The MTD Page
The Mixture Transition Distribution (MTD) model was introduced by
A.E. Raftery (1985) for the modeling of high-order Markov chains.
The principle of the model is to replace the global contribution of
each lagged period to the present by an individual contribution from
each lag to the present. This model is very parsimonious while
having the capability to represent very different situations
including infinite-lag models and spatial dependences.
Since 1985, the MTD model has been developed and generalized in
more than 20 publications. A continuous version of the MTD model
(Le, Martin & Raftery, 1996) has proven to be able to represent
series presenting non-gaussian features. Main applications include the modeling
of wind speed and direction, social behavior, and financial
Two softwares are freely available for the computation of the
finite space version of the MTD model:
Other important references about theoretical developments
- Berchtold, A., A.E. Raftery (2002)
The Mixture Transition Distribution Model for High-Order Markov Chains and Non-Gaussian Time Series. Statistical
Science, 17 (3), 328-356. (This is a review paper including both theoretical aspects and applications of the MTD model.)
- Berchtold, A. (2001)
Estimation in the Mixture Transition Distribution Model.
Journal of Time Series Analysis, 22(4), 379-397. (Description of a new
estimation algorithm for the computation of the finite space version of the MTD
- Le, N.D., R.D. Martin, A.E. Raftery (1996)
Modeling Flat Stretches, Bursts, and Outliers in Time Series
Using Mixture Transition Distribution Models.
Journal of the American Statistical Association, 91,
1504-1515. (Generalization of the MTD model to general state
- Raftery, A.E. (1985)
A model for high-order Markov chains.
Journal of the Royal Statistical Society B, 47 (3),
528-539. (Paper introducing first the MTD model.)
- Adke, S.R., S.R. Deshmukh (1988)
Limit Distribution of a High-Order Markov Chain.
Journal of the Royal Statistical Society B, 50, 105-108.
- Berchtold, A. (1995)
Autoregressive Modelling of Markov Chains.
In Proceedings of the 10th International Workshop on
Statistical Modelling, 19-26. Springer-Verlag, New York.
- Berchtold, A. (1996)
Modélisation autorégressive des chaînes de Markov:
Utilisation d'une matrice différente pour chaque retard.
Revue de Statistique Appliquée, XLIV, 5-25.
- Berchtold, A. (1997)
Swiss Health Insurance System: Mobility and Costs.
Health and System Science, 1, 291-306.
- Berchtold, A. (1998)
Chaînes de Markov et Modèles de Transition: Applications aux
Editions HERMES, Paris.
- Berchtold, A. (1999)
High-Order Extensions of the Double Chain Markov Model.
Report 356, Department of Statistics, University of
- Mehran, F. (1989)
Analysis of Discrete Longitudinal Data : Infinite-Lag Markov
In Statistical Data Analysis and Inference, pp. 533-541.
Y. Dodge Editor, Elsevier Science Publishers.
- Raftery, A.E. (1985)
A new model for discrete-valued time series: autocorrelations
Rassegna di Metodi Statistici ed Applicazioni, 3-4,
- Raftery, A.E. (1993)
Change point and change curve modeling in stochastic processes
and spatial statistics.
Journal of Applied Statistical Science, 1, 403-423.
- Raftery, A.E., S. Tavaré (1994)
Estimation and Modelling Repeated Patterns in High Order
Markov Chains with the Mixture Transition Distribution Model.
Applied Statistics, 43, 179-199.
- Wong, C.S., W.K. Li (2000)
On a mixture autoregression model.
Journal of the Royal Statistical Society B, 62, 95-115.
Last modified: April 6, 2004.