Causality measures

This module provides function to estimate causal influences of signals on each other.

Granger causality

Granger causality is a method to determine causal influence of one signal on another based on autoregressive modelling. It was developed by Nobel prize laureate Clive Granger and has been adopted in various numerical fields ever since (Granger, 1969). In its simplest form, the method tests whether the past values of one signal help to reduce the prediction error of another signal, compared to the past values of the latter signal alone. If it does reduce the prediction error, the first signal is said to Granger cause the other signal. Granger causality analysis can be extended to the spectral domain investigating the influnece signals have onto each other in a frequency resolved manner. It relies on estimating the cross-spectrum of time series and decomposing them into a transfer function and a noise covariance matrix. The method to estimate the spectral Granger causality is non-parametric in the sense that it does not require to fit an autoregressive model (see (Dhamala et al., 2008)).

Limitations

The user must be mindful of the method’s limitations, which are assumptions of covariance stationary data, linearity imposed by the underlying autoregressive modelling as well as the fact that the variables not included in the model will not be accounted for (Seth, 2007). When estimating spectral Granger causality the user must be familiar with basics the multitaper method to estimate power- and cross-spectra (e.g. sampling frequency, DPSS, time-half bandwidth product).

Implementation

The mathematical implementation of Granger causality methods in this module closely follows (Ding et al., 2006). The implementation of spectral Granger causality follows (Dhamala et al., 2008), (Wen et al., 2013) and (Wilson, 1972) for the spectral matrix factorization.

Overview of Functions

Various formulations of Granger causality have been developed. In this module you will find function for time-series data to test pairwise Granger causality (pairwise_granger).

Time-series Granger causality

pairwise_granger(signals, max_order[, ...])	Determine Granger Causality of two time series
conditional_granger(signals, max_order[, ...])	Determine conditional Granger Causality of the second time series on the first time series, given the third time series.

Spectral Granger causality

pairwise_spectral_granger(signal_i, signal_j)

Determine spectral Granger Causality of two signals.

Tutorial

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References

[Granger, 1969]

Clive WJ Granger. Investigating causal relations by econometric models and cross-spectral methods. Econometrica: journal of the Econometric Society, 37:424–438, 1969. doi:10.2307/1912791.

[Dhamala, 2008] (1,2)

Mukeshwar Dhamala, Govindan Rangarajan, and Mingzhou Ding. Analyzing information flow in brain networks with nonparametric granger causality. Neuroimage, 41(2):354–362, 2008.

[Seth, 2007]

A. Seth. Granger causality. Scholarpedia, 2(7):1667, 2007. revision #127333. doi:10.4249/scholarpedia.1667.

[Ding, 2006]

Mingzhou Ding, Yonghong Chen, and Steven L. Bressler. Granger causality: basic theory and application to neuroscience. 2006. arXiv:q-bio/0608035.

[Wen, 2013]

Xiaotong Wen, Govindan Rangarajan, and Mingzhou Ding. Multivariate granger causality: an estimation framework based on factorization of the spectral density matrix. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1997):20110610, 2013.

[Wilson, 1972]

G Tunnicliffe Wilson. The factorization of matricial spectral densities. SIAM Journal on Applied Mathematics, 23(4):420–426, 1972.