elephant.causality.granger.pairwise_granger

elephant.causality.granger.pairwise_granger(signals, max_order, information_criterion='aic')

Determine Granger Causality of two time series

Parameters:

signals(N, 2) np.ndarray or neo.AnalogSignal

A matrix with two time series (second dimension) that have N time points (first dimension).

max_orderint

Maximal order of autoregressive model.

information_criterion{‘aic’, ‘bic’}, optional

A function to compute the information criterion:

• bic for Bayesian information_criterion,

• aic for Akaike information criterion,

Default: ‘aic’

Returns:

Causality

A namedtuple with the following attributes:

directional_causality_x_yfloat

The Granger causality value for X influence onto Y.

directional_causality_y_xfloat

The Granger causality value for Y influence onto X.

instantaneous_causalityfloat

The remaining channel interdependence not accounted for by the directional causalities (e.g. shared input to X and Y).

total_interdependencefloat

The sum of the former three metrics. It measures the dependence of X and Y. If the total interdependence is positive, X and Y are not independent.

Denote covariance matrix of signals

X by C|X - a real number Y by C|Y - a real number (X,Y) by C|XY - a (2 times 2) matrix

directional causality X -> Y given by

log(C|X / C|XY_00)

directional causality Y -> X given by

log(C|Y / C|XY_11)

instantaneous causality of X,Y given by

log(C|XY_00 / C|XY_11)

total interdependence of X,Y given by

log( {C|X cdot C|Y} / det{C|XY} )

Raises:

ValueError

If the provided signal does not have a shape of Nx2.

If the determinant of the prediction error covariance matrix is not positive.

Warns:

UserWarning

If the log determinant of the prediction error covariance matrix is below the tolerance level of 1e-7.

Notes

The formulas used in this implementation follows (Ding et al., 2006). The only difference being that we change the equation 47 in the following way: -R(k) - A(1)R(k - 1) - … - A(m)R(k - m) = 0. This forumlation allows for the usage of R values without transposition (i.e. directly) in equation 48.

Examples

Example 1. Independent variables.

>>> import numpy as np
>>> from elephant.causality.granger import pairwise_granger
>>> pairwise_granger(np.random.uniform(size=(1000, 2)), max_order=2)  # noqa
Causality(directional_causality_x_y=-0.0, directional_causality_y_x=0.0, instantaneous_causality=0.0, total_interdependence=0.0)

Example 2. Dependent variables. Y depends on X but not vice versa.

\[\begin{split}\begin{array}{ll} X_t \sim \mathcal{N}(0, 1) \\ Y_t = 3.5 \cdot X_{t-1} + \epsilon, \; \epsilon \sim\mathcal{N}(0, 1) \end{array}\end{split}\]

In this case, the directional causality is non-zero.

>>> np.random.seed(31)
>>> x = np.random.randn(1001)
>>> y = 3.5 * x[:-1] + np.random.randn(1000)
>>> signals = np.array([x[1:], y]).T  # N x 2 matrix
>>> pairwise_granger(signals, max_order=1)  # noqa
Causality(directional_causality_x_y=2.64, directional_causality_y_x=-0.0, instantaneous_causality=0.0, total_interdependence=2.64)