Two new Markov order estimators

We present two new methods for estimating the order (memory depth) of a finite alphabet Markov chain from observation of a sample path. One method is based on entropy estimation via recurrence times of patterns, and the other relies on a comparison of empirical conditional probabilities. The key to both methods is a qualitative change that occurs when a parameter (a candidate for the order) passes the true order. We also present extensions to order estimation for Markov random fields.

[1]  I. Csiszár,et al.  Consistent estimation of the basic neighborhood of Markov random fields , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[2]  Imre Csiszár.  Large-scale typicality of Markov sample paths and consistency of MDL Order estimators , 2002, IEEE Trans. Inf. Theory.

[3]  I. Csiszar,et al.  The consistency of the BIC Markov order estimator , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[4]  Jorma Rissanen,et al.  The Minimum Description Length Principle in Coding and Modeling , 1998, IEEE Trans. Inf. Theory.

[5]  Ioannis Kontoyiannis,et al.  Asymptotic Recurrence and Waiting Times for Stationary Processes , 1998 .

[6]  P. Shields.  The Ergodic Theory of Discrete Sample Paths , 1996 .

[7]  Philippe Jacquet,et al.  Asymptotic Behavior of the Lempel-Ziv Parsing Scheme and Digital Search Trees , 1995, Theor. Comput. Sci..

[8]  Entropy and recurrence rates for stationary random fields , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

[9]  A. Dembo,et al.  A Topological Criterion for Hypothesis Testing , 1994 .

[10]  N. Fisher,et al.  Probability Inequalities for Sums of Bounded Random Variables , 1994 .

[11]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[12]  Benjamin Weiss,et al.  How Sampling Reveals a Process , 1990 .

[13]  Aaron D. Wyner,et al.  Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression , 1989, IEEE Trans. Inf. Theory.

[14]  Jacob Ziv,et al.  Coding theorems for individual sequences , 1978, IEEE Trans. Inf. Theory.

[15]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[16]  A. J. Sarantakis,et al.  Entropy and data compression , 1970 .

[17]  Kazuoki Azuma.  WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES , 1967 .