14. The new math of EEG: Symbolic transfer entropy, the effects of dimension

Abstract

Symbolic transfer entropy (STE) and transfer entropy (TE) has been proposed as a means to detect exchange of information between two signal sources, such as areas of the brain 1 . The computation of STE and TE assumes that the time series that make up the signals to be analysed have been properly embedded in a higher dimensional space. The significance of dimension is examined. The analysis was conducted using simulated signals consisting of Gaussian noise, coupled Lorenz oscillators, sum-of-sinusoids and a combination of the Lorenz oscillators and sinusoids. Normalized STE was calculated according to 2 using a range of dimensions and delay values for data sets of 2500 data points. The data sets were padded with Gaussian noise in order to induce a time lag between the driver signal and the responding signal. Gaussian noise posed the biggest challenge at D = 3 or 4. In both cases the calculated direction of information flow was + 1 or −1 for 60% of the data sets whereas 60% were correctly labelled a zero at D = 5. Otherwise D = 3 or 4 correctly identified the direction of information flow for the Lorenz data sets and the combination data sets. D = 5 performed best at identifying the lack of net information flow in the sum-of-sinusoids data sets. Increasing delay values, tau, resulted in proportionally greater increases in calculated reverse information flow than forward flow; this effect was more pronounced at higher dimension. NSTE was sensitive to the direction of information flow if as little as 15% the signal and the remainder being background. NSTE may be less sensitive to false neighbours than TE for complex reasons. The correct embedding dimension appears to be highly dependent on the characteristics of the information being analysed.