Abstract
We present a framework for the estimation of transfer entropy (TE) under the conditions typical of physiological system analysis, featuring short multivariate time series and the presence of instantaneous causality (IC). The framework is based on recognizing that TE can be interpreted as the difference between two conditional entropy (CE) terms, and builds on an efficient CE estimator that compensates for the bias occurring for high dimensional conditioning vectors and follows a sequential embedding procedure whereby the conditioning vectors are formed progressively according to a criterion for CE minimization. The issue of IC is faced accounting for zero-lag interactions according to two alternative empirical strategies: if IC is deemed as physiologically meaningful, zero-lag effects are assimilated to lagged effects to make them causally relevant; if not, zero-lag effects are incorporated in both CE terms to obtain a compensation. The resulting compensated TE (cTE) estimator is tested on simulated time series, showing that its utilization improves sensitivity (from 61% to 96%) and specificity (from 5/6 to 0/6 false positives) in the detection of information transfer respectively when instantaneous effect are causally meaningful and non-meaningful. Then, it is evaluated on examples of cardiovascular and neurological time series, supporting the feasibility of the proposed framework for the investigation of physiological mechanisms.
Highlights
Since its first introduction by Schreiber [1], transfer entropy (TE) has been recognized as a powerful tool for detecting the transfer of information between joint processes
The compensation performed in (5) is alternative to the test of timeshifted data recently proposed to detect instantaneous mixing between coupled processes [4]. Note that in both compensations in (4) and (5) instantaneous effects possibly occurring from any scalar element of Z towards Y are conditioned out considering the present term zn, in addition to the past terms z1:n-1, in the two conditional entropy (CE) computations; this is done to avoid that indirect effects x1:n-1→zn→yn were misinterpreted as the presence of predictive information transfer from the system X to the system Y
Our results suggest that the framework proposed in this study for the practical estimation of multivariate TE can successfully deal with the issues arising in the conditions typical of physiological time series analysis
Summary
Since its first introduction by Schreiber [1], transfer entropy (TE) has been recognized as a powerful tool for detecting the transfer of information between joint processes. The formulation of TE does not assume any particular model as underlying the interaction between the considered processes, making it sensitive to all types of dynamical interaction The popularity of this tool has grown even more with the recent elucidation of its close connection with the ubiquitous concept of Granger causality [2], which has led to formally bridge information-theoretic and predictive approaches to the evaluation of directional interactions between processes. One major challenge is the estimation of the probability density functions involved in TE computation from datasets the length of which is limited by experimental constraints and/or by the need for stationarity [13,14] Another critical point is that, to exploit the dynamical information contained in the transition probabilities, one should cover reasonably well the past history of the observed processes; since this corresponds to work with long conditioning vectors represented into high-dimensional spaces, TE estimation from short time series is further hampered, especially in the presence of multiple processes and long memory effects [15]. TE allows to make explicit the problem of disregarding instantaneous causality in the computation of the predictive information transfer in multivariate time series
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