Biclustering ordinal data through a model-based approach

A finite mixture model to simultaneously cluster the rows and columns of a two-mode ordinal data matrix is proposed. Following the Underlying Response Variable (URV) approach, the observed variables are considered as a discretization of latent continuous variables distributed as a mixture of Gaussians. To introduce a partition of the P variables within the g-th component of the mixture, we adopt a factorial representation of the data where a binary row stochastic matrix, representing variable membership, is used to cluster variables. In this way, we associate a component in the finite mixture to a cluster of variables and define a bicluster of units and variables. The number of clusters of variables (and therefore the partition of variables) may vary with clusters of units. Due to the numerical intractability of the likelihood function, estimation of model parameters is based on composite likelihood (CL) methods. It essentially reduces to a computationally efficient Expectation-Maximization type algorithm. The performance of the proposed approach is discussed in both simulated and real datasets.