13 research outputs found
On the accuracy of the Viterbi alignment
In a hidden Markov model, the underlying Markov chain is usually hidden.
Often, the maximum likelihood alignment (Viterbi alignment) is used as its
estimate. Although having the biggest likelihood, the Viterbi alignment can
behave very untypically by passing states that are at most unexpected. To avoid
such situations, the Viterbi alignment can be modified by forcing it not to
pass these states. In this article, an iterative procedure for improving the
Viterbi alignment is proposed and studied. The iterative approach is compared
with a simple bunch approach where a number of states with low probability are
all replaced at the same time. It can be seen that the iterative way of
adjusting the Viterbi alignment is more efficient and it has several advantages
over the bunch approach. The same iterative algorithm for improving the Viterbi
alignment can be used in the case of peeping, that is when it is possible to
reveal hidden states. In addition, lower bounds for classification
probabilities of the Viterbi alignment under different conditions on the model
parameters are studied
MAP segmentation in Bayesian hidden Markov models:a case study
We consider the problem of estimating the maximum posterior probability (MAP)
state sequence for a finite state and finite emission alphabet hidden Markov
model (HMM) in the Bayesian setup, where both emission and transition matrices
have Dirichlet priors. We study a training set consisting of thousands of
protein alignment pairs. The training data is used to set the prior
hyperparameters for Bayesian MAP segmentation. Since the Viterbi algorithm is
not applicable any more, there is no simple procedure to find the MAP path, and
several iterative algorithms are considered and compared. The main goal of the
paper is to test the Bayesian setup against the frequentist one, where the
parameters of HMM are estimated using the training data
Estimation of Viterbi path in Bayesian hidden Markov models
The article studies different methods for estimating the Viterbi path in the Bayesian framework. The Viterbi path is an estimate of the underlying state path in hidden Markov models (HMMs), which has a maximum joint posterior probability. Hence it is also called the maximum a posteriori (MAP) path. For an HMM with given parameters, the Viterbi path can be easily found with the Viterbi algorithm. In the Bayesian framework the Viterbi algorithm is not applicable and several iterative methods can be used instead. We introduce a new EM-type algorithm for finding the MAP path and compare it with various other methods for finding the MAP path, including the variational Bayes approach and MCMC methods. Examples with simulated data are used to compare the performance of the methods. The main focus is on non-stochastic iterative methods and our results show that the best of those methods work as well or better than the best MCMC methods. Our results demonstrate that when the primary goal is segmentation, then it is more reasonable to perform segmentation directly by considering the transition and emission parameters as nuisance parameters.Peer reviewe
Rank Covariance Matrix For A Partially Known Covariance Matrix Rank Covariance Matrix For A Partially Known Covariance Matrix
Abstract Classical multivariate methods are often based on the sample covariance matrix, which is very sensitive to outlying observations. One alternative to the covariance matrix is the ane equivariant rank covariance matrix (RCM) that has been studied for example in Visuri et al. (2003). In this article we assume that the covariance matrix is partially known and study how to estimate the corresponding RCM. We use the properties that the RCM is ane equivariant and that the RCM is proportional to the inverse of the regular covariance matrix, and reduce the problem of estimating the RCM to estimating marginal rank covariance matrices. This is a great advantage when the dimension of the original data vectors is large
Rank Estimation in Elliptical Models : Estimation of Structured Rank Covariance Matrices and Asymptotics for Heteroscedastic Linear Regression
This thesis deals with univariate and multivariate rank methods in making statistical inference. It is assumed that the underlying distributions belong to the class of elliptical distributions. The class of elliptical distributions is an extension of the normal distribution and includes distributions with both lighter and heavier tails than the normal distribution. In the first part of the thesis the rank covariance matrices defined via the Oja median are considered. The Oja rank covariance matrix has two important properties: it is affine equivariant and it is proportional to the inverse of the regular covariance matrix. We employ these two properties to study the problem of estimating the rank covariance matrices when they have a certain structure. The second part, which is the main part of the thesis, is devoted to rank estimation in linear regression models with symmetric heteroscedastic errors. We are interested in asymptotic properties of rank estimates. Asymptotic uniform linearity of a linear rank statistic in the case of heteroscedastic variables is proved. The asymptotic uniform linearity property enables to study asymptotic behaviour of rank regression estimates and rank tests. Existing results are generalized and it is shown that the Jaeckel estimate is consistent and asymptotically normally distributed also for heteroscedastic symmetric errors