An EM algorithm for estimation in the mixture transition distribution model

Abstract

The Mixture Transition Distribution (MTD) model was introduced by Raftery to face the need for parsimony in the modeling of high-order Markov chains in discrete time. The particularity of this model comes from the fact that the effect of each lag upon the present is considered separately and additively, so that the number of parameters required is drastically reduced. However, the efficiency for the MTD parameter estimations proposed up to date still remains problematic on account of the large number of constraints on the parameters. In this paper, an iterative procedure, commonly known as Expectation-Maximization (EM) algorithm, is developed cooperating with the principle of Maximum Likelihood Estimation (MLE) to estimate the MTD parameters. Some applications of modeling MTD show the proposed EM algorithm is easier to be used than the algorithm developed by Berchtold. Moreover, the EM Estimations of parameters for high-order MTD models led on DNA sequences outperform the corresponding fully parametrized Markov chain in terms of Bayesian Information Criterion.A software implementation of our algorithm is available in the library seq++ at http://stat.genopole.cnrs.fr/seqpp.

A Novel Estimation Approach for Mixture Transition Distribution Model in High-Order Markov Chains

“…Cases 1-4 use Q 1 , Cases 5-8 use Q 2 , and Cases 9-12 use Q 3 as the true common transition matrix, respectively. It should be noticed that the three input transition matrices, Q 1 , Q 2 , and Q 3 are from Lébre and Bourguignon (2008). Table 1.…”

“…The first one is the epileptic data; the second one is a time series of the twilight song of the wood pewee; and the third one is the Mouse A-Crystallin Gene. These data sets had been used by Chatfield and Lemon (1970), Bishop et al (1975), Raftery and Tavaré (1994), Berchtold (2001), and Lébre and Bourguignon (2008).…”

“…Berchtold (2001) proposed an efficient iterative process with the boundary adjustment in the MLE process which leads to a modification of the Newton's method. With the constraints (2.7), Lébre and Bourguignon (2008) introduced a hidden process for the coefficients of the MTD models and derived an Expectation-Maximization (EM) algorithm for the MLE of the MTD parameters. However, the complexity from the counts of the patterns of sequences is still unsolved for the search of the maximizer globally.…”

Parsimonious Higher Order Markov Models for Rating Transitions

“…However, the meaning of the parameters for this model is not always clear and the efficiency for mixture transition distribution estimations may be problematic on account of the large number of constraints on the parameters. Hence, there may be a need for more sophisticated optimization techniques (Lèbre and Bourguignon, 2008) but not even these algorithms always guarantee convergence. The Markov regression models in Zeger and Qaqish (1988) are another way to reduce the number of parameters.…”

Linear Models for the Impact of Order Flow on Prices II. The Mixture Transition Distribution Model

“…Under the constraints of Eq. 2 and 3, Lèbre and Bourguignon in [25] introduce a hidden process for the coefficients of the MTDg and propose an Expectation-Maximization approach for the parameters estimation. Chen et al in [26] note that all the previous constraints can be rewritten in a boxconstrained form, which is easier to handle.…”