Dimensional Reduction of Conditional Algebraic Multi-Information Via Transcripts

Symbolic representation is a standard and powerful technique in time series analysis. In an ordinal symbolic representation the symbols are the so-called ordinal patterns, which can be identified with permutations. Transcripts exploit the fact that permutations build a group, the transcript of a pair of permutations being the product of the second permutation times the inverse of the first one. This particular setting can be easily generalized to any representation which elements belong to an algebraic group. The dimensional reduction of conditional multi-information via transcripts, proved in this paper, perfectly shows the potential of such algebraic symbolic representations. Specifically, given N+M group-valued random variables, the multi-information of N variables conditioned on the other M variables can also be calculated as a multi-information of N transcripts conditioned on M-1 transcripts, under some restrictions. Such a dimensional reduction can be crucial when estimating a conditional multi-information from short time series. Applications include two popular ordinal indicators of the information flow in coupled time series, namely, symbolic transfer entropy and momentary sorting information transfer. As a by-product of the above results, two new information directionality indicators based on ordinal transcripts are proposed, the simplest one being an (unconditioned) mutual information.