A likelihood-based constrained algorithm for multivariate normal mixture models

Abstract

It is well known that the log-likelihood function for samples coming from normal mixture distributions may present spurious maxima and singularities. For this reason here we reformulate some Hathaway's results and we propose two constrained estimation procedures for multivariate normal mixture modelling according to the likelihood approach. Their perfomances are illustrated on the grounds of some numerical simulations based on the EM algorithm. A comparison between multivariate normal mixtures and the hot-deck approach in missing data imputation is also considered.

A profile likelihood method for normal mixture with unequal variance

“…Similar to the univariate case, one can also put some constraint on the covariance matrix. For example, let k be the minimum of all the eigenvalues of (Hathaway 1985, Ingrassia, 2004. Then one can define the profile log likelihood for k similar to (7) and use it to find the maximum interior mode.…”

Multivariate Models and Algorithms for Learning Correlation Structures from Replicated Molecular Profiling Data

“…To reduce various problems associated with the convergence of EM algorithm, remedies have been proposed by constraining the eigenvalues of the component correlation matrices (Ingrassia, 2004;Ingrassia & Rocci, 2007). For example, the constrained EM algorithm presented in (Ingrassia, 2004) (Hathaway, 1985) is also satisfied, and results in constrained (global) maximization of the likelihood.…”

Estimating Parameters in Muitivariate Normal Mixtures

“…However, a more general approach is to use penalized maximum likelihood estimation(PMLE), where new likelihood function is formulated by adding a penalty term. Several authors including Ciuperca et al (2003), Ingrassia (2004), and Chen and Tan (2009) used this approach and what they used as a constraint was mostly on variances or variance matrices in multivariate cases.…”