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Hidden Markov processes
 IEEE Trans. Inform. Theory
, 2002
"... Abstract—An overview of statistical and informationtheoretic aspects of hidden Markov processes (HMPs) is presented. An HMP is a discretetime finitestate homogeneous Markov chain observed through a discretetime memoryless invariant channel. In recent years, the work of Baum and Petrie on finite ..."
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Cited by 215 (5 self)
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Abstract—An overview of statistical and informationtheoretic aspects of hidden Markov processes (HMPs) is presented. An HMP is a discretetime finitestate homogeneous Markov chain observed through a discretetime memoryless invariant channel. In recent years, the work of Baum and Petrie on finitestate finitealphabet HMPs was expanded to HMPs with finite as well as continuous state spaces and a general alphabet. In particular, statistical properties and ergodic theorems for relative entropy densities of HMPs were developed. Consistency and asymptotic normality of the maximumlikelihood (ML) parameter estimator were proved under some mild conditions. Similar results were established for switching autoregressive processes. These processes generalize HMPs. New algorithms were developed for estimating the state, parameter, and order of an HMP, for universal coding and classification of HMPs, and for universal decoding of hidden Markov channels. These and other related topics are reviewed in this paper. Index Terms—Baum–Petrie algorithm, entropy ergodic theorems, finitestate channels, hidden Markov models, identifiability, Kalman filter, maximumlikelihood (ML) estimation, order estimation, recursive parameter estimation, switching autoregressive processes, Ziv inequality. I.
Theory of classification: A survey of some recent advances
, 2005
"... The last few years have witnessed important new developments in the theory and practice of pattern classification. We intend to survey some of the main new ideas that have led to these recent results. ..."
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Cited by 76 (3 self)
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The last few years have witnessed important new developments in the theory and practice of pattern classification. We intend to survey some of the main new ideas that have led to these recent results.
The consistency of the BIC Markov order estimator.
"... . The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet A) from observation of a sample path x 1 ; x 2 ; : : : ; x n , as that value k = k that minimizes the sum of the negative logarithm of the kth order maximum likelihood and the penalty term jAj ..."
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Cited by 64 (3 self)
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. The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet A) from observation of a sample path x 1 ; x 2 ; : : : ; x n , as that value k = k that minimizes the sum of the negative logarithm of the kth order maximum likelihood and the penalty term jAj k (jAj\Gamma1) 2 log n: We show that k equals the correct order of the chain, eventually almost surely as n ! 1, thereby strengthening earlier consistency results that assumed an apriori bound on the order. A key tool is a strong ratiotypicality result for Markov sample paths. We also show that the Bayesian estimator or minimum description length estimator, of which the BIC estimator is an approximation, fails to be consistent for the uniformly distributed i.i.d. process. AMS 1991 subject classification: Primary 62F12, 62M05; Secondary 62F13, 60J10 Key words and phrases: Bayesian Information Criterion, order estimation, ratiotypicality, Markov chains. 1 Supported in part by a joint N...
Optimal error exponents in hidden Markov models order estimation
 IEEE Trans. Inf. Theory
, 2003
"... Abstract—We consider the estimation of the number of hidden states (the order) of a discretetime finitealphabet hidden Markov model (HMM). The estimators we investigate are related to codebased order estimators: penalized maximumlikelihood (ML) estimators and penalized versions of the mixture es ..."
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Cited by 24 (5 self)
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Abstract—We consider the estimation of the number of hidden states (the order) of a discretetime finitealphabet hidden Markov model (HMM). The estimators we investigate are related to codebased order estimators: penalized maximumlikelihood (ML) estimators and penalized versions of the mixture estimator introduced by Liu and Narayan. We prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works. We prove a version of Stein’s lemma for HMM order estimation and derive an upper bound on underestimation exponents. Then we prove that this upper bound can be achieved by the penalized ML estimator and by the penalized mixture estimator. The proof of the latter result gets around the elusive nature of the ML in HMM by resorting to largedeviation techniques for empirical processes. Finally, we prove that for any consistent HMM order estimator, for most HMM, the overestimation exponent is null. Index Terms—Composite hypothesis testing, error exponents, generalized likelihood ratio testing, hidden Markov model (HMM), large deviations, order estimation, Stein’s lemma.
1 Nonparametric Statistical Inference for Ergodic Processes
"... Abstract—In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goo ..."
