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Entropy 2015, 17(7), 4891-4917; doi:10.3390/e17074891

Informational and Causal Architecture of Discrete-Time Renewal Processes

1
Department of Physics, University of California at Berkeley, Berkeley, CA 94720-5800, USA
2
Complexity Sciences Center and Department of Physics, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA
*
Authors to whom correspondence should be addressed.
Academic Editor: Antonio M. Scarfone
Received: 17 March 2015 / Revised: 1 July 2015 / Accepted: 9 July 2015 / Published: 13 July 2015
(This article belongs to the Section Statistical Mechanics)
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Abstract

Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic for their prediction (the set of causal states), calculate the historical memory capacity required to store those states (statistical complexity), delineate what information is predictable (excess entropy), and decompose the entropy of a single measurement into that shared with the past, future, or both. The causal state equivalence relation defines a new subclass of renewal processes with a finite number of causal states despite having an unbounded interevent count distribution. We use the resulting formulae to analyze the output of the parametrized Simple Nonunifilar Source, generated by a simple two-state hidden Markov model, but with an infinite-state machine presentation. All in all, the results lay the groundwork for analyzing more complex processes with infinite statistical complexity and infinite excess entropy. View Full-Text
Keywords: stationary renewal process; statistical complexity; predictable information; information anatomy; entropy rate stationary renewal process; statistical complexity; predictable information; information anatomy; entropy rate
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Marzen, S.E.; Crutchfield, J.P. Informational and Causal Architecture of Discrete-Time Renewal Processes. Entropy 2015, 17, 4891-4917.

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