Regularities Unseen, Randomness Observed: Levels of Entropy Convergence

Details Regularities Unseen, Randomness Observed: Levels of Entropy Convergence Regularities Unseen, Randomness Observed: Levels of Entropy Convergence

2001-02-09

arxiv.org/abs/cond-mat/0102181v1

Physics

Condensed Matter

Statistical Mechanics

35 pages, 21 fgures, Santa Fe Institute Working Paper 01-02-012

Scientific paper

We study how the Shannon entropy of sequences produced by an information source converges to the source's entropy rate. We synthesize several phenomenological approaches to applying information theoretic measures of randomness and memory to stochastic and deterministic processes by using successive derivatives of the Shannon entropy growth curve. This leads, in turn, to natural measures of apparent memory stored in a source and the amounts of information that must be extracted from observations of a source in order for it to be optimally predicted and for an observer to synchronize to it. One consequence of ignoring these structural properties is that the missed regularities are converted to apparent randomness. We demonstrate that this problem arises particularly for small data sets; e.g., in settings where one has access only to short measurement sequences.

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