Significance of transfer entropy calculations

Question

I have two time-series U and V (approximately 300 samples) having values between 0 and 5. I wanted to calculate the (first order) transfer entropy $I_{uv}$ (of U with the knowledge of V) and $I_{vu}$ (of V with the knowledge of U). Here are the values I got :

$I_{uv}$ = 0.1492 and $I_{vu}$ = 0.1342

Since these values are close to each other, I started questioning their significance. Thus, I calculated the transfer entropy $I_{u-rv}$ of U with the knowledge of a randomized (in time) version of V, and the transfer entropy $I_{v-ru}$ of V with the knowledge of a randomized (in time) version of U. I calculated 500 values of $I_{u-rv}$ and $I_{v-ru}$, which gave me :

mean($I_{u-rv}$ ) = 0.1218 - standard deviation = 0.0198

mean($I_{v-ru}$) = 0.1019 - standard deviation = 0.0198

The values of $I_{uv}$ and $I_{vu}$ lie close to (mean+std) so I would say there is an influence of one time serie on the other. From these information what can really be said about the two time-series ? (I will upload a picture of these time-series as soon as I figure a way to do it)

Answer 1

You are certainly addressing the problem in the correct manner: creating surrogate "source" time series by randomising/shuffling (while leaving the "destination" unchanged), then computing TE for each surrogate, so as to obtain the distribution of surrogate TEs that one would see were there no source-destination relationship (while other aspects of the time-series are controlled). You can then make a statistical comparison of your measured TE to this distribution so as to check if there is a statistically significant source-destination relationship.

This approach is described formally in the following papers:

• Chavez et. al., "Statistical assessment of nonlinear causality: application to epileptic EEG signals", Journal of Neuroscience Methods 124 (2003) 113-128.
• Verdes, P.F., "Assessing causality from multivariate time series", Phys. Rev. E 72(2), 026222–9 (2005)
• Vicente et al., "Transfer entropy—a model-free measure of effective connectivity for the neurosciences", Journal of Computational Neuroscience, Vol. 30, No. 1. (2011), pp. 45-67
• Lizier et al., "Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity", Journal of Computational Neuroscience, Vol. 30, No. 1. (2011), pp. 85-107.

Assuming your surrogate distribution followed a normal distribution, you could do Student's t-test for significance of the measured values; while their p-values are high, neither is statistically significant at alpha=0.05; I would say that you don't have a statistically significant relationship there.