Copyright (C) 2020 Oliver Cliff.

This repository provides MATLAB functions for computing and assessing the linear dependence between multiple autocorrelated time series. This includes various linear dependence measures and the hypothesis tests for inferring their significance, all discussed in our paper in Phys. Rev. Research and arXiv.

The measures implemented are: mutual information, conditional mutual information, Granger causality, and conditional Granger causality (each for univariate and multivariate linear-Gaussian processes). For completeness we have also included Pearson correlation and partial correlation for univariate processes (with a potentially multivariate conditional process).

The code is licensed under the GNU GPL v3 license (or later).

1. Ensure you have MATLAB's Econometrics Toolbox and Signal Processing Toolbox downloaded and installed (for the autocorrelation and filtering functions).
2. Clone (or download) the repository.
3. Add the repository to your path (including the utils folder), e.g., by using: addpath(genpath('/path/to/repository')).
4. Documentation found in help for each function. The main functions used are mvmi.m (mutual information), mvgc.m (Granger causality), and pcorr.m (Pearson/partial correlation). All three allow adding a conditional process and can optionally output a p-value. The p-values are either generated from the finite/asymptotic tests (i.e., the chi-square and F-tests), or the modified tests (modified F- and lambda-tests) that we derive in our paper.
5. Demos are included in the demos subfolder, including all experiments from the paper. Start with demos/demoForPaper.m or demos/hcpCaseStudy.m to see all the results from the paper. As an example of modifying Student's t-tests for Pearson correlation, we have also included a demo on inferring cross-correlations between univariate autocorrelated processes in demos/demoCrossCorr.m.

Please cite your use of this code as:

Oliver M. Cliff, Leonardo Novelli, Ben D Fulcher, James M. Shine, Joseph T. Lizier, "Assessing the significance of directed and multivariate measures of linear dependence between time series," Phys. Rev. Research 3, 013145 (2021).

This project was in part supported through:

• The Australian Research Council DECRA grant DE160100630.
• A University of Sydney Robinson Fellowship and NHMRC Project Grant 1156536.
• The University of Sydney Research Accelerator (SOAR) Fellowship program.
• The Centre for Translational Data Science at The University of Sydney’s Research Incubator Funding Scheme.