 +5829,25 @@ 
      Submodules
+
      
+
      
 
      idtxl.network_analysis module
      
 
      Parent class for network inference and network comparison.
      
 
      
-
      
-class idtxl.network_analysis.NetworkAnalysis[source]
      
+
      
+class idtxl.network_analysis.NetworkAnalysis[source]
      
 
      Bases: object
      
 
      Provide an analysis setup for network inference or comparison.
      
 
      The class provides routines to check user input and set defaults.
      
-
      
-
      
-property current_value
      
+
      
+
      
+property current_value
      
 
      Get index of the current_value.
      
 
      
 
 
      
-
      
-resume_checkpoint(file_path)[source]
      
+
      
+resume_checkpoint(file_path)[source]
      
 
      Resume analysis from a checkpoint saved to disk.
      
 
      
 
      Args:
      
@@ -4844,33 +5858,33 @@ 
      Submodules
-
      
-property selected_vars_full
      
+
      
+
      
+property selected_vars_full
      
 
      List of indices of the full conditional set.
      
 
      
 
-
      
-
      
-property selected_vars_sources
      
+
      
+
      
+property selected_vars_sources
      
 
      List of indices of source samples in the conditional set.
      
 
      
 
-
      
-
      
-property selected_vars_target
      
+
      
+
      
+property selected_vars_target
      
 
      List of indices of target samples in the conditional set.
      
 
      
 
 
      
 
-
      
-
      
+
+
      
 
      idtxl.network_inference module
      
 
      Parent class for all network inference.
      
 
      
-
      
-class idtxl.network_inference.NetworkInference[source]
      
+
      
+class idtxl.network_inference.NetworkInference[source]
      
 
      Bases: idtxl.network_analysis.NetworkAnalysis
      
 
      Parent class for network inference algorithms.
      
 
      Hold variables that are relevant for network inference using for example
@@ -4896,50 +5910,50 @@ 
      Submodules
-
      
-class idtxl.network_inference.NetworkInferenceBivariate[source]
      
+
      
+class idtxl.network_inference.NetworkInferenceBivariate[source]
      
 
      Bases: idtxl.network_inference.NetworkInference
      
 
      Parent class for bivariate network inference algorithms.
      
 
      
 
 
      
-
      
-class idtxl.network_inference.NetworkInferenceMI[source]
      
+
      
+class idtxl.network_inference.NetworkInferenceMI[source]
      
 
      Bases: idtxl.network_inference.NetworkInference
      
 
      Parent class for mutual information network inference algorithms.
      
 
      
 
 
      
-
      
-class idtxl.network_inference.NetworkInferenceMultivariate[source]
      
+
      
+class idtxl.network_inference.NetworkInferenceMultivariate[source]
      
 
      Bases: idtxl.network_inference.NetworkInference
      
 
      Parent class for multivariate network inference algorithms.
      
 
      
 
 
      
-
      
-class idtxl.network_inference.NetworkInferenceTE[source]
      
+
      
+class idtxl.network_inference.NetworkInferenceTE[source]
      
 
      Bases: idtxl.network_inference.NetworkInference
      
 
      Parent class for transfer entropy network inference algorithms.
      
 
      
 
-
      
-
      
+
+
      
 
      idtxl.single_process_analysis module
      
 
      Parent class for analysis of single processes in the network.
      
 
      
-
      
-class idtxl.single_process_analysis.SingleProcessAnalysis[source]
      
+
      
+class idtxl.single_process_analysis.SingleProcessAnalysis[source]
      
 
      Bases: idtxl.network_analysis.NetworkAnalysis
      
 
      
 
-
      
-
      
+
+
      
 
      idtxl.network_comparison module
      
 
      Perform inference statistics on groups of data.
      
 
      
-
      
-class idtxl.network_comparison.NetworkComparison[source]
      
+
      
+class idtxl.network_comparison.NetworkComparison[source]
      
 
      Bases: idtxl.network_analysis.NetworkAnalysis
      
 
      Set up network comparison between two experimental conditions.
      
 
      The class provides methods for the comparison of networks inferred from
@@ -4985,8 +5999,8 @@ 
      Submodules
-
+
 
      Calculate the information transfer for whole links into a target.
      
 
      Calculate the information transfer for whole links as the joint
 information transfer from all variables selected for a single source
@@ -5016,8 +6030,8 @@ 
      Submodules
-
      
-compare_between(settings, network_set_a, network_set_b, data_set_a, data_set_b)[source]
      
+
      
+compare_between(settings, network_set_a, network_set_b, data_set_a, data_set_b)[source]
      
 
      Compare networks inferred under two conditions between subjects.
      
 
      Compare two sets of networks inferred from two sets of data recorded
 under different experimental conditions within multiple subjects, i.e.,
@@ -5050,8 +6064,8 @@ 
      Submodules
-
+
 
      Compare two links within the same network.
      
 
      Compare two links within the same network. Check if information
 transfer is different from information transfer in a second link.
      
@@ -5100,8 +6114,8 @@ 
      Submodules
-
      
-compare_within(settings, network_a, network_b, data_a, data_b)[source]
      
+
      
+compare_within(settings, network_a, network_b, data_a, data_b)[source]
      
 
      Compare networks inferred under two conditions within one subject.
      
 
      Compare two networks inferred from data recorded under two different
 experimental conditions within one subject (units of observations are
@@ -5150,30 +6164,30 @@ 
      Submodules
+
      
+
      
 
      idtxl.results module
      
 
      Provide results class for IDTxl network analysis.
      
 
      
-
      
-class idtxl.results.AdjacencyMatrix(n_nodes, weight_type)[source]
      
+
      
+class idtxl.results.AdjacencyMatrix(n_nodes, weight_type)[source]
      
 
      Bases: object
      
 
      Adjacency matrix representing inferred networks.
      
 
      
-
      
-add_edge(i, j, weight)[source]
      
+
      
+add_edge(i, j, weight)[source]
      
 
      Add weighted edge (i, j) to adjacency matrix.
      
 
      
 
 
      
-
      
-add_edge_list(i_list, j_list, weights)[source]
      
+
      
+add_edge_list(i_list, j_list, weights)[source]
      
 
      Add multiple weighted edges (i, j) to adjacency matrix.
      
 
      
 
 
      
-
      
-get_edge_list()[source]
      
+
      
+get_edge_list()[source]
      
 
      Return list of weighted edges.
      
 
      
 
      Returns
      
@@ -5185,33 +6199,33 @@ 
      Submodules
-
      
-n_edges()[source]
      
+
      
+n_edges()[source]
      
 
      
 
 
      
-
      
-n_nodes()[source]
      
+
      
+n_nodes()[source]
      
 
      Return number of nodes.
      
 
      
 
 
      
-
      
-print_matrix()[source]
      
+
      
+print_matrix()[source]
      
 
      Print weight and edge matrix.
      
 
      
 
 
      
 
 
      
-
      
-class idtxl.results.DotDict[source]
      
+
      
+class idtxl.results.DotDict[source]
      
 
      Bases: dict
      
 
      Dictionary with dot-notation access to values.
      
 
      Provides the same functionality as a regular dict, but also allows
 accessing values using dot-notation.
      
 
      Example:
      
-
      >>> from idtxl.results import DotDict
+
      >>> from idtxl.results import DotDict
 >>> d = DotDict({'a': 1, 'b': 2})
 >>> d.a
 >>> # Out: 1
@@ -5222,8 +6236,8 @@ 
      Submodules
-
      
-class idtxl.results.Results(n_nodes, n_realisations, normalised)[source]
      
+
      
+class idtxl.results.Results(n_nodes, n_realisations, normalised)[source]
      
 
      Bases: object
      
 
      Parent class for results of network analysis algorithms.
      
 
      Provide a container for results of network analysis algorithms, e.g.,
@@ -5250,8 +6264,8 @@ 
      Submodules
-
      
-combine_results(*results)[source]
      
+
      
+combine_results(*results)[source]
      
 
      Combine multiple (partial) results objects.
      
 
      Combine a list of partial network analysis results into a single
 results object (e.g., results from analysis parallelized over
@@ -5283,8 +6297,8 @@ 
      Submodules
-
      
-class idtxl.results.ResultsMultivariatePID(n_nodes, n_realisations, normalised)[source]
      
+
      
+class idtxl.results.ResultsMultivariatePID(n_nodes, n_realisations, normalised)[source]
      
 
      Bases: idtxl.results.ResultsNetworkAnalysis
      
 
      Store results of Multivariate Partial Information Decomposition (PID)
 analysis.
      
@@ -5325,8 +6339,8 @@ 
      Submodules
-
      
-get_single_target(target)[source]
      
+
      
+get_single_target(target)[source]
      
 
      Return results for a single target in the network.
      
 
      Results for single targets include for each target
      
 
        
@@ -5362,12 +6376,12 @@ 
        Submodules
-
        
-class idtxl.results.ResultsNetworkAnalysis(n_nodes, n_realisations, normalised)[source]
        
+
        
+class idtxl.results.ResultsNetworkAnalysis(n_nodes, n_realisations, normalised)[source]
        
 
        Bases: idtxl.results.Results
        
 
        
-
        
-get_single_target(target, fdr=True)[source]
        
+
        
+get_single_target(target, fdr=True)[source]
        
 
        Return results for a single target in the network.
        
 
        Return results for individual processes, contains for each process
        
 
        Results for single targets include for each target
        
@@ -5413,8 +6427,8 @@ 
        Submodules
-
        
-get_target_sources(target, fdr=True)[source]
        
+
        
+get_target_sources(target, fdr=True)[source]
        
 
        Return list of sources (parents) for given target.
        
 
        
 
        Args:
        
@@ -5428,17 +6442,17 @@ 
        Submodules
-
        
-property targets_analysed
        
+
        
+
        
+property targets_analysed
        
 
        Get index of the current_value.
        
