Figure 3.

Graphical visualization of TE. (A) Coupled systems X Y. To test directed interaction X Y we predict a future Y(t + u) (star) once from past values (circles) of Y alone: <a onClick="popup('http://www.biomedcentral.com/1471-2202/12/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2202/12/119/mathml/M4">View MathML</a>, once from past values of Y and X: <a onClick="popup('http://www.biomedcentral.com/1471-2202/12/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2202/12/119/mathml/M5">View MathML</a>. d - embedding dimension, τ - embedding lag. (B) Embedding. Y(t + u), Y(t), X(t) - coordinates in the embedding space, repetition of embedding for all t gives an estimate of the probability p(Y(t + u), Y(t),X(t)) (part C, embedding dimensions limited to 1).(C) p(Y(t + u)|Y(t),X(t)) - probability to observe Y(t+u) after Y(t) and X(t) were observed. This probability can be used for a prediction of the future of Y from the past of X and Y. Here, p(Y(t + u)|Y(t), X(t)) is obtained by a binning approach. We compute p(Y(t + u) ± Δ, Y(t) ± Δ,X(t) ± Δ), let Δ → 0 and normalize by p(Y(t),X(t))). TRENTOOL computes these densities via a Kernel-estimator. (D) p(Y(t + u)|Y(i)) predicts Y(t + u) from Y(t), without knowing about X(t). It predicts the future of Y from the past of Y alone. (E) If the past of X is irrelevant for prediction, the conditional distributions p(Y(t + u)|Y(t), X(t)), should be all equal to p(Y(t + u)|Y(t)). Differences indicate directed interaction from X to Y. Their weighted sum is transfer entropy.

Lindner et al. BMC Neuroscience 2011 12:119   doi:10.1186/1471-2202-12-119
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