Inferring the directionality of coupling with conditional mutual information

Uncovering the directionality of coupling is a significant step in understanding drive-response relationships in complex systems. In this paper, we discuss a nonparametric method for detecting the directionality of coupling based on the estimation of information theoretic functionals. We consider several different methods for estimating conditional mutual information. The behavior of each estimator with respect to its free parameter is shown using a linear model where an analytical estimate of conditional mutual information is available. Numerical experiments in detecting coupling directionality are performed using chaotic oscillators, where the influence of the phase extraction method and relative frequency ratio is investigated.

Figure 1

Dependence of the mean conditional mutual information estimate by the $B$-spline method (top) and by the equiquantal estimator (bottom) on the number of samples in the time series for different numbers of bins.

Figure 2

Dependence of the mean conditional mutual information estimated with the $k$-nearest-neighbor estimator on the number of samples in the time series for different $k$.

Figure 3

Dependence of the mean conditional mutual information estimated with the cross-redundancies estimator on the number of samples in the time series for different neighborhood sizes.

Figure 4

Rössler 1:1 experiment. Dependence of the false positive rate of the cross-redundancy method for various neighbor sizes. Only $ϵ\ge 0.1$ yielded acceptable false positive levels in the experiment.

Figure 5

Rössler 1:1 experiment. Dependence of the false positive rate of the $k\text{NN}$ method for various nearest-neighbor counts. The false positive rate is zero for $k\in \left\{16,32,64\right\}$.

Figure 6

Rössler 5:1 experiment. Histogram of conditional mutual information estimates for the model data and for the permutation surrogates where either one of the time series or both of them were randomized. The coupling is unidirectional from the faster system to the slower system. In the direction of coupling all the surrogates generate very similar distributions of the CMI index (top). However, in the reverse direction when the fast system is also randomized, the width of the distribution of the surrogates is clearly smaller than that of the distribution arising from the original data (bottom).