A Method for Estimating the Entropy Rate of Hidden Markov Processes PDF Download
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Author: Katy Marchand
Publisher:
ISBN:
Category :
Languages : en
Pages : 148
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Author: Katy Marchand
Publisher:
ISBN:
Category :
Languages : en
Pages : 148
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Book Description
Author: Nicholas F. Travers
Publisher:
ISBN: 9781303444012
Category :
Languages : en
Pages :
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Book Description
There is no general closed form expression for the entropy rate of a hidden Markov process. However, the finite length block estimates h(t) often converge to the true entropy rate h quite rapidly. We establish exponential bounds on the rate of convergence of the block estimates for finite hidden Markov models under several different conditions, including exactness, unifilarity, and a flag-state condition. In the case of unifilar hidden Markov models, we also give exponential bounds on the L1 and a.s. decay of the state uncertainty U(t).
Author: Brian Marcus
Publisher: Cambridge University Press
ISBN: 1139495747
Category : Mathematics
Languages : en
Pages : 279
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Book Description
This collection of research and survey papers sets out the theory of hidden Markov processes, in particular addressing a central problem of the subject: computation of the Shannon entropy rate of an HMP. Connections are drawn between approaches from various disciplines, whilst recent research results and open problems are described.
Author: Andrew M. Fraser
Publisher: SIAM
ISBN: 0898717744
Category : Mathematics
Languages : en
Pages : 142
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Book Description
This text provides an introduction to hidden Markov models (HMMs) for the dynamical systems community. It is a valuable text for third or fourth year undergraduates studying engineering, mathematics, or science that includes work in probability, linear algebra and differential equations. The book presents algorithms for using HMMs, and it explains the derivation of those algorithms. It presents Kalman filtering as the extension to a continuous state space of a basic HMM algorithm. The book concludes with an application to biomedical signals. This text is distinctive for providing essential introductory material as well as presenting enough of the theory behind the basic algorithms so that the reader can use it as a guide to developing their own variants.
Author: M. Vidyasagar
Publisher: Princeton University Press
ISBN: 1400850517
Category : Mathematics
Languages : en
Pages : 303
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Book Description
This book explores important aspects of Markov and hidden Markov processes and the applications of these ideas to various problems in computational biology. The book starts from first principles, so that no previous knowledge of probability is necessary. However, the work is rigorous and mathematical, making it useful to engineers and mathematicians, even those not interested in biological applications. A range of exercises is provided, including drills to familiarize the reader with concepts and more advanced problems that require deep thinking about the theory. Biological applications are taken from post-genomic biology, especially genomics and proteomics. The topics examined include standard material such as the Perron-Frobenius theorem, transient and recurrent states, hitting probabilities and hitting times, maximum likelihood estimation, the Viterbi algorithm, and the Baum-Welch algorithm. The book contains discussions of extremely useful topics not usually seen at the basic level, such as ergodicity of Markov processes, Markov Chain Monte Carlo (MCMC), information theory, and large deviation theory for both i.i.d and Markov processes. The book also presents state-of-the-art realization theory for hidden Markov models. Among biological applications, it offers an in-depth look at the BLAST (Basic Local Alignment Search Technique) algorithm, including a comprehensive explanation of the underlying theory. Other applications such as profile hidden Markov models are also explored.
Author: Robert James Elliott
Publisher: New York : Springer-Verlag
ISBN:
Category : Distribution (Probability theory).
Languages : en
Pages : 388
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Book Description
The authors begin with discrete time and discrete state spaces. From there, they proceed to cover continuous time, and progress from linear models to nonlinear models, and from completely known models to only partially known models.