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Cited by 21 (19 self)
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Abstract—In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goodnessoffit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance. Index Terms—Nonparametric hypothesis testing, stationary ergodic processes, goodnessoffit test, process classification, change point problem. I.
Testing composite hypotheses about discretevalued stationary processes
 In ITW : 291– 295
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On hypotheses testing for ergodic processes
 In Proceedgings of Information Theory Workshop (2008
, 1998
"... We propose a method for statistical analysis of time series, that allows us to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, we consider three problems: goodnessoffit (or identity ..."
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Cited by 15 (15 self)
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We propose a method for statistical analysis of time series, that allows us to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, we consider three problems: goodnessoffit (or identity) testing, process classification, and the change point problem. For each of the problems we construct a test that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance.
NUMBER OF HIDDEN STATES AND MEMORY: A JOINT ORDER ESTIMATION PROBLEM FOR MARKOV CHAINS WITH MARKOV REGIME
"... Abstract. This paper deals with order identification for Markov chains with Markov regime (MCMR) in the context of finite alphabets. We define the joint order of a MCMR process in terms of the number k of states of the hidden Markov chain and the memory m of the conditional Markov chain. We study th ..."
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Cited by 5 (4 self)
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Abstract. This paper deals with order identification for Markov chains with Markov regime (MCMR) in the context of finite alphabets. We define the joint order of a MCMR process in terms of the number k of states of the hidden Markov chain and the memory m of the conditional Markov chain. We study the properties of penalized maximum likelihood estimators for the unknown order (k, m) of an observed MCMR process, relying on information theoretic arguments. The novelty of our work relies in the joint estimation of two structural parameters. Furthermore, the different models in competition are not nested. In an asymptotic framework, we prove that a penalized maximum likelihood estimator is strongly consistent without prior bounds on k and m. We complement our theoretical work with a simulation study of its behaviour. We also study numerically the behaviour of the BIC criterion. A theoretical proof of its consistency seems to us presently out of reach for MCMR, as such a result does not yet exist in the simpler case where m = 0 (that is for hidden Markov models). Résumé. Ce travail porte sur l’identification de l’ordre d’une chaîne de Markov à régime Markovien (MCMR) sur un alphabet fini. L’ordre d’une MCMR est défini comme le couple (k, m) où k est le nombre d’états de la chaîne cachée et m la mémoire de la chaîne de Markov conditionnelle. Nous étudions des estimateurs du maximum de vraisemblance pénalisée en utilisant des techniques issues de
Discrete Universal Filtering via Hidden Markov Modelling
"... Abstract — We consider the discrete universal filtering problem, where the components of a discrete signal emitted by an unknown source and corrupted by a known DMC are to be causally estimated. We derive a family of filters which we show to be universally asymptotically optimal in the sense of achi ..."
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Cited by 3 (2 self)
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Abstract — We consider the discrete universal filtering problem, where the components of a discrete signal emitted by an unknown source and corrupted by a known DMC are to be causally estimated. We derive a family of filters which we show to be universally asymptotically optimal in the sense of achieving the optimum filtering performance when the clean signal is stationary, ergodic, and satisfies an additional mild positivity condition. Our schemes are based on approximating the noisy signal by a hidden Markov process (HMP) via maximumlikelihood (ML) estimation, followed by use of the wellknown forward recursions for HMP state estimation. We show that as the data length increases, and as the number of states in the HMP approximation increases, our family of filters attain the performance of the optimal distributiondependent filter. I.
Uniform hypothesis testing for ergodic time series distributions
, 2011
"... Given a discretevalued sample X1,...,Xn we wish to decide whether it was generated by a distribution belonging to a family H0, or it was generated by a distribution belonging to a family H1. In this work we assume that all distributions are stationary ergodic, and do not make any further assumption ..."
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Given a discretevalued sample X1,...,Xn we wish to decide whether it was generated by a distribution belonging to a family H0, or it was generated by a distribution belonging to a family H1. In this work we assume that all distributions are stationary ergodic, and do not make any further assumptions (e.g. no independence or mixing rate assumptions). We would like to have a test whose probability of error (both Type I and Type II) is uniformly bounded. More precisely, we require that for each ε there exist a sample size n such that probability of error is upperbounded by ε for samples longer than n. We find some necessaryand some sufficient conditions on H0 and H1 under which a consistent test (with this notion of consistency) exists. These conditions are topological, with respect to the topology of distributional distance. 1