 
        
 
 
        
 
 
        
-
        
-class idtxl.results.ResultsNetworkComparison(n_nodes, n_realisations, normalised)[source]
        
+
        
+class idtxl.results.ResultsNetworkComparison(n_nodes, n_realisations, normalised)[source]
        
 
        Bases: idtxl.results.ResultsNetworkAnalysis
        
 
        Store results of network comparison.
        
 
        Provide a container for results of network comparison algorithms.
        
@@ -5484,8 +6498,8 @@ 
        Submodules
-
        
-get_adjacency_matrix(weights='comparison')[source]
        
+
        
+get_adjacency_matrix(weights='comparison')[source]
        
 
        Return adjacency matrix.
        
 
        Return adjacency matrix resulting from network inference.
 Multiple options for the weights are available.
        
@@ -5519,8 +6533,8 @@ 
        Submodules
-
        
-get_single_target(target)[source]
        
+
        
+get_single_target(target)[source]
        
 
        Return results for a single target in the network.
        
 
        Results for single targets include for each target
        
 
          
@@ -5551,8 +6565,8 @@ 
          Submodules
-
          
-get_target_sources(target)[source]
          
+
          
+get_target_sources(target)[source]
          
 
          Return list of sources (parents) for given target.
          
 
          
 
          Args:
          
@@ -5564,8 +6578,8 @@ 
          Submodules
-
          
-print_edge_list(weights='comparison')[source]
          
+
          
+print_edge_list(weights='comparison')[source]
          
 
          Print results of network comparison to console.
          
 
          Print results of network comparison to console. Output looks like this:
          
 
          >>> 0 -> 1, diff_abs = 0.2
@@ -5590,8 +6604,8 @@ 
          Submodules
-
          
-class idtxl.results.ResultsNetworkInference(n_nodes, n_realisations, normalised)[source]
          
+
          
+class idtxl.results.ResultsNetworkInference(n_nodes, n_realisations, normalised)[source]
          
 
          Bases: idtxl.results.ResultsNetworkAnalysis
          
 
          Store results of network inference.
          
 
          Provide a container for results of network inference algorithms, e.g.,
@@ -5629,8 +6643,8 @@ 
          Submodules
-
          
-get_adjacency_matrix(weights, fdr=True)[source]
          
+
          
+get_adjacency_matrix(weights, fdr=True)[source]
          
 
          Return adjacency matrix.
          
 
          Return adjacency matrix resulting from network inference. The adjacency
 matrix can either be generated from FDR-corrected results or
@@ -5677,8 +6691,8 @@ 
          Submodules
-
          
-get_target_delays(target, criterion='max_te', fdr=True)[source]
          
+
          
+get_target_delays(target, criterion='max_te', fdr=True)[source]
          
 
          Return list of information-transfer delays for a given target.
          
 
          Return a list of information-transfer delays for a given target.
 Information-transfer delays are determined by the lag of the variable
@@ -5713,8 +6727,8 @@ 
          Submodules
-
          
-print_edge_list(weights, fdr=True)[source]
          
+
          
+print_edge_list(weights, fdr=True)[source]
          
 
          Print results of network inference to console.
          
 
          Print edge list resulting from network inference to console.
 Output may look like this:
          
@@ -5744,8 +6758,8 @@ 
          Submodules
-
          
-class idtxl.results.ResultsPID(n_nodes, n_realisations, normalised)[source]
          
+
          
+class idtxl.results.ResultsPID(n_nodes, n_realisations, normalised)[source]
          
 
          Bases: idtxl.results.ResultsNetworkAnalysis
          
 
          Store results of Partial Information Decomposition (PID) analysis.
          
 
          Provide a container for results of Partial Information Decomposition (PID)
@@ -5783,8 +6797,8 @@ 
          Submodules
-
          
-get_single_target(target)[source]
          
+
          
+get_single_target(target)[source]
          
 
          Return results for a single target in the network.
          
 
          Results for single targets include for each target
          
 
          
 
+
          
+
          
+class idtxl.results.ResultsSingleProcessRudelt(processes)[source]
          
+
          Bases: object
          
+
          Store results of single process analysis.
          
+
          Provides a container for the results Rudelt optimization algorithm. To
+obtain results for individual processes, call the .get_single_process()
+method (see docstring for details).
          
+
          Note that for convenience all dictionaries in this class can additionally
+be accessed using dot-notation:
          
+
          >>> res_network.settings.estimation_method
+
          
+
          
+
          or
          
+
          >>> res_network.settings['estimation_method'].
+
          
 
          
-
          
+
          
+
          Attributes:
          
+
          settingsdict
          settings used for estimation of information theoretic measures
          
+
          
+
          data_propertiesdict
          
+
          data properties, contains
            
+
          • n_processes : int - total number of processes analysed
            • 
+
          
+
          
+
          
+
          
+
          processes_analysedlist
          list of analysed processes
          
+
          
+
          
+
          
+
          
+
          
+
          
+get_single_process(process)[source]
          
+
          Return results for a single process.
          
+
          Return results for individual processes, contains for each process
          
+
          
+
          Args:
          
+
          processint
          process id
          
+
          
+
          
+
          
+
          Returns:
          
+
          dict
          results for single process. Note that for convenience
+dictionary entries can either be accessed via keywords
+(result[‘selected_vars’]) or via dot-notation
+(result.selected_vars). Contains keys
          
+
          
+
          Processint
          Process that was optimized
          
+
          
+
          estimation_methodString
          Estimation method that was used for optimization
          
+
          
+
          T_Dfloat
          Estimated optimal value for the temporal depth TD
          
+
          
+
          tau_R :
          Information timescale tau_R, a characteristic timescale of history
+dependence similar to an autocorrelation time.
          
+
          
+
          R_totfloat
          Estimated value for the total history dependence Rtot,
          
+
          
+
          AIS_totfloat
          Estimated value for the total active information storage
          
+
          
+
          opt_number_of_bins_dint
          Number of bins d for the embedding that yields (R̂tot ,T̂D)
          
+
          
+
          opt_scaling_kint
          Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
          
+
          
+
          opt_first_bin_sizeint
          Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
          
+
          
+
          history_dependencearray with floating-point values
          Estimated history dependence for each embedding
          
+
          
+
          firing_ratefloat
          Firing rate of the neuron/ spike train
          
+
          
+
          recording_lengthfloat
          Length of the recording (in seconds)
          
+
          
+
          H_spikingfloat
          Entropy of the spike times
          
+
          
+
          
+
          if analyse_auto_MI was set to True additionally:
+auto_MI : dict
          
+
          
+
          numpy array of MI values for each delay
          
+
          
+
          
+
          auto_MI_delayslist of int
          list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay
          
+
          
+
          
+
          
+
          
+
          
+
          
+
          
+
+
          
+
          
+property processes_analysed
          
+
          Get index of the current_value.
          
+
          
+
+
          
+
+
      
+
      
 
      idtxl.stats module
      
 
      Provide statistics functions.
      
 
      
-
      
-idtxl.stats.ais_fdr(settings=None, *results)[source]
      
+
      
+idtxl.stats.ais_fdr(settings=None, *results)[source]
      
 
      Perform FDR-correction on results of network AIS estimation.
      
 
      Perform correction of the false discovery rate (FDR) after estimation of
 active information storage (AIS) for all processes in the network. FDR
@@ -5984,8 +7099,8 @@ 
      Submodules
-
      
-idtxl.stats.check_n_perm(n_perm, alpha)[source]
      
+
      
+idtxl.stats.check_n_perm(n_perm, alpha)[source]
      
 
      Check if no. permutations is big enough to obtain the requested alpha.
      
 
      
 
      Note:
      The no. permutations must be big enough to theoretically allow for the
@@ -5997,8 +7112,8 @@ 
      Submodules
-
      
-idtxl.stats.max_statistic(analysis_setup, data, candidate_set, te_max_candidate, conditional=None)[source]
      
+
      
+idtxl.stats.max_statistic(analysis_setup, data, candidate_set, te_max_candidate, conditional)[source]
      
 
      Perform maximum statistics for one candidate source.
      
 
      Test if a transfer entropy value is significantly bigger than the maximum
 values obtained from surrogates of all remanining candidates.
      
@@ -6022,9 +7137,8 @@ 
      Submodulesfloat
      transfer entropy value to be tested
      
 
      
-
      conditionalnumpy array [optional]
      realisations of conditional, 2D numpy array where array dimensions
-represent [realisations x variable dimension] (default=None, no
-conditioning performed)
      
+
      conditionalnumpy array
      realisations of conditional, 2D numpy array where array dimensions
+represent [realisations x variable dimension]
      
 
      
 
      
 
      
@@ -6047,8 +7161,8 @@ 
      Submodules
-
      
-idtxl.stats.max_statistic_sequential(analysis_setup, data)[source]
      
+
      
+idtxl.stats.max_statistic_sequential(analysis_setup, data)[source]
      
 
      Perform sequential maximum statistics for a set of candidate sources.
      
 
      Test multivariate/bivariate MI/TE values against surrogates. Test highest
 TE/MI value against distribution of highest surrogate values, second
@@ -6103,8 +7217,8 @@ 
      Submodules
-
      
-idtxl.stats.max_statistic_sequential_bivariate(analysis_setup, data)[source]
      
+
      
+idtxl.stats.max_statistic_sequential_bivariate(analysis_setup, data)[source]
      
 
      Perform sequential maximum statistics for a set of candidate sources.
      
 
      Test multivariate/bivariate MI/TE values against surrogates. Test highest
 TE/MI value against distribution of highest surrogate values, second
@@ -6159,8 +7273,8 @@ 
      Submodules
-
      
-idtxl.stats.mi_against_surrogates(analysis_setup, data)[source]
      
+
      
+idtxl.stats.mi_against_surrogates(analysis_setup, data)[source]
      
 
      Test estimated mutual information for significance against surrogate data.
      
