Reconciling emergences: Identifying causal emergence in multivariate data

This code provides basic functionality to test hypotheses about causal emergence in time series data. This repository contains the companion software for the paper:

Rosas*, Mediano*, et al. (2020). Reconciling emergences: An information-theoretic approach to identify causal emergence in multivariate data. PLoS Computational Biology, 16(12): e1008289. DOI: 10.1371/journal.pcbi.1008289

Please cite the paper (and give us a shout!) if you use this software. Please contact Pedro Mediano for bug reports, pull requests, and feature requests.

Basic theory

The paper provides a mathematical account of emergence in complex dynamical systems. In a nutshell, a macroscopic feature V of a multivariate system X is said to be causally emergent if it contains some information about the future of X that no microscopic element of X has on its own. Furthermore, it is possible to distinguish between two kinds of emergence: downward causation (when macroscopic features have an effect on microscopic elements), and causal decoupling (when macroscopic features have an effect on other macroscopic features). These intuitions are formulated rigorously using the Partial Information Decomposition and the Integrated Information Decomposition frameworks.

Unfortunately, information decomposition is in general difficult and computationally expensive (for now). Fortunately, the theory also provides a set of practical criteria (Ψ, Δ, and Γ) to detect emergence which can be used when a given macroscopic feature V is known. In particular:

• If Ψ > 0, then V is causally emergent.
• If Δ > 0, then V shows downward causation.
• If Ψ > 0 and Γ = 0, then V shows causal decoupling.

Note, however, that the converse does not hold: a negative Ψ does not imply that V is not causally emergent -- see section III.A of the paper for details.

Finally, note that for the assumptions of the theory to hold, the candidate emergent feature V has to be a supervenient feature of X -- in other words, V(t,:) has to be a (possibly stochastic) function of X(t,:), and nothing else.

Code examples

As a simple example, consider the bivariate binary system from Fig. 1 (right) from the paper:

T = 1000;
X = zeros([T,2]);
for t=2:T
  X(t,1) = xor(X(t-1,1), X(t-1,2));
  X(t,2) = rand < 0.5;
end

V = xor(X(:,1), X(:,2));
psi   = EmergencePsi(X, V);
delta = EmergenceDelta(X, V);

You will see that, as expected, Ψ is close to zero but Δ is positive and close to one.

Note, however, that the values are not exactly zero or one, as any estimation on a finite dataset is bound to have some bias and variance. To estimate the variance and correct the bias, it is common to perform surrogate data tests by time shuffling. Assuming that T = size(X,1) and that nb_surr is a large(-ish) integer, this can be done by:

surr_psi = arrayfun(@(j) EmergencePsi(X(randperm(T),:), V(randperm(T),:)), 1:nb_surr);

With this, the bias-corrected Ψ is psi - mean(surr_psi) and its standard deviation is std(surr_psi).

Download and installation

None, really. Just download this folder (with Github's zip download or with git clone), add it to your Octave/Matlab path, and enjoy.

Note, however, that if you use Octave you will need to install the statistics package. You can do this simply by running pkg install -forge io statistics (and remember to run pkg load statistics every time you start a new session).

Tests are provided in the tests/ subfolder. To run them in Matlab, run runtests('tests/') from this repository's root folder.

Use in Python

All functions in this repository are Octave-friendly, which means they can be easily called from Python through the wholesome oct2py package. With a functional Octave and Python installation, you can simply run (from the root folder of the repo):

import numpy.random as rn
from oct2py import Oct2Py

oc = Oct2Py()
oc.EmergencePsi(rn.randn(100,2), rn.randn(100,1))

Licence

This software is distributed under the modified 3-clause BSD Licence.

Further reading

• P. Mediano*, F. Rosas*, et al. (2019). Beyond integrated information: A taxonomy of information dynamics phenomena. arXiv:1909.02297

• P. Williams and R. Beer (2010). Nonnegative decomposition of multivariate information. arXiv:1004.2515

(C) Pedro Mediano and Fernando Rosas, 2020