Pub Date : 2019-01-04DOI: 10.1201/9780429055911-10
D. Rossell, M. Steel
{"title":"Continuous Mixtures with Skewness and Heavy Tails","authors":"D. Rossell, M. Steel","doi":"10.1201/9780429055911-10","DOIUrl":"https://doi.org/10.1201/9780429055911-10","url":null,"abstract":"","PeriodicalId":12943,"journal":{"name":"Handbook of Mixture Analysis","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80437927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-04DOI: 10.1201/9780429055911-15
K. Mengersen, Earl W. Duncan, Julyan Arbel, C. Alston-Knox, Nicole M White
This chapter describes the middle ground and include activities that have a commercial focus. It shows the wide diversity of applications of mixture models to problems in industry, and the potential advantages of these approaches, through a series of case studies. The chapter focuses on the iconic and pervasive need for process monitoring, and reviews a range of mixture approaches that have been proposed to tackle complex multimodal and dynamic or online processes. It also focuses on mixture approaches to resource allocation, applied here in a spatial health context but applicable more generally. The chapter provides a more detailed description of a multivariate Gaussian mixture approach to a biosecurity risk assessment problem, using big data in the form of satellite imagery. It argues that a detailed description of a mixture model, this time using a nonparametric formulation, for assessing an industrial impact, notably the influence of a toxic spill on soil biodiversity.
{"title":"Applications in Industry","authors":"K. Mengersen, Earl W. Duncan, Julyan Arbel, C. Alston-Knox, Nicole M White","doi":"10.1201/9780429055911-15","DOIUrl":"https://doi.org/10.1201/9780429055911-15","url":null,"abstract":"This chapter describes the middle ground and include activities that have a commercial focus. It shows the wide diversity of applications of mixture models to problems in industry, and the potential advantages of these approaches, through a series of case studies. The chapter focuses on the iconic and pervasive need for process monitoring, and reviews a range of mixture approaches that have been proposed to tackle complex multimodal and dynamic or online processes. It also focuses on mixture approaches to resource allocation, applied here in a spatial health context but applicable more generally. The chapter provides a more detailed description of a multivariate Gaussian mixture approach to a biosecurity risk assessment problem, using big data in the form of satellite imagery. It argues that a detailed description of a mixture model, this time using a nonparametric formulation, for assessing an industrial impact, notably the influence of a toxic spill on soil biodiversity.","PeriodicalId":12943,"journal":{"name":"Handbook of Mixture Analysis","volume":"203 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73258683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Determining the number G of components in a finite mixture distribution is an important and difficult inference issue. This is a most important question, because statistical inference about the resulting model is highly sensitive to the value of G. Selecting an erroneous value of G may produce a poor density estimate. This is also a most difficult question from a theoretical perspective as it relates to unidentifiability issues of the mixture model. This is further a most relevant question from a practical viewpoint since the meaning of the number of components G is strongly related to the modelling purpose of a mixture distribution. We distinguish in this chapter between selecting G as a density estimation problem in Section 2 and selecting G in a model-based clustering framework in Section 3. Both sections discuss frequentist as well as Bayesian approaches. We present here some of the Bayesian solutions to the different interpretations of picking the "right" number of components in a mixture, before concluding on the ill-posed nature of the question.
{"title":"Model Selection for Mixture Models – Perspectives and Strategies","authors":"G. Celeux, Sylvia Fruewirth-Schnatter, C. Robert","doi":"10.1201/9780429055911-7","DOIUrl":"https://doi.org/10.1201/9780429055911-7","url":null,"abstract":"Determining the number G of components in a finite mixture distribution is an important and difficult inference issue. This is a most important question, because statistical inference about the resulting model is highly sensitive to the value of G. Selecting an erroneous value of G may produce a poor density estimate. This is also a most difficult question from a theoretical perspective as it relates to unidentifiability issues of the mixture model. This is further a most relevant question from a practical viewpoint since the meaning of the number of components G is strongly related to the modelling purpose of a mixture distribution. We distinguish in this chapter between selecting G as a density estimation problem in Section 2 and selecting G in a model-based clustering framework in Section 3. Both sections discuss frequentist as well as Bayesian approaches. We present here some of the Bayesian solutions to the different interpretations of picking the \"right\" number of components in a mixture, before concluding on the ill-posed nature of the question.","PeriodicalId":12943,"journal":{"name":"Handbook of Mixture Analysis","volume":"96 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76880940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. Celeux, K. Kamary, G. Malsiner‐Walli, J. Marin, C. Robert
This chapter surveys the most standard Monte Carlo methods available for simulating from a posterior distribution associated with a mixture and conducts some experiments about the robustness of the Gibbs sampler in high dimensional Gaussian settings. This is a chapter prepared for the forthcoming 'Handbook of Mixture Analysis'.
{"title":"Computational Solutions for Bayesian Inference in Mixture Models","authors":"G. Celeux, K. Kamary, G. Malsiner‐Walli, J. Marin, C. Robert","doi":"10.1201/9780429055911-5","DOIUrl":"https://doi.org/10.1201/9780429055911-5","url":null,"abstract":"This chapter surveys the most standard Monte Carlo methods available for simulating from a posterior distribution associated with a mixture and conducts some experiments about the robustness of the Gibbs sampler in high dimensional Gaussian settings. This is a chapter prepared for the forthcoming 'Handbook of Mixture Analysis'.","PeriodicalId":12943,"journal":{"name":"Handbook of Mixture Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82526376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-01DOI: 10.1201/9780429055911-16
F. Forbes
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Mixture Models for Image Analysis Florence Forbes
{"title":"Mixture Models for Image Analysis","authors":"F. Forbes","doi":"10.1201/9780429055911-16","DOIUrl":"https://doi.org/10.1201/9780429055911-16","url":null,"abstract":"HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Mixture Models for Image Analysis Florence Forbes","PeriodicalId":12943,"journal":{"name":"Handbook of Mixture Analysis","volume":"273 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91434987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mixture models extend the toolbox of clustering methods available to the data analyst. They allow for an explicit definition of the cluster shapes and structure within a probabilistic framework and exploit estimation and inference techniques available for statistical models in general. In this chapter an introduction to cluster analysis is provided, model-based clustering is related to standard heuristic clustering methods and an overview on different ways to specify the cluster model is given. Post-processing methods to determine a suitable clustering, infer cluster distribution characteristics and validate the cluster solution are discussed. The versatility of the model-based clustering approach is illustrated by giving an overview on the different areas of applications.