 
      Shuffle realisations of the current value (point to be predicted) and re-
 calculate mutual information (MI) for shuffled data. The actual estimated
@@ -6202,8 +7316,8 @@ 
      Submodules
-
      
-idtxl.stats.min_statistic(analysis_setup, data, candidate_set, te_min_candidate, conditional=None)[source]
      
+
      
+idtxl.stats.min_statistic(analysis_setup, data, candidate_set, te_min_candidate, conditional=None)[source]
      
 
      Perform minimum statistics for one candidate source.
      
 
      Test if a transfer entropy value is significantly bigger than the minimum
 values obtained from surrogates of all remanining candidates.
      
@@ -6252,8 +7366,8 @@ 
      Submodules
-
      
-idtxl.stats.network_fdr(settings=None, *results)[source]
      
+
      
+idtxl.stats.network_fdr(settings=None, *results)[source]
      
 
      Perform FDR-correction on results of network inference.
      
 
      Perform correction of the false discovery rate (FDR) after network
 analysis. FDR correction can either be applied at the target level
@@ -6297,8 +7411,8 @@ 
      Submodules
-
      
-idtxl.stats.omnibus_test(analysis_setup, data)[source]
      
+
      
+idtxl.stats.omnibus_test(analysis_setup, data)[source]
      
 
      Perform an omnibus test on identified conditional variables.
      
 
      Test the joint information transfer from all identified sources to the
 current value conditional on candidates in the target’s past. To test for
@@ -6342,8 +7456,8 @@ 
      Submodules
-
      
-idtxl.stats.syn_shd_against_surrogates(analysis_setup, data)[source]
      
+
      
+idtxl.stats.syn_shd_against_surrogates(analysis_setup, data)[source]
      
 
      Test the shared/synergistic information in the PID estimate.
      
 
      Shuffle realisations of the target and re-calculate PID, in particular the
 synergistic and shared information from shuffled data. The original
@@ -6382,8 +7496,8 @@ 
      Submodules
-
      
-idtxl.stats.unq_against_surrogates(analysis_setup, data)[source]
      
+
      
+idtxl.stats.unq_against_surrogates(analysis_setup, data)[source]
      
 
      Test the unique information in the PID estimate against surrogate data.
      
 
      Shuffle realisations of both sources individually and re-calculate PID,
 in particular the unique information from shuffled data. The original
@@ -6421,13 +7535,13 @@ 
      Submodules
+
      
+
      
 
      idtxl.visualise_graph module
      
 
      Plot results of network inference.
      
 
      
-
      
-idtxl.visualise_graph.plot_mute_graph()[source]
      
+
      
+idtxl.visualise_graph.plot_mute_graph()[source]
      
 
      Plot MuTE example network.
      
 
      Network of 5 AR-processes, which is used as an example the paper
 on the MuTE toolbox (Montalto, PLOS ONE, 2014, eq. 14). The
@@ -6450,8 +7564,8 @@ 
      Submodules
-
      
-idtxl.visualise_graph.plot_network(results, weights, fdr=True)[source]
      
+
      
+idtxl.visualise_graph.plot_network(results, weights, fdr=True)[source]
      
 
      Plot network of multivariate TE between processes.
      
 
      Plot graph of the network of (multivariate) interactions between processes
 (e.g., multivariate TE). The function uses  the networkx class for directed
@@ -6525,8 +7639,8 @@ 
      Submodules
-
      
-idtxl.visualise_graph.plot_network_comparison(results)[source]
      
+
      
+idtxl.visualise_graph.plot_network_comparison(results)[source]
      
 
      Plot results of network comparison.
      
 
      Plot results of network comparison. Produces a figure with five subplots,
 where the first plot shows the network graph of the union network, the
@@ -6551,8 +7665,8 @@ 
      Submodules
-
      
-idtxl.visualise_graph.plot_selected_vars(results, target, sign_sources=True, display_edge_labels=False, fdr=True)[source]
      
+
      
+idtxl.visualise_graph.plot_selected_vars(results, target, sign_sources=True, display_edge_labels=False, fdr=True)[source]
      
 
      Plot network of a target process and single variables.
      
 
      Plot graph of the network of (multivariate) interactions between source
 variables and the target. The function uses the networkx class for directed
@@ -6582,8 +7696,8 @@ 
      Submodules
+
      
+
      
 
      Module contents
      
 
      IDTxl: Information Dynamics Toolkit xl.
      
 
      IDTxl is a comprehensive software package for efficient inference of networks
@@ -6613,8 +7727,8 @@ 
      Submodules
      
@@ -6623,8 +7737,9 @@ 
      Submodules
         
      
-  
      Table of Contents
      
-  
     
        
@@ -49,11 +50,11 @@ 
      Navigation
      
         
      
           
      
             
-  
      
+  
      
 
      The Data Class
      
 
      
-
      
-class idtxl.data.Data(data=None, dim_order='psr', normalise=True, seed=None)[source]
      
+
      
+class idtxl.data.Data(data=None, dim_order='psr', normalise=True, seed=None)[source]
      
 
      Store data for information dynamics estimation.
      
 
      Data takes a 1- to 3-dimensional array representing realisations of random
 variables in dimensions: processes, samples (over time), and replications.
@@ -116,15 +117,15 @@ 
      The Data Class
-
      
-property data
      
+
      
+
      
+property data
      
 
      Return data array.
      
 
      
 
 
      
-
      
-generate_logistic_maps_data(n_samples=1000, n_replications=10, coefficient_matrices=array([[[0.5, 0.0], [0.4, 0.5]]]), noise_std=0.1)[source]
      
+
      
+generate_logistic_maps_data(n_samples=1000, n_replications=10, coefficient_matrices=array([[[0.5, 0.], [0.4, 0.5]]]), noise_std=0.1)[source]
      
 
      Generate discrete-time coupled-logistic-maps time series.
      
 
      Generate data and overwrite the instance’s current data.
      
 
      The implemented logistic map function is f(x) = 4 * x * (1 - x).
      
@@ -150,8 +151,8 @@ 
      The Data Class
-
      
-generate_mute_data(n_samples=1000, n_replications=10)[source]
      
+
      
+generate_mute_data(n_samples=1000, n_replications=10)[source]
      
 
      Generate example data for a 5-process network.
      
 
      Generate example data and overwrite the instance’s current data. The
 network is used as an example the paper on the MuTE toolbox (Montalto,
@@ -185,8 +186,8 @@ 
      The Data Class
-
      
-generate_var_data(n_samples=1000, n_replications=10, coefficient_matrices=array([[[0.5, 0.0], [0.4, 0.5]]]), noise_std=0.1)[source]
      
+
      
+generate_var_data(n_samples=1000, n_replications=10, coefficient_matrices=array([[[0.5, 0.], [0.4, 0.5]]]), noise_std=0.1)[source]
      
 
      Generate discrete-time VAR (vector autoregressive) time series.
      
 
      Generate data and overwrite the instance’s current data.
      
 
      
@@ -211,8 +212,8 @@ 
      The Data Class
-
      
-get_realisations(current_value, idx_list, shuffle=False)[source]
      
+
      
+get_realisations(current_value, idx_list, shuffle=False)[source]
      
 
      Return realisations for a list of indices.
      
 
      Return realisations for indices in list. Optionally, realisations can
 be shuffled to create surrogate data for statistical testing. For
@@ -303,20 +304,20 @@ 
      The Data Class
-
      
-get_seed()[source]
      
+
      
+get_seed()[source]
      
 
      return the initial seed of the data
      
 
      
 
 
      
-
      
-get_state()[source]
      
+
      
+get_state()[source]
      
 
      return the current state of the random seed
      
 
      
 
 
      
-
      
-n_realisations(current_value=None)[source]
      
+
      
+n_realisations(current_value=None)[source]
      
 
      Number of realisations over samples and replications.
      
 
      
 
      Args:
      
@@ -332,14 +333,14 @@ 
      The Data Class
-
      
-n_realisations_repl()[source]
      
+
      
+n_realisations_repl()[source]
      
 
      Number of realisations over replications.
      
 
      
 
 
      
-
      
-n_realisations_samples(current_value=None)[source]
      
+
      
+n_realisations_samples(current_value=None)[source]
      
 
      Number of realisations over samples.
      
 
      
 
      Args:
      
@@ -354,8 +355,8 @@ 
      The Data Class
-
      
-permute_replications(current_value, idx_list)[source]
      
+
      
+permute_replications(current_value, idx_list)[source]
      
 
      Return realisations with permuted replications (time stays intact).
      
 
      Create surrogate data by permuting realisations over replications while
 keeping the temporal structure (order of samples) intact. Return
@@ -444,8 +445,8 @@ 
      The Data Class
-
      
-permute_samples(current_value, idx_list, perm_settings)[source]
      
+
      
+permute_samples(current_value, idx_list, perm_settings)[source]
      
 
      Return realisations with permuted samples (repl. stays intact).
      
 
      Create surrogate data by permuting realisations over samples (time)
 while keeping the order of replications intact. Surrogates can be
@@ -655,8 +656,8 @@ 
      The Data Class
-
      
-set_data(data, dim_order)[source]
      
+
      
+set_data(data, dim_order)[source]
      
 
      Overwrite data in an existing Data object.
      
 
      
 
      Args:
      
@@ -672,8 +673,8 @@ 
      The Data Class
-
      
-slice_permute_replications(process)[source]
      
+
      
+slice_permute_replications(process)[source]
      
 
      Return data slice with permuted replications (time stays intact).
      
