## Order Estimation for a Special Class of Hidden Markov Sources and Binary Renewal Processes (2002)

Venue: | IEEE Trans. Inform. Theory |

Citations: | 5 - 0 self |

### BibTeX

@ARTICLE{Khudanpur02orderestimation,

author = {Sanjeev Khudanpur and Prakash Narayan},

title = {Order Estimation for a Special Class of Hidden Markov Sources and Binary Renewal Processes},

journal = {IEEE Trans. Inform. Theory},

year = {2002},

volume = {48},

pages = {2002}

}

### OpenURL

### Abstract

We consider the estimation of the order, i.e., the number of hidden states, of a special class of discrete-time finite-alphabet hidden Markov sources. This class can be characterized in terms of equivalent renewal processes. No a priori bound is assumed on the maximum permissible order. An order estimator based on renewal types is constructed, and is shown to be strongly consistent by computing the precise asymptotics of the probability of estimation error. The probability of underestimation of the true order decays exponentially in the number of observations while the probability of overestimation goes to zero sufficiently fast. It is further shown that this estimator has the best possible error exponent in a large class of estimators. Our results are also valid for the general class of binary independent-renewal processes with finite mean renewal times.

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Citation Context ... notion of order corresponds to the depth of memory of the Markov process. Exceptions include the estimator based on Lempel–Ziv parsing in [17] and that based on the Bayesian Information Criterion i=-=n [6] (cf-=-. also [3]). Another set of results of interest to us appear in [10], where a strongly consistent order estimator is presented, which additionally possesses the best “error exponent,” i.e., the be... |

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Citation Context ...pper bound on the model order. A number of strongly consistent estimators have also been proposed for the order estimation for a hidden Markov source (cf. e.g., several relevant results cited in [2], =-=[15]-=-), where order now refers to the number of states of the underlying Markov chain. Most of these estimators also assume a prior bound on model order. Furthermore, while order estimators with exponentia... |

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Citation Context ...ect to the measure € .An alternative equivalent characterization of h@€ �€ A is as the € -almost sure limit (in �) of‘ I € @ˆ A � ��� € @ˆ “ [9], [14]. A The following la=-=rge deviations result, due to [1] (cf. also [8]), is relev-=-ant in characterizing the above-mentioned candidate for the optimal error exponent. Let e� &ˆ � , � aIYPY FFF, be a sequence of sets for which ��� ��� �3I € @e�A b HX (60) The... |

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Citation Context ...processes considered earlier, this candidate does indeed coincide with the known optimal error exponents. To this end, the divergence between two stationary and ergodic HMSs € and € on ˆ I is def=-=ined [9], [14] as h@€ �€ A 1 a ��� �3I i€ ��-=-��� € @ˆ�CI�ˆ � A € @ˆ�CI�ˆ � A (59) where i€ denotes expectation with respect to the measure € .An alternative equivalent characterization of h@€ �€ A is as the € -a... |

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Citation Context ...rror exponent; furthermore, such an estimator does not require a prior bound on the model order. This class of processes lends itself to an analysis based on the notion of renewal types introduced in =-=[7]-=-, which is then exploited in constructing an estimator along the lines of that proposed in [10]. In fact, our results are valid for the general class of binary independent-renewal processes with finit... |

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Citation Context ...t order estimator based on “mixture” probabilities, with an exponentially decaying underestimation probability, was presented: however, the corresponding exponent was not explicitly characterized.=-= In [13]-=-, another strongly consistent order estimator, based on maximum-likelihood parameter estimates, was presented, also with an exponentially decaying underestimation probability. It had been unclear if t... |