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Symmetry
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The Mixture Transition Distribution Model and Other Models for High-Order...
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The Mixture Transition Distribution Model and Other Models for High-Order Dependencies

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 11346

Special Issue Editor

Prof. Dr. André Berchtold

E-Mail Website
Guest Editor
Institute of Social Sciences & NCCR LIVES, University of Lausanne, Lausanne, Switzerland
Interests: Markovian models; missing data; life course analysis; adolescence

Special Issue Information

Dear Colleagues,

High-order Markov chains are very useful for the analysis of complex temporal relationships, but they generally reqire a very high number of parameters. A good approach is then to approximate them, and since its introduction in 1985 by Adrian Raftery, the mixture transition distribution (MTD) model attracted much attention, thanks to its parsimony and versatility. The MTD model has been developed and improved in various ways. Now, it can be used to represent and analyze categorical and continuous variables, covariates can be added as additional explanatory terms to the model, and it can also be combined with a hidden or a double-chain Markov model in order to consider the latent phenomena. The MTD model can be estimated through the standard EM algorithm, but ad hoc optimization algorithms were also developed for special situations. Finally, many applied papers have been published in fields as diverse as health, neural networks, finance, marketing, life course, and weather, among others.

Given its base concept, the MTD model introduces a kind of symmetrical relationship between the past and the present—the form of the contribution of each past event to the present is similar, only the respective importance of each contribution is different.

Given the large interest existing around the MTD model principles, the main objectives of this Special Issue are to take stock of 35 years of use of the model, to introduce new directions for future developments, to compare it with alternative specifications of high-order models, and to think about new fields of application. Therefore, this Special Issue welcomes all papers related to the MTD model, demonstrating new theoretical developments, new applications, or both. Papers showing the combined use of the MTD model with other statistical tools are especially welcome, as well as use of the MTD model in mixed-method research.

Prof. Dr. André Berchtold
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mixture transition distribution (MTD) model
  • Markov chain
  • high-order model
  • hidden model
  • optimization
  • combination of models
  • applications

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Published Papers (3 papers)

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Research

10 pages, 1533 KiB  
Article
The Deep Learning LSTM and MTD Models Best Predict Acute Respiratory Infection among Under-Five-Year Old Children in Somaliland
by Mohamed Yusuf Hassan
Symmetry 2021, 13(7), 1156; https://doi.org/10.3390/sym13071156 - 28 Jun 2021
Cited by 6 | Viewed by 2390
Abstract
The most effective techniques for predicting time series patterns include machine learning and classical time series methods. The aim of this study is to search for the best artificial intelligence and classical forecasting techniques that can predict the spread of acute respiratory infection [...] Read more.
The most effective techniques for predicting time series patterns include machine learning and classical time series methods. The aim of this study is to search for the best artificial intelligence and classical forecasting techniques that can predict the spread of acute respiratory infection (ARI) and pneumonia among under-five-year old children in Somaliland. The techniques used in the study include seasonal autoregressive integrated moving averages (SARIMA), mixture transitions distribution (MTD), and long short term memory (LSTM) deep learning. The data used in the study were monthly observations collected from five regions in Somaliland from 2011–2014. Prediction results from the three best competing models are compared by using root mean square error (RMSE) and absolute mean deviation (MAD) accuracy measures. Results have shown that the deep learning LSTM and MTD models slightly outperformed the classical SARIMA model in predicting ARI values. Full article
(This article belongs to the Special Issue The Mixture Transition Distribution Model and Other Models for High-Order Dependencies)
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22 pages, 911 KiB  
Article
The Use of a Hidden Mixture Transition Distribution Model in Clustering Few but Long Continuous Sequences: An Illustration with Cognitive Skills Data
by Zhivko Taushanov and Paolo Ghisletta
Symmetry 2020, 12(10), 1618; https://doi.org/10.3390/sym12101618 - 29 Sep 2020
Cited by 2 | Viewed by 3677
Abstract
In accordance with the theme of this special issue, we present a model that indirectly discovers symmetries and asymmetries between past and present assessments within continuous sequences. More specifically, we present an alternative use of a latent variable version of the Mixture Transition [...] Read more.
In accordance with the theme of this special issue, we present a model that indirectly discovers symmetries and asymmetries between past and present assessments within continuous sequences. More specifically, we present an alternative use of a latent variable version of the Mixture Transition Distribution (MTD) model, which allows for clustering of continuous longitudinal data, called the Hidden MTD (HMTD) model. We compare the HMTD and its clustering performance to the popular Growth Mixture Model (GMM), as well as to the recently introduced GMM based on individual case residuals (ICR-GMM). The GMM and the ICR-GMM contrast with HMTD, because they are based on an explicit change function describing the individual sequences on the dependent variable (here, we implement a non-linear exponential change function). This paper has three objectives. First, it introduces the HMTD. Second, we present the GMM and the ICR-GMM and compare them to the HMTD. Finally, we apply the three models and comment on how the conclusions differ depending on the clustering model, when using a specific dataset in psychology, which is characterized by a small number of sequences (n = 102), but that are relatively long (for the domains of psychology and social sciences: t = 20). We use data from a learning experiment, in which healthy adults (19–80 years old) were asked to perform a perceptual–motor skills over 20 trials. Full article
(This article belongs to the Special Issue The Mixture Transition Distribution Model and Other Models for High-Order Dependencies)
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13 pages, 318 KiB  
Article
Handling Covariates in Markovian Models with a Mixture Transition Distribution Based Approach
by Danilo Bolano
Symmetry 2020, 12(4), 558; https://doi.org/10.3390/sym12040558 - 4 Apr 2020
Cited by 4 | Viewed by 4042
Abstract
This paper presents and discusses the use of a Mixture Transition Distribution-like model (MTD) to account for covariates in Markovian models. The MTD was introduced in 1985 by Raftery as an approximation of higher order Markov chains. In the MTD, each lag is [...] Read more.
This paper presents and discusses the use of a Mixture Transition Distribution-like model (MTD) to account for covariates in Markovian models. The MTD was introduced in 1985 by Raftery as an approximation of higher order Markov chains. In the MTD, each lag is estimated separately using an additive model, which introduces a kind of symmetrical relationship between the past and the present. Here, using an MTD-based approach, we consider each covariate separately, and we combine the effects of the lags and of the covariates by means of a mixture model. This approach has three main advantages. First, no modification of the estimation procedure is needed. Second, it is parsimonious in terms of freely estimated parameters. Third, the weight parameters of the mixture can be used as an indication of the relevance of the covariate in explaining the time dependence between states. An illustrative example taken from life course studies using a 3-state hidden Markov model and a covariate with three levels shows how to interpret the results of such models. Full article
(This article belongs to the Special Issue The Mixture Transition Distribution Model and Other Models for High-Order Dependencies)
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