 
      Create surrogate data by permuting realisations over replications while
 keeping the temporal structure (order of samples) intact. Return
@@ -683,8 +684,8 @@ 
      The Data Class
-
      
-slice_permute_samples(process, perm_settings)[source]
      
+
      
+slice_permute_samples(process, perm_settings)[source]
      
 
      Return slice of data with permuted samples (repl. stays intact).
      
 
      Create surrogate data by permuting data in a slice over samples (time)
 while keeping the order of replications intact. Return slice for the
@@ -883,7 +884,7 @@ 
      The Data Class
      
@@ -892,12 +893,16 @@ 
      The Data Class
         
      
-  
      Previous topic
      
-  
      Welcome to IDTxl’s documentation!
      
-  
      Next topic
      
-  
      Algorithms for network inference
      
+  
+  
   
      
     
      This Page
      
     
     
      
     
   
 
\ No newline at end of file
diff --git a/docs/html/idtxl_estimators.html b/docs/html/idtxl_estimators.html
index 50226472..010a7744 100644
--- a/docs/html/idtxl_estimators.html
+++ b/docs/html/idtxl_estimators.html
@@ -4,11 +4,12 @@
 
   
     
-    
-    Information theoretic estimators — IDTxl 1.2.1 documentation
-    
-    
-    
+    
+
+    Information theoretic estimators — IDTxl 1.3 documentation
+    
+    
+    
     
     
     
@@ -39,7 +40,7 @@ 
      Navigation
      
         
    • 
           previous |
      • 
-        
    • IDTxl 1.2.1 documentation »
      • 
+        
    • IDTxl 1.3 documentation »
      • 
         
    • Information theoretic estimators
      •  
       
     
        
@@ -49,14 +50,14 @@ 
      Navigation
      
         
      
           
      
             
-  
      
+  
      
 
      Information theoretic estimators
      
-
      
+
      
 
      JIDT Estimators (CPU)
      
 
      Provide JIDT estimators.
      
 
      
-
      
-class idtxl.estimators_jidt.JidtDiscrete(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscrete(settings)[source]
      
 
      Abstract class for implementation of discrete JIDT-estimators.
      
 
      Abstract class for implementation of plug-in JIDT-estimators for discrete
 data. Child classes implement estimators for mutual information (MI),
@@ -88,8 +89,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate_surrogates_analytic(n_perm=200, **data)[source]
      
+
      
+estimate_surrogates_analytic(n_perm=200, **data)[source]
      
 
      Return estimate of the analytical surrogate distribution.
      
 
      This method must be implemented because this class’
 is_analytic_null_estimator() method returns true.
      
@@ -114,8 +115,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-abstract get_analytic_distribution(**data)[source]
      
+
      
+abstract get_analytic_distribution(**data)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -137,8 +138,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -151,8 +152,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtDiscreteAIS(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscreteAIS(settings)[source]
      
 
      Calculate AIS with JIDT’s discrete-variable implementation.
      
 
      Calculate the active information storage (AIS) for one process. Call JIDT
 via jpype and use the discrete estimator. See parent class for references.
      
@@ -182,8 +183,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(process, return_calc=False)[source]
      
+
      
+estimate(process, return_calc=False)[source]
      
 
      Estimate active information storage.
      
 
      
 
      Args:
      
@@ -215,8 +216,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-get_analytic_distribution(process)[source]
      
+
      
+get_analytic_distribution(process)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -241,8 +242,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtDiscreteCMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscreteCMI(settings=None)[source]
      
 
      Calculate CMI with JIDT’s implementation for discrete variables.
      
 
      Calculate the conditional mutual information between two variables given
 the third. Call JIDT via jpype and use the discrete estimator. See parent
@@ -276,8 +277,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(var1, var2, conditional=None, return_calc=False)[source]
      
+
      
+estimate(var1, var2, conditional=None, return_calc=False)[source]
      
 
      Estimate conditional mutual information.
      
 
      
 
      Args:
      
@@ -314,8 +315,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-get_analytic_distribution(var1, var2, conditional=None)[source]
      
+
      
+get_analytic_distribution(var1, var2, conditional=None)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -345,8 +346,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtDiscreteMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscreteMI(settings=None)[source]
      
 
      Calculate MI with JIDT’s discrete-variable implementation.
      
 
      Calculate the mutual information (MI) between two variables. Call JIDT via
 jpype and use the discrete estimator. See parent class for references.
      
@@ -379,8 +380,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(var1, var2, return_calc=False)[source]
      
+
      
+estimate(var1, var2, return_calc=False)[source]
      
 
      Estimate mutual information.
      
 
      
 
      Args:
      
@@ -414,8 +415,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-get_analytic_distribution(var1, var2)[source]
      
+
      
+get_analytic_distribution(var1, var2)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -442,8 +443,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtDiscreteTE(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtDiscreteTE(settings)[source]
      
 
      Calculate TE with JIDT’s implementation for discrete variables.
      
 
      Calculate the transfer entropy between two time series processes.
 Call JIDT via jpype and use the discrete estimator. Transfer entropy is
@@ -488,8 +489,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(source, target, return_calc=False)[source]
      
+
      
+estimate(source, target, return_calc=False)[source]
      
 
      Estimate transfer entropy from a source to a target variable.
      
 
      
 
      Args:
      
@@ -523,8 +524,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-get_analytic_distribution(source, target)[source]
      
+
      
+get_analytic_distribution(source, target)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -551,8 +552,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtEstimator(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtEstimator(settings=None)[source]
      
 
      Abstract class for implementation of JIDT estimators.
      
 
      Abstract class for implementation of JIDT estimators, child classes
 implement estimators for mutual information (MI), conditional mutual
@@ -588,8 +589,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -602,8 +603,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtGaussian(CalcClass, settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussian(CalcClass, settings)[source]
      
 
      Abstract class for implementation of JIDT Gaussian-estimators.
      
 
      Abstract class for implementation of JIDT Gaussian-estimators, child
 classes implement estimators for mutual information (MI), conditional
@@ -629,10 +630,10 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate_surrogates_analytic(n_perm=200, **data)[source]
      
-
      Estimate the surrogate distribution analytically.
-This method must be implemented because this class’
+
      
+estimate_surrogates_analytic(n_perm=200, **data)[source]
      
+
      Estimate the surrogate distribution analytically.
      
+
      This method must be implemented because this class’
 is_analytic_null_estimator() method returns true
      
 
      
 
      Args:
      
@@ -655,8 +656,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-get_analytic_distribution(**data)[source]
      
+
      
+get_analytic_distribution(**data)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -678,8 +679,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -692,8 +693,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtGaussianAIS(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussianAIS(settings)[source]
      
 
      Calculate active information storage with JIDT’s Gaussian implementation.
      
 
      Calculate active information storage (AIS) for some process using JIDT’s
 implementation of the Gaussian estimator. AIS is defined as the
@@ -727,8 +728,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(process)[source]
      
+
      
+estimate(process)[source]
      
 
      Estimate active information storage.
      
 
      
 
      Args:
      
@@ -750,8 +751,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtGaussianCMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussianCMI(settings=None)[source]
      
 
      Calculate conditional mutual infor with JIDT’s Gaussian implementation.
      
 
      Computes the differential conditional mutual information of two
 multivariate sets of observations, conditioned on another, assuming that
@@ -782,8 +783,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(var1, var2, conditional=None)[source]
      
+
      
+estimate(var1, var2, conditional=None)[source]
      
 
      Estimate conditional mutual information.
      
 
      
 
      Args:
      
@@ -808,8 +809,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-get_analytic_distribution(var1, var2, conditional=None)[source]
      
+
      
+get_analytic_distribution(var1, var2, conditional=None)[source]
      
 
      Return a JIDT AnalyticNullDistribution object.
      
 
      Required so that our estimate_surrogates_analytic method can use the
 common_estimate_surrogates_analytic() method, where data is formatted
@@ -838,8 +839,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtGaussianMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussianMI(settings=None)[source]
      
 
      Calculate mutual information with JIDT’s Gaussian implementation.
      
 
      Calculate the mutual information between two variables. Call JIDT via jpype
 and use the Gaussian estimator. See parent class for references.
      
@@ -866,8 +867,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(var1, var2)[source]
      
+
      
+estimate(var1, var2)[source]
      
 
      Estimate mutual information.
      
 
      
 
      Args:
      
@@ -891,8 +892,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtGaussianTE(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtGaussianTE(settings)[source]
      
 
      Calculate transfer entropy with JIDT’s Gaussian implementation.
      
 
      Calculate transfer entropy between a source and a target variable using
 JIDT’s implementation of the Gaussian estimator. Transfer entropy is
@@ -935,8 +936,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(source, target)[source]
      
+
      
+estimate(source, target)[source]
      
 
      Estimate transfer entropy from a source to a target variable.
      
 
      
 
      Args:
      
@@ -960,8 +961,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtKraskov(CalcClass, settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskov(CalcClass, settings=None)[source]
      
 
      Abstract class for implementation of JIDT Kraskov-estimators.
      
 
      Abstract class for implementation of JIDT Kraskov-estimators, child classes
 implement estimators for mutual information (MI), conditional mutual
@@ -1001,8 +1002,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -1015,8 +1016,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtKraskovAIS(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskovAIS(settings)[source]
      
 
      Calculate active information storage with JIDT’s Kraskov implementation.
      
 
      Calculate active information storage (AIS) for some process using JIDT’s
 implementation of the Kraskov type 1 estimator. AIS is defined as the
@@ -1062,8 +1063,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(process)[source]
      
+
      
+estimate(process)[source]
      
 
      Estimate active information storage.
      
 
      
 
      Args:
      
@@ -1085,8 +1086,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtKraskovCMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskovCMI(settings=None)[source]
      
 
      Calculate conditional mutual inform with JIDT’s Kraskov implementation.
      
