Search for the Best Basis Representation of a Binary Dataset with up to n=64 variables

or: Best Independent Minimally Complex Models (MCM)

This program searches for the Best Basis representation for a chosen binary dataset, while taking into account possible high-order patterns of the data. It was develop for the paper Statistical Inference of Minimally Complex Models [1]. More details on the general algorithm can be found in the paper.

[1] C. de Mulatier, P. P. Mazza, M. Marsili, Statistical Inference of Minimally Complex Models, arXiv:2008.00520

General information

The Best basis for a binary data with n variables is the one for which the independent model formed by n field operators has the largest log-likelihood (and therefore the largest log-evidence, as all independent model with the same number of operators are equivalent -- see Ref[1]).

There are three main functions that you can use to search for the best basis from the main.cpp:

1. Exhaustive Search: This function will compute all $2^n-1$ operators and will search for the best basis among them with a Greedy approach (i.e. rank them from the most to least biased and extract the set of n most bias independent operators starting from the most biased one):

vector<Operator64> BestBasis_ExhaustiveSearch(vector<pair<uint64_t, unsigned int>> Nvect, unsigned int n, unsigned int N, bool bool_print = false)

Note: we advise doing such search only for small systems (up to ~15 variables).

2. Search in a fixed representation up to order kmax: This function searches for the best Basis among all operators up to order kmax in the a given representation (which is the representation used when storing the data in Nvect -- by default, this is the original representation of the data):

BestBasisSearch_FixedRepresentation(vector<pair<uint64_t, unsigned int>> Nvect, unsigned int n, unsigned int N, unsigned int k_max, unsigned int B_it, bool bool_print = false)

If you take the largest order to be equal to the number of variables (kmax = n), then this function will perform an exhaustive search for the best basis among all possible operators.

3. Search in varying representations: This is the recommended approach when the number of variables exceeds $n\simeq 15$ - $20$. A priori, this heuristic approach is able to explore possible basis interactions of any high order.

vector<Operator64> BestBasisSearch_Final(vector<pair<uint64_t, unsigned int>> Nvect, unsigned int n, unsigned int N, unsigned int k_max, bool bool_print = false)

This function performs the search procedure described in Ref.[1]. The program first searches for the best basis up to order k_max; the data is then successively transformed in the representation given by the previously found best basis, and the program search for the new best basis in this representation. The algorithm stops when the new basis found is the identity (i.e. the basis has not changed).

Requirements

The code uses the C++11 version of C++.

Usage with Makefile:

Run the following commands in your terminal, from the main folder (folder containing the makefile document):

• To compile: make

• To Execute: make run . This will use the datafile and variables that are specified in the makefile.

To change datafile: open the makefile and replace the values of the two following variables at the very top of the file (an example is provided):

• datafile: name of your datafile; this file must be placed in the folder INPUT
• n: number of variables in your file; maximum possible value n = 128.

You can also execute the code by running in your terminal the command (from the main folder):

./BestBasis.out  datafilename  n

where you must replace datafilename by the name of your datafile and n by your number of variables.

• To clean: make clean (to use only once you're done using the code)

Examples

All the functions that can be called from int main() are declared at the beginning of the main.cpp file. The most useful functions are described in the section "General information" above.

In the INPUT folder, we provided two examples:

• the binary dataset SCOTUS_n9_N895_Data.dat, which is the dataset of votes of the US Supreme court analysed in Ref.[3] and used as an example in Ref.[1].
• the binary dataset Big5-IPC1_VS3_Ne5.dat: this is a binarized version of the first 100 000 samples of the Big 5 dataset [4], which has 50 variables. See paper [1] for comments on the Best Basis obtained for this dataset.

Each of these two datasets can be the one run as an example by commenting/uncommenting the correct datafile choice at the beginning of the makefile.

For hands-on and simple tests of the program, please check the examples in the function int main() of the main.cpp file.

[3] E.D. Lee, C.P. Broedersz, W. Bialek, Statistical Mechanics of the US Supreme Court. J Stat Phys 160, 275–301 (2015).

[4] Raw data from Open-Source Psychometrics Project in the line indicated as "Answers to the IPIP Big Five Factor Markers"; here is a direct link to the same zipfile.

License

This code is an open source project under the GNU GPLv3.