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67
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 258 (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.
A tutorial introduction to the minimum description length principle
 in Advances in Minimum Description Length: Theory and Applications. 2005
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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 93 (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.
Datadriven calibration of penalties for leastsquares regression
, 2008
"... Penalization procedures often suffer from their dependence on multiplying factors, whose optimal values are either unknown or hard to estimate from data. We propose a completely datadriven calibration algorithm for these parameters in the leastsquares regression framework, without assuming a parti ..."
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Cited by 55 (14 self)
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Penalization procedures often suffer from their dependence on multiplying factors, whose optimal values are either unknown or hard to estimate from data. We propose a completely datadriven calibration algorithm for these parameters in the leastsquares regression framework, without assuming a particular shape for the penalty. Our algorithm relies on the concept of minimal penalty, recently introduced by Birgé and Massart (2007) in the context of penalized least squares for Gaussian homoscedastic regression. On the positive side, the minimal penalty can be evaluated from the data themselves, leading to a datadriven estimation of an optimal penalty which can be used in practice; on the negative side, their approach heavily relies on the homoscedastic Gaussian nature of their stochastic framework. The purpose of this paper is twofold: stating a more general heuristics for designing a datadriven penalty (the slope heuristics) and proving that it works for penalized leastsquares regression with a random design, even for heteroscedastic nonGaussian data. For technical reasons, some exact mathematical results will be proved only for regressogram binwidth selection. This is at least a first step towards further results, since the approach and the method that we use are indeed general.
Context tree estimation for not necessarily finite memory processes, via BIC and MDL
 IEEE Trans. Inf. Theory
, 2006
"... The concept of context tree, usually defined for finite memory processes, is extended to arbitrary stationary ergodic processes (with finite alphabet). These context trees are not necessarily complete, and may be of infinite depth. The familiar BIC and MDL principles are shown to provide strongly co ..."
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Cited by 54 (1 self)
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The concept of context tree, usually defined for finite memory processes, is extended to arbitrary stationary ergodic processes (with finite alphabet). These context trees are not necessarily complete, and may be of infinite depth. The familiar BIC and MDL principles are shown to provide strongly consistent estimators of the context tree, via optimization of a criterion for hypothetical context trees of finite depth, allowed to grow with the sample size n as o(log n). Algorithms are provided to compute these estimators in O(n) time, and to compute them online for all i ≤ n in o(n log n) time.
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 28 (6 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.
Application of Kolmogorov complexity and universal codes to identity testing and nonparametric testing of serial independence for time series
, 2006
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Activity modeling using event probability sequences
 IEEE Transactions on image processing
"... Abstract—Changes in motion properties of trajectories provide useful cues for modeling and recognizing human activities. We associate an event with significant changes that are localized in time and space, and represent activities as a sequence of such events. The localized nature of events allows f ..."
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Cited by 17 (0 self)
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Abstract—Changes in motion properties of trajectories provide useful cues for modeling and recognizing human activities. We associate an event with significant changes that are localized in time and space, and represent activities as a sequence of such events. The localized nature of events allows for detection of subtle changes or anomalies in activities. In this paper, we present a probabilistic approach for representing events using the hidden Markov model (HMM) framework. Using trained HMMs for activities, an event probability sequence is computed for every motion trajectory in the training set. It reflects the probability of an event occurring at every time instant. Though the parameters of the trained HMMs depend on viewing direction, the event probability sequences are robust to changes in viewing direction. We describe sufficient conditions for the existence of view invariance. The usefulness of the proposed event representation is illustrated using activity recognition and anomaly detection. Experiments using the indoor University of Central Florida human action dataset, the Carnegie Mellon University Credo Intelligence, Inc., Motion Capture dataset, and the outdoor Transportation Security Administration airport tarmac surveillance dataset show encouraging results. Index Terms—Activity modeling, event detection, hidden Markov model (HMM). I.
Universal Codes as a Basis for Time Series Testing
 Statistical Methodology
, 2006
"... We suggest a new approach to hypothesis testing for ergodic and stationary processes. In contrast to standard methods, the suggested approach gives a possibility to make tests, based on any lossless data compression method even if the distribution law of the codeword lengths is not known. We apply t ..."
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Cited by 16 (7 self)
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We suggest a new approach to hypothesis testing for ergodic and stationary processes. In contrast to standard methods, the suggested approach gives a possibility to make tests, based on any lossless data compression method even if the distribution law of the codeword lengths is not known. We apply this approach to the following four problems: goodnessoffit testing (or identity testing), testing for independence, testing of serial independence and homogeneity testing and suggest nonparametric statistical tests for these problems. It is important to note that practically used socalled archivers can be used for suggested testing.