A Fortran program blue called EMMIX-WIRE (EM-based MIXture analysis WIth Random Effects) is available on request from the corresponding author.

Finite mixture models are being commonly used in a wide range of applications in practice concerning density estimation and clustering. An attractive feature of this approach to clustering is that it provides a sound statistical framework in which to assess the important question of how many clusters there are in the data and their validity. We consider the applications of normal mixture models to high-dimensional data of a continuous nature. One way to handle the fitting of normal mixture models is to adopt mixtures of factor analyzers. However, for extremely high-dimensional data, some variable-reduction method needs to be used in conjunction with the latter model such as with the procedure called EMMIX-GENE. It was developed for the clustering of microarray data in bioinformatics, but is applicable to other types of data. We shall also consider the mixture procedure EMMIX-WIRE (based on mixtures of normal components with random effects), which is suitable for clustering high-dimensional data that may be structured (correlated and and replicated) as in longitudinal studies.

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Time-course experiments with microarrays are often used to study dynamic biological systems and genetic regulatory networks (GRNs) that model how genes influence each other in cell-level development of organisms. The inference for GRNs provides important insights into the fundamental biological processes such as growth and is useful in disease diagnosis and genomic drug design. Due to the experimental design, multilevel data hierarchies are often present in time-course gene expression data. Most existing methods, however, ignore the dependency of the expression measurements over time and the correlation among gene expression profiles. Such independence assumptions violate regulatory interactions and can result in overlooking certain important subject effects and lead to spurious inference for regulatory networks or mechanisms. In this paper, a multilevel mixed-effects model is adopted to incorporate data hierarchies in the analysis of time-course data, where temporal and subject effects are both assumed to be random. The method starts with the clustering of genes by fitting the mixture model within the multilevel random-effects model framework using the expectation-maximization (EM) algorithm. The network of regulatory interactions is then determined by searching for regulatory control elements (activators and inhibitors) shared by the clusters of co-expressed genes, based on a time-lagged correlation coefficients measurement. The method is applied to two real time-course datasets from the budding yeast (Saccharomyces cerevisiae) genome. It is shown that the proposed method provides clusters of cell-cycle regulated genes that are supported by existing gene function annotations, and hence enables inference on regulatory interactions for the genetic network.

We supplement the article of Meng (2006) on the EM algorithm and its applications, providing also an update on its more recent developments and applications. The expectation–maximization algorithm, popularly known as the EM algorithm, is a general‐purpose algorithm for maximum‐likelihood estimation in a wide variety of situations best described as incomplete‐data problems. The name EM algorithm was given by Dempster et al . (1997) in a celebrated paper read before the Royal Statistical Society in 1976 and published in its journal in 1977.