Abstract. The mixture transition distribution model (MTD) was introduced in 1985 by Raftery for the modeling of high-order Markov chains with a finite state space. Since then it has been generalized and successfully applied to a range of situations, including the analysis of wind directions, DNA sequences and social behavior. Here we review the MTD model and the developments since 1985. We first introduce the basic principle and then we present several extensions, including general state spaces and spatial statistics. Following that, we review methods for estimating the model parameters. Finally, a review of different types of applications shows the practical interest of the MTD model. Key words and phrases: Mixture transition distribution (MTD) model, Markov chains, high-order dependences, time series, GMTD model, EM algorithm,

We consider Bayesian nonparametric inference for continuous-valued partially exchangeable data, when the partition of the observations into groups is unknown. This includes change-point problems and mixture models. As the prior, we consider a mixture of products of Dirichlet processes. We show that the discreteness of the Dirichlet process can have a large effect on inference (posterior distributions and Bayes factors), leading to conclusions that can be different from those that result from a reasonable parametric model. When the observed data are all distinct, the effect of the prior on the posterior is to favor more evenly balanced partitions, and its effect on Bayes factors is to favor more groups. In a hierarchical model with a Dirichlet process as the second-stage prior, the prior can also have a large effect on inference, but in the opposite direction, towards more unbalanced partitions. (~) 1997 Elsevier Science B.V.

This paper introduces a new iterative algorithm for the estimation of the Mixture Transition Distribution model (MTD). It does not require the use of any speci c external optimization procedure and can therefore be programmed in any computing language. Comparisons with previously published results show that this new algorithm performs at least as good or better than other methods. The choice of initial values is also discussed. The MTD model was designed for the modeling of high-order Markov chains and already proved to be a useful tool for the analysis of di erenttypes of time-series such as wind speeds and wind directions. In this paper, we also propose to use this it for the modeling of onedimensional spatial data. An application using a DNA sequence shows that this approach

The Mixture Transition Distribution model (MTD) was introduced by Raftery (1985) for the modeling of high-order Markov chains with a finite state space. Since then, it has been generalized and successfully applied to a range of situations including the analysis of wind direction, DNA and social behavior. Here we review the MTD model and the developments since 1985. We rst introduce the basic principle and then we present several extensions including general state spaces and spatial statistics. We then review methods for estimating the model parameters. Finally, a review

In the past, several authors have found evidence for the existence of a priority pattern of acquisition for durable goods, as well as for financial services. Its usefulness lies in the fact that if the position of a particular customer in this acquisition sequence is known, one can predict what service will be acquired next by that customer. In this paper, we analyse purchase sequences of financial services to identify cross-selling opportunities as part of a CRM (customer relationship management). Hereby, special attention is paid to transitions, which might encourage bank- or insurance only customers to become financial services customers. We introduce the Mixture Transition Distribution model (MTD) as a parsimonious alternative to the Markov model for use in the analysis of marketing problems. An interesting extension on the MTD model is the MTDg model, which is able to represent situations where the relationship between each lag and the current state differs. We illustrate the MTD and MTDg model on acquisition sequences of customers of a major financial-services company and compare the fit of these models with that of the corresponding Markov model. Our results are in favor of the MTD and MTDg models. Therefore, the MTD as well as the MTDg transition matrices are investigated in order to reveal cross-sell opportunities. The results are of great value to the product managers as they clarify the customer flows among product groups. In some cases, the lag-specific transition matrices of the MTDg model are better for the guidance of cross-sell actions than the general transition matrix of the MTD model.