A family of mixture models for biclustering

Abstract

Biclustering is used for simultaneous clustering of the observations and variables when there is no group structure known a priori. It is being increasingly used in bioinformatics, text analytics, and so on. Previously, biclustering has been introduced in a model‐based clustering framework by utilizing a structure similar to a mixture of factor analyzers. In such models, observed variables X are modeled using a latent variable U that is assumed to be from Nfalse(0,Ifalse). Clustering of variables are introduced by imposing constraints on the entries of the factor loading matrix to be 0 and 1 that results in block diagonal covariance matrices. However, this approach is overly restrictive as off‐diagonal elements in the blocks of the covariance matrices can only be 1 which can lead to unsatisfactory model fit on complex data. Here, the latent variable U is assumed to be from a Nfalse(0,Tfalse) where T is a diagonal matrix. This ensures that the off‐diagonal terms in the block matrices within the covariance matrices are non‐zero and not restricted to be 1. This leads to a superior model fit on complex data. A family of models is developed by imposing constraints on the components of the covariance matrix. For parameter estimation, an alternating expectation conditional maximization (AECM) algorithm is used. Finally, the proposed method is illustrated using simulated and real datasets.

Estimation of Gaussian Bi-Clusters with General Block-Diagonal Covariance Matrix and Applications

“…Motivated by the approach in Tu and Subedi (2022), we decided to adopt a greedy algorithm for estimating D = [d 1 , . .…”

“…d K ] in (4). To initialize we follow Tu and Subedi (2022) and set D to the first K principal components of the correlation matrix S .…”

“…They partition columns of the dataset twice -once by means and another by variances. Others (Martella et al, 2008;Wong et al, 2017;Tu and Subedi, 2022) leveraged the bi-clustering framework rooted in the mixtures of factor analyzers (McNicholas and Murphy, 2008) approach . The advantage of the factor analyzer structure is that it decreases the number of estimated parameters by assuming a latent low-dimensional representation of high-dimensional variables.…”

“…For example, Martella et al (2008) and Wong et al (2017) require the within block offdiagonal elements of the covariance matrix to 1. Tu and Subedi (2022) relaxed some constraints by characterizing covariance to allow the all of the off-diagonal entries of i-th covariance block to be equal to a positive constant. This is illustrated in the left panel of Figure 1.…”

“…This is illustrated in the left panel of Figure 1. In addition, the parameter estimation utilized in Martella et al (2008); Wong et al (2017); Tu and Subedi (2022) has limitations due to the complexity of finding column groupings.…”