Mixture Model - History

History

Mixture distributions and the problem of mixture decomposition, that is the identification of its constituent components and the parameters thereof, has been cited in the literature as far back as 1846 (Quetelet in McLaughlan, 2000) although common reference is made to the work of Karl Pearson (1894) as the first author to explicitly address the decomposition problem in characterising non-normal attributes of forehead to body length ratios in female shore crab populations. The motivation for this work was provided by the zoologist Walter Frank Raphael Weldon who had speculated in 1893 (in Tarter and Lock) that asymmetry in the histogram of these ratios could signal evolutionary divergence. Pearson’s approach was to fit a univariate mixture of two normals to the data by choosing the five parameters of the mixture such that the empirical moments matched that of the model.

While his work was successful in identifying two potentially distinct sub-populations and in demonstrating the flexibility of mixtures as a moment matching tool, the formulation required the solution of a 9th degree (nonic) polynomial which at the time posed a significant computational challenge.

Subsequent works focused on addressing these problems, but it was not until the advent of the modern computer and the popularisation of Maximum Likelihood (ML) parameterisation techniques that research really took off. Since that time there has been a vast body of research on the subject spanning areas such as Fisheries research, Agriculture, Botany, Economics, Medicine, Genetics, Psychology, Palaeontology, Electrophoresis, Finance, Sedimentology/Geology and Zoology.