Abstract
The notion of defining a cluster as a component in a mixture model was put forth by Tiedeman in 1955; since then, the use of mixture models for clustering has grown into an important subfield of classification. Considering the volume of work within this field over the past decade, which seems equal to all of that which went before, a review of work to date is timely. First, the definition of a cluster is discussed and some historical context for model-based clustering is provided. Then, starting with Gaussian mixtures, the evolution of model-based clustering is traced, from the famous paper by Wolfe in 1965 to work that is currently available only in preprint form. This review ends with a look ahead to the next decade or so.
Highlights
The notion of defining a cluster as a component in a mixture model was put forth by Tiedeman in 1955; since the use of mixture models for clustering has grown into an important subfield of classification
The best place to start is at the beginning, which consists in a question: what is a cluster? Before positing an answer, some historical context is helpful
It will be interesting to explore the use of mixtures of multivariate power exponential (MPE) as an alternative to t-mixtures; whereas t-mixtures essentially allow more dispersion about the mean when compared with Gaussian mixtures, mixtures of MPEs would allow both more and less dispersion
Summary
The best place to start is at the beginning, which consists in a question: what is a cluster? Before positing an answer, some historical context is helpful. Wolfe (1970) draws an analogy between his approach for Gaussian mixtures with common covariance matrices and one of the criteria described by Friedman and Rubin (1967) This and other work on parameter estimation in Gaussian model-based clustering—e.g., Edwards and Cavalli-Sforza (1965), Baum et al (1970), Scott and Symons (1971), Orchard and Woodbury (1972), and Sundberg (1974)—effectively culminated in the landmark paper by Dempster, Laird and Rubin (1977), wherein the expectation-maximization (EM) algorithm is introduced; see Titterington et al (1985, Section 4.3.2) and McNicholas (2016, Chapter 2). An interesting perspective on the ICL is presented by Baudry (2015), who discusses the conditional classification likelihood and related ideas
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