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Predictability of behavior has emerged an an important characteristic in manyfields including biology, medicine, and marketing. Behavior can be recorded asa sequence of actions performed by an individual over a given time period. Thissequence of actions can often be modeled as a stationary time-homogeneousMarkov chain and the predictability of the individual's behavior can bequantified by the entropy rate of the process. This paper provides acomprehensive investigation of three estimators of the entropy rate of finiteMarkov processes and a bootstrap procedure for providing standard errors. Thefirst two methods directly estimate the entropy rate through estimates of thetransition matrix and stationary distribution of the process; the methodsdiffer in the technique used to estimate the stationary distribution. The thirdmethod is related to the sliding-window Lempel-Ziv (SWLZ) compressionalgorithm. The first two methods achieve consistent estimates of the trueentropy rate for reasonably short observed sequences, but are limited byrequiring a priori specification of the order of the process. The method basedon the SWLZ algorithm does not require specifying the order of the process andis optimal in the limit of an infinite sequence, but is biased for shortsequences. When used together, the methods can provide a clear picture of theentropy rate of an individual's behavior.

Estimating the Entropy Rate of Finite Markov Chains with Application to Behavior Studies