 
      Calculate the conditional mutual information (CMI) between three variables.
 Call JIDT via jpype and use the Kraskov 1 estimator. If no conditional is
@@ -1127,8 +1128,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(var1, var2, conditional=None)[source]
      
+
      
+estimate(var1, var2, conditional=None)[source]
      
 
      Estimate conditional mutual information.
      
 
      
 
      Args:
      
@@ -1155,8 +1156,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtKraskovMI(settings=None)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskovMI(settings=None)[source]
      
 
      Calculate mutual information with JIDT’s Kraskov implementation.
      
 
      Calculate the mutual information between two variables. Call JIDT via jpype
 and use the Kraskov 1 estimator. See parent class for references.
      
@@ -1195,8 +1196,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(var1, var2)[source]
      
+
      
+estimate(var1, var2)[source]
      
 
      Estimate mutual information.
      
 
      
 
      Args:
      
@@ -1220,8 +1221,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-class idtxl.estimators_jidt.JidtKraskovTE(settings)[source]
      
+
      
+class idtxl.estimators_jidt.JidtKraskovTE(settings)[source]
      
 
      Calculate transfer entropy with JIDT’s Kraskov implementation.
      
 
      Calculate transfer entropy between a source and a target variable using
 JIDT’s implementation of the Kraskov type 1 estimator. Transfer entropy is
@@ -1266,8 +1267,8 @@ 
      JIDT Estimators (CPU)
 
      
 
      
-
      
-estimate(source, target)[source]
      
+
      
+estimate(source, target)[source]
      
 
      Estimate transfer entropy from a source to a target variable.
      
 
      
 
      Args:
      
@@ -1291,8 +1292,8 @@ 
      JIDT Estimators (CPU)
      
 
 
      
-
      
-idtxl.estimators_jidt.common_estimate_surrogates_analytic(estimator, n_perm=200, **data)[source]
      
+
      
+idtxl.estimators_jidt.common_estimate_surrogates_analytic(estimator, n_perm=200, **data)[source]
      
 
      Estimate the surrogate distribution analytically for JidtEstimator.
      
 
      Estimate the surrogate distribution analytically for a JidtEstimator
 which is_analytic_null_estimator(), by sampling estimates at random
@@ -1318,12 +1319,12 @@ 
      JIDT Estimators (CPU)
 
      
 
-
      
-
      
+
      
+
      
 
      OpenCL Estimators (GPU)
      
 
      
-
      
-class idtxl.estimators_opencl.OpenCLKraskov(settings=None)[source]
      
+
      
+class idtxl.estimators_opencl.OpenCLKraskov(settings=None)[source]
      
 
      Abstract class for implementation of OpenCL estimators.
      
 
      Abstract class for implementation of OpenCL estimators, child classes
 implement estimators for mutual information (MI) and conditional mutual
@@ -1375,8 +1376,8 @@ 
      OpenCL Estimators (GPU)
 
      
 
      
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -1387,8 +1388,8 @@ 
      OpenCL Estimators (GPU)
      
 
 
      
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -1401,8 +1402,8 @@ 
      OpenCL Estimators (GPU)
      
 
 
      
-
      
-class idtxl.estimators_opencl.OpenCLKraskovCMI(settings=None)[source]
      
+
      
+class idtxl.estimators_opencl.OpenCLKraskovCMI(settings=None)[source]
      
 
      Calculate conditional mutual inform with OpenCL Kraskov implementation.
      
 
      Calculate the conditional mutual information (CMI) between three variables
 using OpenCL GPU-code. If no conditional is given (is None), the function
@@ -1434,8 +1435,8 @@ 
      OpenCL Estimators (GPU)
 
      
 
      
-
      
-estimate(var1, var2, conditional=None, n_chunks=1)[source]
      
+
      
+estimate(var1, var2, conditional=None, n_chunks=1)[source]
      
 
      Estimate conditional mutual information.
      
 
      If conditional is None, the mutual information between var1 and var2 is
 calculated.
      
@@ -1470,8 +1471,8 @@ 
      OpenCL Estimators (GPU)
      
 
 
      
-
      
-class idtxl.estimators_opencl.OpenCLKraskovMI(settings=None)[source]
      
+
      
+class idtxl.estimators_opencl.OpenCLKraskovMI(settings=None)[source]
      
 
      Calculate mutual information with OpenCL Kraskov implementation.
      
 
      Calculate the mutual information (MI) between two variables using OpenCL
 GPU-code. See parent class for references.
      
@@ -1503,8 +1504,8 @@ 
      OpenCL Estimators (GPU)
 
      
 
      
-
      
-estimate(var1, var2, n_chunks=1)[source]
      
+
      
+estimate(var1, var2, n_chunks=1)[source]
      
 
      Estimate mutual information.
      
 
      
 
      Args:
      
@@ -1534,8 +1535,8 @@ 
      OpenCL Estimators (GPU)
      
 
-
      
-
      
+
+
      
 
      PID Estimators
      
 
      Partical information decomposition for discrete random variables.
      
 
      This module provides an estimator for partial information decomposition
@@ -1543,8 +1544,8 @@ 
      PID Estimatorshttp://doi.org/10.3390/e16042161
      
 
      
-
      
-class idtxl.estimators_pid.SydneyPID(settings)[source]
      
+
      
+class idtxl.estimators_pid.SydneyPID(settings)[source]
      
 
      Estimate partial information decomposition of discrete variables.
      
 
      Fast implementation of the BROJA partial information decomposition (PID)
 estimator for discrete data (Bertschinger, 2014). The estimator does not
@@ -1603,8 +1604,8 @@ 
      PID Estimators
-
      
-estimate(s1, s2, t)[source]
      
+
      
+estimate(s1, s2, t)[source]
      
 
      
 
      Args:
      
 
      s1numpy array
      1D array containing realizations of a discrete random variable
      
@@ -1625,8 +1626,8 @@ 
      PID Estimators
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -1637,8 +1638,8 @@ 
      PID Estimators
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -1651,8 +1652,8 @@ 
      PID Estimators
-
      
-class idtxl.estimators_pid.TartuPID(settings)[source]
      
+
      
+class idtxl.estimators_pid.TartuPID(settings)[source]
      
 
      Estimate partial information decomposition for two inputs and one output
      
 
      Implementation of the partial information decomposition (PID) estimator for
 discrete data. The estimator finds shared information, unique information
@@ -1684,8 +1685,8 @@ 
      PID Estimators
-
      
-estimate(s1, s2, t)[source]
      
+
      
+estimate(s1, s2, t)[source]
      
 
      
 
      Args:
      
 
      s1numpy array
      1D array containing realizations of a discrete random variable
      
@@ -1705,8 +1706,8 @@ 
      PID Estimators
-
      
-is_analytic_null_estimator()[source]
      
+
      
+is_analytic_null_estimator()[source]
      
 
      Indicate if estimator supports analytic surrogates.
      
 
      Return true if the estimator implements estimate_surrogates_analytic()
 where data is formatted as per the estimate method for this estimator.
      
@@ -1717,8 +1718,8 @@ 
      PID Estimators
-
      
-is_parallel()[source]
      
+
      
+is_parallel()[source]
      
 
      Indicate if estimator supports parallel estimation over chunks.
      
 
      Return true if the supports parallel estimation over chunks, where a
 chunk is one independent data set.
      
@@ -1730,8 +1731,8 @@ 
      PID Estimators
      
@@ -1740,8 +1741,9 @@ 
      PID Estimators
         
      
-  
      Table of Contents
      
-  
     
        
@@ -49,32 +50,32 @@ 
      Navigation
      
         
      
           
      
             
-  
      
+  
      
 
      Helper functions
      
-
      
+
      
 
      utils module
      
 
      Provide IDTxl utility functions.
      
 
      
-
      
-idtxl.idtxl_utils.argsort_descending(a)[source]
      
+
      
+idtxl.idtxl_utils.argsort_descending(a)[source]
      
 
      Sort array in descending order and return sortind indices.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.autocorrelation(x)[source]
      
+
      
+idtxl.idtxl_utils.autocorrelation(x)[source]
      
 
      Calculate autocorrelation of a vector.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.calculate_mi(corr)[source]
      
+
      
+idtxl.idtxl_utils.calculate_mi(corr)[source]
      
 
      Calculate mutual information from correlation coefficient.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.combine_discrete_dimensions(a, numBins)[source]
      
+
      
+idtxl.idtxl_utils.combine_discrete_dimensions(a, numBins)[source]
      
 
      Combine multi-dimensional discrete variable into a single dimension.
      
 
      Combine all dimensions for a discrete variable down into a single
 dimensional value for each sample. This is done basically by multiplying
@@ -100,8 +101,8 @@ 
      utils module
-
      
-idtxl.idtxl_utils.conflicting_entries(dict_1, dict_2)[source]
      
+
      
+idtxl.idtxl_utils.conflicting_entries(dict_1, dict_2)[source]
      
 
      Test two dictionaries for unequal entries.
      
 
      Note that only keys that are present in both dicts are compared. If one
 dictionary contains an entry not present in the other dictionary, the
@@ -109,8 +110,8 @@ 
      utils module
-
      
-idtxl.idtxl_utils.discretise(a, numBins)[source]
      
+
      
+idtxl.idtxl_utils.discretise(a, numBins)[source]
      
 
      Discretise continuous data.
      
 
      Discretise continuous data into discrete values (with 0 as lowest) by
 evenly partitioning the range of the data, one dimension at a time.
@@ -134,8 +135,8 @@ 
      utils module
-
      
-idtxl.idtxl_utils.discretise_max_ent(a, numBins)[source]
      
+
      
+idtxl.idtxl_utils.discretise_max_ent(a, numBins)[source]
      
 
      Discretise continuous data using maximum entropy partitioning.
      