{"title":"Model-Based Clustering","authors":"Bettina Grun","doi":"10.1201/9780429055911-8","DOIUrl":"https://doi.org/10.1201/9780429055911-8","url":null,"abstract":"Mixture models extend the toolbox of clustering methods available to the data analyst. They allow for an explicit definition of the cluster shapes and structure within a probabilistic framework and exploit estimation and inference techniques available for statistical models in general. In this chapter an introduction to cluster analysis is provided, model-based clustering is related to standard heuristic clustering methods and an overview on different ways to specify the cluster model is given. Post-processing methods to determine a suitable clustering, infer cluster distribution characteristics and validate the cluster solution are discussed. The versatility of the model-based clustering approach is illustrated by giving an overview on the different areas of applications.","PeriodicalId":12943,"journal":{"name":"Handbook of Mixture Analysis","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75213909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mixture models have been around for over 150 years, as an intuitively simple and practical tool for enriching the collection of probability distributions available for modelling data. In this chapter we describe the basic ideas of the subject, present several alternative representations and perspectives on these models, and discuss some of the elements of inference about the unknowns in the models. Our focus is on the simplest set-up, of finite mixture models, but we discuss also how various simplifying assumptions can be relaxed to generate the rich landscape of modelling and inference ideas traversed in the rest of this book.
{"title":"Introduction to Finite Mixtures","authors":"P. Green","doi":"10.1201/9780429055911-1","DOIUrl":"https://doi.org/10.1201/9780429055911-1","url":null,"abstract":"Mixture models have been around for over 150 years, as an intuitively simple and practical tool for enriching the collection of probability distributions available for modelling data. In this chapter we describe the basic ideas of the subject, present several alternative representations and perspectives on these models, and discuss some of the elements of inference about the unknowns in the models. Our focus is on the simplest set-up, of finite mixture models, but we discuss also how various simplifying assumptions can be relaxed to generate the rich landscape of modelling and inference ideas traversed in the rest of this book.","PeriodicalId":12943,"journal":{"name":"Handbook of Mixture Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73775193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-09-01DOI: 10.1201/9780429055911-13
S. Kaufmann
Markov models introduce persistence in the mixture distribution. In time series analysis, the mixture components relate to different persistent states characterizing the state-specific time series process. Model specification is discussed in a general form. Emphasis is put on the functional form and the parametrization of timeinvariant and time-varying specifications of the state transition distribution. The concept of mean-square stability is introduced to discuss the condition under which Markov switching processes have finite first and second moments in the indefinite future. Not surprisingly, a time series process may be mean-square stable even if it switches between bounded and unbounded state-specific processes. Surprisingly, switching between stable state-specific processes is neither necessary nor sufficient to obtain a mean-square stable time series process. Model estimation proceeds by data augmentation. We derive the basic forward-filtering backward-smoothing/sampling algorithm to infer on the latent state indicator in maximum likelihood and Bayesian estimation procedures. Emphasis is again laid on the state transition distribution. We discuss the specification of state-invariant prior parameter distributions and posterior parameter inference under either a logit or probit functional form of the state transition distribution. With simulated data, we show that the estimation of parameters under a probit functional form is more efficient. However, a probit functional form renders estimation extremely slow if more than two states drive the time series process. Finally, various applications illustrate how to obtain informative switching in Markov switching models with time-invariant and time-varying transition distributions.
{"title":"Hidden Markov Models in Time Series, with Applications in Economics","authors":"S. Kaufmann","doi":"10.1201/9780429055911-13","DOIUrl":"https://doi.org/10.1201/9780429055911-13","url":null,"abstract":"Markov models introduce persistence in the mixture distribution. In time series analysis, the mixture components relate to different persistent states characterizing the state-specific time series process. Model specification is discussed in a general form. Emphasis is put on the functional form and the parametrization of timeinvariant and time-varying specifications of the state transition distribution. The concept of mean-square stability is introduced to discuss the condition under which Markov switching processes have finite first and second moments in the indefinite future. Not surprisingly, a time series process may be mean-square stable even if it switches between bounded and unbounded state-specific processes. Surprisingly, switching between stable state-specific processes is neither necessary nor sufficient to obtain a mean-square stable time series process. Model estimation proceeds by data augmentation. We derive the basic forward-filtering backward-smoothing/sampling algorithm to infer on the latent state indicator in maximum likelihood and Bayesian estimation procedures. Emphasis is again laid on the state transition distribution. We discuss the specification of state-invariant prior parameter distributions and posterior parameter inference under either a logit or probit functional form of the state transition distribution. With simulated data, we show that the estimation of parameters under a probit functional form is more efficient. However, a probit functional form renders estimation extremely slow if more than two states drive the time series process. Finally, various applications illustrate how to obtain informative switching in Markov switching models with time-invariant and time-varying transition distributions.","PeriodicalId":12943,"journal":{"name":"Handbook of Mixture Analysis","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87574695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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