11 research outputs found
An improved bound for the exponential stability of predictive filters of hidden Markov models
We consider hidden Markov processes in discrete time with a finite state space X and a general observation or read-out space Y, which is assumed to be a Polish space. It is well-known that in the statistical analysis of HMMs the so-called predictive filter plays a fundamental role. A useful result establishing the exponential stability of the predictive filter with respect to perturbations of its initial condition was given in the paper of LeGland and Mevel, MCSS, 2000, in the case, when the assumed transition probability matrix was primitive. The main technical result of the present paper is the extension of the cited result by showing that the random constant and the deterministic positive exponent showing up in the inequality stating exponential stability can be chosen so that for any prescribed s exceeding 1 the s-th exponential moment of the random constant is finite. An application of this result to the estimation of HMMs with primitive transition probability densities will be also briefly presented
Limit Theorems in Hidden Markov Models
In this paper, under mild assumptions, we derive a law of large numbers, a
central limit theorem with an error estimate, an almost sure invariance
principle and a variant of Chernoff bound in finite-state hidden Markov models.
These limit theorems are of interest in certain ares in statistics and
information theory. Particularly, we apply the limit theorems to derive the
rate of convergence of the maximum likelihood estimator in finite-state hidden
Markov models.Comment: 35 page
Maximum likelihood estimation in possibly misspecified dynamic models with time-inhomogeneous Markov Regimes
This paper considers maximum likelihood (ML) estimation in a large class of models with hidden Markov regimes. We investigate consistency and local asymptotic normality of the ML estimator under general conditions which allow for autoregressive dynamics in the observable process, time-inhomogeneous Markov regime sequences, and possible model misspecification. A Monte Carlo study examines the finite-sample properties of the ML estimator. An empirical application is also discussed
Optimization of Information Rate Upper and Lower Bounds for Channels with Memory
We consider the problem of minimizing upper bounds and maximizing lower
bounds on information rates of stationary and ergodic discrete-time channels
with memory. The channels we consider can have a finite number of states, such
as partial response channels, or they can have an infinite state-space, such as
time-varying fading channels. We optimize recently-proposed information rate
bounds for such channels, which make use of auxiliary finite-state machine
channels (FSMCs). Our main contribution in this paper is to provide iterative
expectation-maximization (EM) type algorithms to optimize the parameters of the
auxiliary FSMC to tighten these bounds. We provide an explicit, iterative
algorithm that improves the upper bound at each iteration. We also provide an
effective method for iteratively optimizing the lower bound. To demonstrate the
effectiveness of our algorithms, we provide several examples of partial
response and fading channels, where the proposed optimization techniques
significantly tighten the initial upper and lower bounds. Finally, we compare
our results with an improved variation of the \emph{simplex} local optimization
algorithm, called \emph{Soblex}. This comparison shows that our proposed
algorithms are superior to the Soblex method, both in terms of robustness in
finding the tightest bounds and in computational efficiency. Interestingly,
from a channel coding/decoding perspective, optimizing the lower bound is
related to increasing the achievable mismatched information rate, i.e., the
information rate of a communication system where the decoder at the receiver is
matched to the auxiliary channel, and not to the original channel.Comment: Submitted to IEEE Transactions on Information Theory, November 24,
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Uncertainty quantification and its properties for hidden Markov models with application to condition based maintenance
Condition-based maintenance (CBM) can be viewed as a transformation of data gathered from a piece of equipment into information about its condition, and further into decisions on what to do with the equipment. Hidden Markov model (HMM) is a useful framework to probabilistically model the condition of complex engineering systems with partial observability of the underlying states. Condition monitoring and prediction of such type of system requires accurate knowledge of HMM that describes the degradation of such a system with data collected from the sensors mounted on it, as well as understanding of the uncertainty of the HMMs identified from the available data. To that end, this thesis proposes a novel HMM estimation scheme based on the principles of Bayes theorem. The newly proposed Bayesian estimation approach for estimating HMM parameters naturally yields information about model parametric uncertainties via posterior distributions of HMM parameters emanating from the estimation process. In addition, a novel condition monitoring scheme based on uncertain
HMMs of the degradation process is proposed and demonstrated on a large dataset obtained from a semiconductor manufacturing facility. Portion of the data was used to build operating mode specific HMMs of machine degradation via the newly proposed Bayesian estimation process, while the remainder of the data was used for monitoring of machine condition using the uncertain degradation HMMs yielded by Bayesian estimation. Comparison with a traditional signature-based statistical monitoring method showed that the newly proposed approach effectively utilizes the fact that its parameters are uncertain themselves, leading to orders of magnitude fewer false alarms. This methodology is further extended to address the practical issue that maintenance interventions are usually imperfect. We propose both a novel non-ergodic and non-homogeneous HMM that assumes imperfect maintenances and a novel process monitoring method capable of monitoring the hidden states considering model uncertainty. Significant improvement in both the log-likelihood of estimated HMM parameters and monitoring performance were observed, compared to those obtained using degradation HMMs that always assumed perfect maintenance.
Finally, behavior of the posterior distribution of parameters of unidirectional non- ergodic HMMs modeling in this thesis for degradation was theoretically analyzed in terms of their evolution as more data become available in the estimation process. The convergence problem is formulated as a Bernstein-von Mises theorem (BvMT), and under certain regularity conditions, the sequence of posterior distributions is proven to converge to a Gaussian distribution with variance matrix being the inverse of the Fisher information matrix. An example of a unidirectional HMM is presented for which the regularity conditions are verified, and illustrations of expected theoretical results are given using simulation. The understanding of such convergence of posterior distributions
enables one to determine when Bayesian estimation of degradation HMMs is justified and converges toward true model parameters, as well as how much data one then needs to achieve desired accuracy of the resulting model. Understanding of these issues is of utmost important if HMMs are to be used for degradation modeling and monitoring.Operations Research and Industrial Engineerin