 
      Discretise continuous data into discrete values (with 0 as lowest) by
 making a maximum entropy partitioning, one dimension at a time. Adapted
@@ -159,20 +160,20 @@ 
      utils module
-
      
-idtxl.idtxl_utils.equal_dicts(dict_1, dict_2)[source]
      
+
      
+idtxl.idtxl_utils.equal_dicts(dict_1, dict_2)[source]
      
 
      Test two dictionaries for equality.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.print_dict(d, indent=4)[source]
      
+
      
+idtxl.idtxl_utils.print_dict(d, indent=4)[source]
      
 
      Use Python’s pretty printer to print dictionaries to the console.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.remove_column(a, j)[source]
      
+
      
+idtxl.idtxl_utils.remove_column(a, j)[source]
      
 
      Remove a column from a numpy array.
      
 
      This is faster than logical indexing (‘25 times faster’), because it does
 not make copies, see
@@ -189,8 +190,8 @@ 
      utils module
-
      
-idtxl.idtxl_utils.remove_row(a, i)[source]
      
+
      
+idtxl.idtxl_utils.remove_row(a, i)[source]
      
 
      Remove a row from a numpy array.
      
 
      This is faster than logical indexing (‘25 times faster’), because it does
 not make copies, see
@@ -207,8 +208,8 @@ 
      utils module
-
      
-idtxl.idtxl_utils.separate_arrays(idx_all, idx_single, a)[source]
      
+
      
+idtxl.idtxl_utils.separate_arrays(idx_all, idx_single, a)[source]
      
 
      Separate a single column from all other columns in a 2D-array.
      
 
      Return the separated single column and the remaining columns of a 2D-
 array.
      
@@ -235,14 +236,14 @@ 
      utils module
-
      
-idtxl.idtxl_utils.sort_descending(a)[source]
      
+
      
+idtxl.idtxl_utils.sort_descending(a)[source]
      
 
      Sort array in descending order.
      
 
      
 
 
      
-
      
-idtxl.idtxl_utils.standardise(a, dimension=0, df=1)[source]
      
+
      
+idtxl.idtxl_utils.standardise(a, dimension=0, df=1)[source]
      
 
      Z-standardise a numpy array along a given dimension.
      
 
      Standardise array along the axis defined in dimension using the denominator
 (N - df) for the calculation of the standard deviation.
      
@@ -265,8 +266,8 @@ 
      utils module
-
      
-idtxl.idtxl_utils.swap_chars(s, i_1, i_2)[source]
      
+
      
+idtxl.idtxl_utils.swap_chars(s, i_1, i_2)[source]
      
 
      Swap to characters in a string.
      
 
      
 
      Example:
      >>> print(swap_chars('heLlotHere', 2, 6))
@@ -277,13 +278,13 @@ 
      utils module
+
      
+
      
 
      stats module
      
 
      Provide statistics functions.
      
 
      
-
      
-idtxl.stats.ais_fdr(settings=None, *results)[source]
      
+
      
+idtxl.stats.ais_fdr(settings=None, *results)[source]
      
 
      Perform FDR-correction on results of network AIS estimation.
      
 
      Perform correction of the false discovery rate (FDR) after estimation of
 active information storage (AIS) for all processes in the network. FDR
@@ -323,8 +324,8 @@ 
      stats module
-
      
-idtxl.stats.check_n_perm(n_perm, alpha)[source]
      
+
      
+idtxl.stats.check_n_perm(n_perm, alpha)[source]
      
 
      Check if no. permutations is big enough to obtain the requested alpha.
      
 
      
 
      Note:
      The no. permutations must be big enough to theoretically allow for the
@@ -336,8 +337,8 @@ 
      stats module
-
      
-idtxl.stats.max_statistic(analysis_setup, data, candidate_set, te_max_candidate, conditional=None)[source]
      
+
      
+idtxl.stats.max_statistic(analysis_setup, data, candidate_set, te_max_candidate, conditional)[source]
      
 
      Perform maximum statistics for one candidate source.
      
 
      Test if a transfer entropy value is significantly bigger than the maximum
 values obtained from surrogates of all remanining candidates.
      
@@ -361,9 +362,8 @@ 
      stats modulefloat
      transfer entropy value to be tested
      
 
      
-
      conditionalnumpy array [optional]
      realisations of conditional, 2D numpy array where array dimensions
-represent [realisations x variable dimension] (default=None, no
-conditioning performed)
      
+
      conditionalnumpy array
      realisations of conditional, 2D numpy array where array dimensions
+represent [realisations x variable dimension]
      
 
      
 
      
 
      
@@ -386,8 +386,8 @@ 
      stats module
-
      
-idtxl.stats.max_statistic_sequential(analysis_setup, data)[source]
      
+
      
+idtxl.stats.max_statistic_sequential(analysis_setup, data)[source]
      
 
      Perform sequential maximum statistics for a set of candidate sources.
      
 
      Test multivariate/bivariate MI/TE values against surrogates. Test highest
 TE/MI value against distribution of highest surrogate values, second
@@ -442,8 +442,8 @@ 
      stats module
-
      
-idtxl.stats.max_statistic_sequential_bivariate(analysis_setup, data)[source]
      
+
      
+idtxl.stats.max_statistic_sequential_bivariate(analysis_setup, data)[source]
      
 
      Perform sequential maximum statistics for a set of candidate sources.
      
 
      Test multivariate/bivariate MI/TE values against surrogates. Test highest
 TE/MI value against distribution of highest surrogate values, second
@@ -498,8 +498,8 @@ 
      stats module
-
      
-idtxl.stats.mi_against_surrogates(analysis_setup, data)[source]
      
+
      
+idtxl.stats.mi_against_surrogates(analysis_setup, data)[source]
      
 
      Test estimated mutual information for significance against surrogate data.
      
 
      Shuffle realisations of the current value (point to be predicted) and re-
 calculate mutual information (MI) for shuffled data. The actual estimated
@@ -541,8 +541,8 @@ 
      stats module
-
      
-idtxl.stats.min_statistic(analysis_setup, data, candidate_set, te_min_candidate, conditional=None)[source]
      
+
      
+idtxl.stats.min_statistic(analysis_setup, data, candidate_set, te_min_candidate, conditional=None)[source]
      
 
      Perform minimum statistics for one candidate source.
      
 
      Test if a transfer entropy value is significantly bigger than the minimum
 values obtained from surrogates of all remanining candidates.
      
@@ -591,8 +591,8 @@ 
      stats module
-
      
-idtxl.stats.network_fdr(settings=None, *results)[source]
      
+
      
+idtxl.stats.network_fdr(settings=None, *results)[source]
      
 
      Perform FDR-correction on results of network inference.
      
 
      Perform correction of the false discovery rate (FDR) after network
 analysis. FDR correction can either be applied at the target level
@@ -636,8 +636,8 @@ 
      stats module
-
      
-idtxl.stats.omnibus_test(analysis_setup, data)[source]
      
+
      
+idtxl.stats.omnibus_test(analysis_setup, data)[source]
      
 
      Perform an omnibus test on identified conditional variables.
      
 
      Test the joint information transfer from all identified sources to the
 current value conditional on candidates in the target’s past. To test for
@@ -681,8 +681,8 @@ 
      stats module
-
      
-idtxl.stats.syn_shd_against_surrogates(analysis_setup, data)[source]
      
+
      
+idtxl.stats.syn_shd_against_surrogates(analysis_setup, data)[source]
      
 
      Test the shared/synergistic information in the PID estimate.
      
 
      Shuffle realisations of the target and re-calculate PID, in particular the
 synergistic and shared information from shuffled data. The original
@@ -721,8 +721,8 @@ 
      stats module
-
      
-idtxl.stats.unq_against_surrogates(analysis_setup, data)[source]
      
+
      
+idtxl.stats.unq_against_surrogates(analysis_setup, data)[source]
      
 
      Test the unique information in the PID estimate against surrogate data.
      
 
      Shuffle realisations of both sources individually and re-calculate PID,
 in particular the unique information from shuffled data. The original
@@ -760,8 +760,8 @@ 
      stats module
      
@@ -770,8 +770,9 @@ 
      stats module
         
      
-  
      Table of Contents
      
-  
     
        
@@ -49,11 +50,11 @@ 
      Navigation
      
         
      
           
      
             
-  
      
+  
      
 
      Network comparison
      
 
      
-
      
-class idtxl.network_comparison.NetworkComparison[source]
      
+
      
+class idtxl.network_comparison.NetworkComparison[source]
      
 
      Set up network comparison between two experimental conditions.
      
 
      The class provides methods for the comparison of networks inferred from
 data recorded under two experimental conditions A and B. Four statistical
@@ -98,8 +99,8 @@ 
      Network comparison
-
      
-calculate_link_te(data, target, sources='all')[source]
      
+
      
+calculate_link_te(data, target, sources='all')[source]
      
 
      Calculate the information transfer for whole links into a target.
      
 
      Calculate the information transfer for whole links as the joint
 information transfer from all variables selected for a single source
@@ -129,8 +130,8 @@ 
      Network comparison
-
      
-compare_between(settings, network_set_a, network_set_b, data_set_a, data_set_b)[source]
      
+
      
+compare_between(settings, network_set_a, network_set_b, data_set_a, data_set_b)[source]
      
 
      Compare networks inferred under two conditions between subjects.
      
 
      Compare two sets of networks inferred from two sets of data recorded
 under different experimental conditions within multiple subjects, i.e.,
@@ -163,8 +164,8 @@ 
      Network comparison
-
      
-compare_links_within(settings, link_a, link_b, network, data)[source]
      
+
      
+compare_links_within(settings, link_a, link_b, network, data)[source]
      
 
      Compare two links within the same network.
      
 
      Compare two links within the same network. Check if information
 transfer is different from information transfer in a second link.
      
@@ -213,8 +214,8 @@ 
      Network comparison
-
      
-compare_within(settings, network_a, network_b, data_a, data_b)[source]
      
+
      
+compare_within(settings, network_a, network_b, data_a, data_b)[source]
      
 
      Compare networks inferred under two conditions within one subject.
      
