Abstract
This paper presents a review about the usage of eigenvalues restrictions for constrained parameter estimation in mixtures of elliptical distributions according to the likelihood approach. The restrictions serve a twofold purpose: to avoid convergence to degenerate solutions and to reduce the onset of non interesting (spurious) local maximizers, related to complex likelihood surfaces. The paper shows how the constraints may play a key role in the theory of Euclidean data clustering. The aim here is to provide a reasoned survey of the constraints and their applications, considering the contributions of many authors and spanning the literature of the last 30 years.
Highlights
Finite mixture distributions play a central role in statistical modelling, as they combine much of the flexibility of non parametric models with nice analytical properties of parametric models, see e.g. Titterington et al (1985), Lindsay (1995), McLachlan and Peel (2000)
The last part of the paper is devoted to the role of eigenvalues in parsimonious models: Gaussian parsimonious clustering models are summarized in Section 5 while in Section 6 mixture of factor analyzers are presented
The EM algorithm is usually implemented for parameter estimation in mixture modeling, see e.g. McLachlan and Krishnan (2008b)
Summary
Finite mixture distributions play a central role in statistical modelling, as they combine much of the flexibility of non parametric models with nice analytical properties of parametric models, see e.g. Titterington et al (1985), Lindsay (1995), McLachlan and Peel (2000). For parameter estimation in mixture models several approaches may be considered, as the ones exposed in McLachlan and Krishnan (2008a). The maximum likelihood (ML) framework is among the most commonly used approaches to mixture parameter estimation, and it is the approach we consider here. This paper presents a review about the usage of eigenvalues restrictions for constrained estimation, that serves a twofold purpose: to avoid convergence to degenerate solutions and to reduce the onset of non interesting (spurious) maximizers, related to complex likelihood surfaces. The last part of the paper is devoted to the role of eigenvalues in parsimonious models: Gaussian parsimonious clustering models are summarized in Section 5 while in Section 6 mixture of factor analyzers are presented.
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