Author: Halina Frydman
Publisher:
ISBN:
Category :
Languages : en
Pages : 17
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Book Description
We consider the process dYt = ut dt + dWt , where u is a processnot necessarily adapted to F Y (the filtration generated by the process Y)and W is a Brownian motion. We obtain a general representation for thelikelihood ratio of the law of the Y process relative to Brownian measure.This representation involves only one basic filter (expectation of u conditionalon observed process Y). This generalizes the result of Kailath and Zakai[Ann.Math. Statist. 42 (1971) 130acirc;Ĩquot;140] where it is assumed that the process uis adapted to F Y . In particular, we consider the model in which u is afunctional of Y and of a random element X which is independent of theBrownian motion W. For example, X could be a diffusion or a Markov chain.This result can be applied to the estimation of an unknown multidimensionalparameter Icirc;cedil; appearing in the dynamics of the process u based on continuousobservation of Y on the time interval [0,T ]. For a specific hidden diffusionfinancial model in which u is an unobserved mean-reverting diffusion, wegive an explicit form for the likelihood function of Icirc;cedil;. For this model we alsodevelop a computationally explicit Eacirc;Ĩquot;M algorithm for the estimation of Icirc;cedil;. Incontrast to the likelihood ratio, the algorithm involves evaluation of a numberof filtered integrals in addition to the basic filter.
Author: Sebastien Roland
Publisher:
ISBN:
Category :
Languages : en
Pages :
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This paper is concerned with the estimation of coefficients of continuous-time hidden Markov models when the observations are sampled at discrete times, with a view towards financial applications. These estimates are commonly computed from discretely and frequently-sampled returns. However, recent findings indicate that these estimators are not robust when the frequency increases due to market microstructure. The present work attempts to reconcile continuous-time modeling and discret-time observations. To this end, we propose a model where all the coefficients of the asset log-price Y are unobservable and follow a Markov process X, which represents the hidden market factors which affect Y. We also suppose that stock prices are observed only discretely at random times T. Under the above setting, the inference problem can be treated as a non-linear filtering problem for X by considering measurements given by the random measure associated to (T(k),Y(k))(k0) From a numerical perspective, we develop and compare optimization methods by means of maximum likelihood and Bayesian paradigm so as to compute the state and parameters estimates. Eventually, we provide empirical evidence of the performance of these approaches on simulated and empirical data sets of index returns.
Author: Jörn Dannemann
Publisher: Cuvillier Verlag
ISBN: 3736932472
Category : Business & Economics
Languages : en
Pages : 140
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Book Description
Hidden Markov models (HMMs) and other latent variable models form complex, flexible frameworks for univariate and multivariate data structures. In the last two decades models with latent variables have entered almost all fields of statistical applications. It is common for these models that unobserved variables are introduced to model a complex data structure given by the observables. A major advantage of latent structures is the principle simplicity and the accessibility to practitioners as well as their application-driven interpretations rather than black box systems. In this dissertation the statistical methodology of HMMs and related models is extended in certain aspects and illustrated by several applications from various fields, including epileptic seizures, financial time series and a dental health trail. We first investigate testing problems for HMMs under nonstandard conditions, namely when the true parameter lies on the boundary. In practical applications of HMMs, non-standard testing problems are frequently encountered, e.g. testing for the probability of staying in a certain unobserved state being zero. We derive the relevant asymptotic distribution theory for the likelihood ratio test in HMMs under these conditions. A number of examples with particular relevance in the HMM framework are examined. Secondly, we are concerned with testing for the number of states in HMMs and switching regression models, in particular, testing for two states in an HMM, and testing for two components in switching regression models. The specification of the number of states is very important in all models with discrete latent variables, and performing statistical testing of such hypotheses is one way to deal with this problem. For testing for homogeneity or for two components in finite mixtures the modified likelihood ratio test is a well-developed method. Based on this approach we propose a test for two states in HMMs. Testing for two states is of primary interest in particular for HMMs, since a two-state HMM represents the smallest non-trivial member of this model class. We derive the asymptotic distribution for the modified likelihood ratio test with independence assumption under the hypothesis of a two-state HMM. In addition, we propose a test for two components in switching regression models with independent or Markov-dependent regime. In the third part we depart from the classical parametric framework and relax the parametric assumptions, aiming for more flexible models, which reduce systematic errors caused by model misspecification and give rise to model validation techniques. We propose a parametric as well as a semiparametric approach to this problem. In particular, the latter one introduces a new flavor to hidden Markov modeling by linking recently developed semiparametric mixture models to the HMM framework. We discuss identifiability and propose an estimation procedure to semiparametric two-state HMMs based on the expectation-maximization algorithm. This enables extensions of modern estimation techniques in semiparametric mixtures like log-concave density estimation to HMMs.
Author: Robert James Elliott
Publisher:
ISBN: 9780540943647
Category : Markov processes
Languages : en
Pages : 361
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