 
      Compare two networks inferred from data recorded under two different
 experimental conditions within one subject (units of observations are
@@ -263,7 +264,7 @@ 
      Network comparison
      
@@ -272,12 +273,16 @@ 
      Network comparison
         
      
-  
      Previous topic
      
-  
      Algorithms for the analysis of node dynamics
      
-  
      Next topic
      
-  
      The Results Class
      
+  
+  
      
+    
      Next topic
      
+    
      The Results Class
      
+  
      
   
      
     
      This Page
      
     
     
      
     
   
 
\ No newline at end of file
diff --git a/docs/html/idtxl_network_inference.html b/docs/html/idtxl_network_inference.html
index 504a7ebe..1d9fc815 100644
--- a/docs/html/idtxl_network_inference.html
+++ b/docs/html/idtxl_network_inference.html
@@ -4,11 +4,12 @@
 
   
     
-    
-    Algorithms for network inference — IDTxl 1.2.1 documentation
-    
-    
-    
+    
+
+    Algorithms for network inference — IDTxl 1.3 documentation
+    
+    
+    
     
     
     
@@ -39,7 +40,7 @@ 
      Navigation
      
         
    • 
           previous |
      • 
-        
    • IDTxl 1.2.1 documentation »
      • 
+        
    • IDTxl 1.3 documentation »
      • 
         
    • Algorithms for network inference
      •  
       
     
        
@@ -49,13 +50,13 @@ 
      Navigation
      
         
      
           
      
             
-  
      
+  
      
 
      Algorithms for network inference
      
-
      
+
      
 
      Multivariate Transfer Entropy (mTE)
      
 
      
-
      
-class idtxl.multivariate_te.MultivariateTE[source]
      
+
      
+class idtxl.multivariate_te.MultivariateTE[source]
      
 
      Perform network inference using multivariate transfer entropy.
      
 
      Perform network inference using multivariate transfer entropy (TE). To
 perform network inference call analyse_network() on the whole network or a
@@ -78,8 +79,9 @@ 
      Multivariate Transfer Entropy (mTE)http://doi.org/10.1103/PhysRevE.83.051112
      
 
-
      
-
      Attributes:
      
+
      Attributes:
      
+
      
+
      
 
      source_setlist
      indices of source processes tested for their influence on the
 target
      
 
      
@@ -107,11 +109,10 @@ 
      Multivariate Transfer Entropy (mTE)bool
      statistical significance of the over-all TE
      
 
      
 
      
-
      
-
      
+
      
 
      
-
      
-analyse_network(settings, data, targets='all', sources='all')[source]
      
+
      
+analyse_network(settings, data, targets='all', sources='all')[source]
      
 
      Find multivariate transfer entropy between all nodes in the network.
      
 
      Estimate multivariate transfer entropy (TE) between all nodes in the
 network or between selected sources and targets.
      
@@ -173,8 +174,8 @@ 
      Multivariate Transfer Entropy (mTE)
-
      
-analyse_single_target(settings, data, target, sources='all')[source]
      
+
      
+analyse_single_target(settings, data, target, sources='all')[source]
      
 
      Find multivariate transfer entropy between sources and a target.
      
 
      Find multivariate transfer entropy (TE) between all source processes
 and the target process. Uses multivariate, non-uniform embedding found
@@ -283,12 +284,12 @@ 
      Multivariate Transfer Entropy (mTE)
+
      
+
      
 
      Bivariate Transfer Entropy (bTE)
      
 
      
-
      
-class idtxl.bivariate_te.BivariateTE[source]
      
+
      
+class idtxl.bivariate_te.BivariateTE[source]
      
 
      Perform network inference using bivariate transfer entropy.
      
 
      Perform network inference using bivariate transfer entropy (TE). To
 perform network inference call analyse_network() on the whole network or a
@@ -343,8 +344,8 @@ 
      Bivariate Transfer Entropy (bTE)
-
      
-analyse_network(settings, data, targets='all', sources='all')[source]
      
+
      
+analyse_network(settings, data, targets='all', sources='all')[source]
      
 
      Find bivariate transfer entropy between all nodes in the network.
      
 
      Estimate bivariate transfer entropy (TE) between all nodes in the
 network or between selected sources and targets.
      
@@ -405,8 +406,8 @@ 
      Bivariate Transfer Entropy (bTE)
-
      
-analyse_single_target(settings, data, target, sources='all')[source]
      
+
      
+analyse_single_target(settings, data, target, sources='all')[source]
      
 
      Find bivariate transfer entropy between sources and a target.
      
 
      Find bivariate transfer entropy (TE) between all potential source
 processes and the target process. Uses bivariate, non-uniform embedding
@@ -519,12 +520,12 @@ 
      Bivariate Transfer Entropy (bTE)
+
      
+
      
 
      Multivariate Mutual Information (mMI)
      
 
      
-
      
-class idtxl.multivariate_mi.MultivariateMI[source]
      
+
      
+class idtxl.multivariate_mi.MultivariateMI[source]
      
 
      Perform network inference using multivariate mutual information.
      
 
      Perform network inference using multivariate mutual information (MI). To
 perform network inference call analyse_network() on the whole network or a
@@ -571,8 +572,8 @@ 
      Multivariate Mutual Information (mMI)
-
      
-analyse_network(settings, data, targets='all', sources='all')[source]
      
+
      
+analyse_network(settings, data, targets='all', sources='all')[source]
      
 
      Find multivariate mutual information between nodes in the network.
      
 
      Estimate multivariate mutual information (MI) between all nodes in the
 network or between selected sources and targets.
      
@@ -640,8 +641,8 @@ 
      Multivariate Mutual Information (mMI)
-
      
-analyse_single_target(settings, data, target, sources='all')[source]
      
+
      
+analyse_single_target(settings, data, target, sources='all')[source]
      
 
      Find multivariate mutual information between sources and a target.
      
 
      Find multivariate mutual information (MI) between all source processes
 and the target process. Uses multivariate, non-uniform embedding found
@@ -752,12 +753,12 @@ 
      Multivariate Mutual Information (mMI)
+
      
+
      
 
      Bivariate Mutual Information (bMI)
      
 
      
-
      
-class idtxl.bivariate_mi.BivariateMI[source]
      
+
      
+class idtxl.bivariate_mi.BivariateMI[source]
      
 
      Perform network inference using bivariate mutual information.
      
 
      Perform network inference using bivariate mutual information (MI). To
 perform network inference call analyse_network() on the whole network or a
@@ -806,8 +807,8 @@ 
      Bivariate Mutual Information (bMI)
-
      
-analyse_network(settings, data, targets='all', sources='all')[source]
      
+
      
+analyse_network(settings, data, targets='all', sources='all')[source]
      
 
      Find bivariate mutual information between all nodes in the network.
      
 
      Estimate bivariate mutual information (MI) between all nodes in the
 network or between selected sources and targets.
      
@@ -871,8 +872,8 @@ 
      Bivariate Mutual Information (bMI)
-
      
-analyse_single_target(settings, data, target, sources='all')[source]
      
+
      
+analyse_single_target(settings, data, target, sources='all')[source]
      
 
      Find bivariate mutual information between sources and a target.
      
 
      Find bivariate mutual information (MI) between all potential source
 processes and the target process. Uses bivariate, non-uniform embedding
@@ -986,8 +987,8 @@ 
      Bivariate Mutual Information (bMI)
      
@@ -996,8 +997,9 @@ 
      Bivariate Mutual Information (bMI)
         
      
-  
      Table of Contents
      
-  
     
        
@@ -49,13 +50,13 @@ 
      Navigation
      
         
      
           
      
             
-  
      
+  
      
 
      Algorithms for the analysis of node dynamics
      
-
      
+
      
 
      Active Information Storage (AIS)
      
 
      
-
      
-class idtxl.active_information_storage.ActiveInformationStorage[source]
      
+
      
+class idtxl.active_information_storage.ActiveInformationStorage[source]
      
 
      Estimate active information storage in individual processes.
      
 
      Estimate active information storage (AIS) in individual processes of the
 network. To perform AIS estimation call analyse_network() on the whole
@@ -97,8 +98,8 @@ 
      Active Information Storage (AIS)
-
      
-analyse_network(settings, data, processes='all')[source]
      
+
      
+analyse_network(settings, data, processes='all')[source]
      
 
      Estimate active information storage for multiple network processes.
      
 
      Estimate active information storage for all or a subset of processes in
 the network.
      
@@ -156,8 +157,8 @@ 
      Active Information Storage (AIS)
-
      
-analyse_single_process(settings, data, process)[source]
      
+
      
+analyse_single_process(settings, data, process)[source]
      
 
      Estimate active information storage for a single process.
      
 
      Estimate active information storage for one process in the network.
 Uses non-uniform embedding found through information maximisation. This
@@ -229,12 +230,12 @@ 
      Active Information Storage (AIS)
+
      
+
      
 
      Multivariate Partial Information Decomposition (PID)
      
 
      
-
      
-class idtxl.multivariate_pid.MultivariatePID[source]
      
+
      
+class idtxl.multivariate_pid.MultivariatePID[source]
      
 
      Perform partial information decomposition for individual processes.
      
 
      Perform partial information decomposition (PID) for multiple source
 processes (up to 4 sources) and a target process in the network.
@@ -265,8 +266,8 @@ 
      Multivariate Partial Information Decomposition (PID)
 
      
 
      
-
      
-analyse_network(settings, data, targets, sources)[source]
      
+
      
+analyse_network(settings, data, targets, sources)[source]
      
 
      Estimate partial information decomposition for network nodes.
      
 
      Estimate, for multiple nodes (target processes), the partial
 information decomposition (PID) for multiple source processes
@@ -330,8 +331,8 @@ 
      Multivariate Partial Information Decomposition (PID)
      
 
 
      
-
      
-analyse_single_target(settings, data, target, sources)[source]
      
+
      
+analyse_single_target(settings, data, target, sources)[source]
      
 
      Estimate partial information decomposition for a network node.
      
 
      Estimate partial information decomposition (PID) for multiple source
 processes (up to 4 sources) and a target process in the network.
      
@@ -392,12 +393,12 @@ 
      Multivariate Partial Information Decomposition (PID)
      
 
-
      
-
      
+
      
+
      
 
      Bivariate Partial Information Decomposition (PID)
      
 
      
-
      
-class idtxl.bivariate_pid.BivariatePID[source]
      
+
      
+class idtxl.bivariate_pid.BivariatePID[source]
      
 
      Perform partial information decomposition for individual processes.
      
 
      Perform partial information decomposition (PID) for two source processes
 and one target process in the network. Estimate unique, shared, and
@@ -428,8 +429,8 @@ 
      Bivariate Partial Information Decomposition (PID)
 
      
 
      
-
      
-analyse_network(settings, data, targets, sources)[source]
      
+
      
+analyse_network(settings, data, targets, sources)[source]
      
 
      Estimate partial information decomposition for network nodes.
      
 
      Estimate partial information decomposition (PID) for multiple nodes in
 the network.
      
@@ -492,8 +493,8 @@ 
      Bivariate Partial Information Decomposition (PID)
      
 
 
      
-
      
-analyse_single_target(settings, data, target, sources)[source]
      
+
      
+analyse_single_target(settings, data, target, sources)[source]
      
 
      Estimate partial information decomposition for a network node.
      
 
      Estimate partial information decomposition (PID) for a target node in
 the network.
      
@@ -557,32 +558,1165 @@ 
      Bivariate Partial Information Decomposition (PID)
      
 
-
      
-
      
+
      
+
      
+
      History-dependence estimator for neural spiking data
      
+
      
+
      Embedding optimization
      
+
      
+
      
+class idtxl.embedding_optimization_ais_Rudelt.OptimizationRudelt(settings=None)[source]
      
+
      Optimization of embedding parameters of spike times using the history dependence estimators
      
+
      References:
      
+
      
+
      
+
      [1]: L. Rudelt, D. G. Marx, M. Wibral, V. Priesemann: Embedding
      optimization reveals long-lasting history dependence in
+neural spiking activity, 2021, PLOS Computational Biology, 17(6)
      
+
      
+
      
+
      [2]: https://github.com/Priesemann-Group/hdestimator
      
+
      
+
      implemented in idtxl by Michael Lindner, Göttingen 2021
      
+
      
+
      Args:
      
+
      settingsdict
        
+
      • 
+
        estimation_methodstring
        The method to be used to estimate the history dependence ‘bbc’ or ‘shuffling’.
        
+
        
+
        
+
        • 
+
      • 
+
        embedding_step_sizefloat
        Step size delta t (in seconds) with which the window is slid through the data.
+(default: 0.005)
        
+
        
+
        
+
        • 
+
      • 
+
        embedding_number_of_bins_setlist of integer values
        Set of values for d, the number of bins in the embedding.
+(default: [1, 2, 3, 4, 5])
        
+
        
+
        
+
        • 
+
      • 
+
        embedding_past_range_setlist of floating-point values
        Set of values for T, the past range (in seconds) to be used for embeddings.
+(default: [0.005, 0.00561, 0.00629, 0.00706, 0.00792, 0.00889,
+0.00998, 0.01119, 0.01256, 0.01409, 0.01581, 0.01774, 0.01991,
+0.02233, 0.02506, 0.02812, 0.03155, 0.0354, 0.03972, 0.04456,
+0.05, 0.0561, 0.06295, 0.07063, 0.07924, 0.08891, 0.09976,
+0.11194, 0.12559, 0.14092, 0.15811, 0.17741, 0.19905, 0.22334,
+0.25059, 0.28117, 0.31548, 0.35397, 0.39716, 0.44563, 0.5,
+0.56101, 0.62946, 0.70627, 0.79245, 0.88914, 0.99763, 1.11936,
+1.25594, 1.40919, 1.58114, 1.77407, 1.99054, 2.23342, 2.50594,
+2.81171, 3.15479, 3.53973, 3.97164, 4.45625, 5.0])
        
+
        
+
        
+
        • 
+
      • 
+
        embedding_scaling_exponent_setdict
        Set of values for kappa, the scaling exponent for the bins in the embedding.
+Should be a python-dictionary with the three entries ‘number_of_scalings’, ‘min_first_bin_size’ and
+‘min_step_for_scaling’.
+defaults: {‘number_of_scalings’: 10, ‘min_first_bin_size’: 0.005, ‘min_step_for_scaling’: 0.01})
        
+
        
+
        
+
        • 
+
      • 
+
        bbc_tolerancefloat
        The tolerance for the Bayesian Bias Criterion. Influences which embeddings are
+discarded from the analysis.
+(default: 0.05)
        
+
        
+
        
+
        • 
+
      • 
+
        return_averaged_Rbool
        Return R_tot as the average over R(T) for T in [T_D, T_max], instead of R_tot = R(T_D).
+If set to True, the setting for number_of_bootstraps_R_tot (see below) is ignored and set to 0
+and CI bounds are not calculated.
+(default: True)
        
+
        
+
        
+
        • 
+
      • 
+
        timescale_minimum_past_rangefloat
        Minimum past range T_0 (in seconds) to take into consideration for the estimation of the
+information timescale tau_R.
+(default: 0.01)
        
+
        
+
        
+
        • 
+
      • 
+
        number_of_bootstraps_R_maxint
        The number of bootstrap re-shuffles that should be used to determine the optimal
+embedding. (Bootstrap the estimates of R_max to determine R_tot.)
+These are computed during the ‘history-dependence’ task because they are essential
+to obtain R_tot.
+(default: 250)
        
+
        
+
        
+
        • 
+
      • 
+
        number_of_bootstraps_R_totint
        The number of bootstrap re-shuffles that should be used to estimate the confidence
+interval of the optimal embedding. (Bootstrap the estimates of R_tot = R(T_D) to
+obtain a confidence interval for R_tot.).
+These are computed during the ‘confidence-intervals’ task.
+The setting return_averaged_R (see above) needs to be set to False for this setting
+to take effect.
+(default: 250)
        
+
        
+
        
+
        • 
+
      • 
+
        number_of_bootstraps_nonessentialint
        The number of bootstrap re-shuffles that should be used to estimate the confidence
+intervals for embeddings other than the optimal one. (Bootstrap the estimates of
+R(T) for all other T.)
+(These are not necessary for the main analysis and therefore default to 0.)
        
+
        
+
        
+
        • 
+
      • 
+
        symbol_block_lengthint
        The number of symbols that should be drawn in each block for bootstrap resampling
+If it is set to None (recommended), the length is automatically chosen, based
+on heuristics
+(default: None)
        
+
        
+
        
+
        • 
+
      • 
+
        bootstrap_CI_use_sdbool
        Most of the time we observed normally-distributed bootstrap replications,
+so it is sufficient (and more efficient) to compute confidence intervals
+based on the standard deviation
+(default: True)
        
+
        
+
        
+
        • 
+
      • 
+
        bootstrap_CI_percentile_lofloat
        The lower percentile for the confidence interval.
+This has no effect if bootstrap_CI_use_sd is set to True
+(default: 2.5)
        
+
        
+
        
+
        • 
+
      • 
+
        bootstrap_CI_percentile_hifloat
        The upper percentiles for the confidence interval.
+This has no effect if bootstrap_CI_use_sd is set to True
+(default: 97.5)
        
+
        
+
        
+
        • 
+
      • 
+
        analyse_auto_MIbool
        perform calculation of auto mutual information of the spike train
+(default: True)
+If set to True:
        
+
          
+
        • 
+
          auto_MI_bin_size_setlist of floating-point values
          Set of values for the sizes of the bins (in seconds).
+(default: [0.005, 0.01, 0.025, 0.05, 0.25, 0.5])
          
+
          
+
          
+
          • 
+
        • 
+
          auto_MI_max_delayint
          The maximum delay (in seconds) between the past bin and the response.
+(default: 5)
          
+
          
+
          
+
          • 
+
        
+
        
+
        
+
        • 
+
      • 
+
        visualizationbool
        create .eps output image showing the optimization values and graphs for
+the history dependence and the auto mutual information
+(default: False)
+if set to True:
        
+
          
+
        • 
+
          output_pathString
          Path where the .eps images should be saved
          
+
          
+
          
+
          • 
+
        • 
+
          output_prefixString
          Prefix of the output images
+e.g. <output_prefix>_process0.eps
          
+
          
+
          
+
          • 
+
        
+
        
+
        
+
        • 
+
      • 
+
        debug: bool
        show values while calculating
+(default: False)
        
+
        
+
        
+
        • 
+
      
+
      
+
      
+
      
+
      
+
      
+
      
+analyse_auto_MI(spike_times)[source]
      
+
      Get the auto MI for the spike times.  If it is available from file, load
+it, else compute it.
      
+
      
 
+
      
+
      
+check_inputs()[source]
      
+
      Check input settings for completeness
      
+
      
 
-            
      
-          
      
-        
      
-      
      
-      
+      
      
+      
      		   
        idtxl.data
        
         
+       idtxl.embedding_optimization_ais_Rudelt
+       
      
      
      
        
           
@@ -109,12 +114,12 @@ 
      Python Module Index
      
      
      
        
           
-       idtxl.idtxl_exceptions
+       idtxl.estimators_Rudelt
        
      
      
      
        
           
-       idtxl.idtxl_import
+       idtxl.idtxl_exceptions

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