Matrix factorisation methods applied in microarray data analysis

Abstract

Numerous methods have been applied to microarray data to group genes into clusters that show similar expression patterns. These methods assign each gene to a single group, which does not reflect the widely held view among biologists that most, if not all, genes in eukaryotes are involved in multiple biological processes and therefore will be multiply regulated. Here, we review several methods of matrix factorisation that identify patterns of behaviour in transcriptional response and assign genes to multiple patterns. We focus on these methods rather than traditional clustering methods applied to microarray data, which assign one gene to one cluster.

Figure 1

Matrix Factorization. The data matrix, D, is modeled as arising from the multiplication of a set of patterns, the rows of P, and the assignment of genes to those patterns with varying strengths, the columns of A.

Figure 2

Singular Value Decomposition. The initial matrix D is decomposed into the product of the left singular matrix U, the diagonal matrix of ordered singular values S, and the right singular matrix V^T. The vectors v_k are the eigenassays, while vectors u_I are the eigengenes.

Figure 3

Truncated Singular Value Decomposition. It is possible to discard smaller singular values, keeping only the first p singular values that keep most of the expression information.

Figure 4

Atomic Domain. Bayesian Decomposition utilizes two atomic domains, one that maps to the A matrix and one to the P matrix. The kernel functions, K, allow an atom to map to multiple matrix elements, introducing correlations in the model. Here, K₁ is a simple mapping, with all the amplitude of an atom going to a single matrix element, while K₂ represents a kernel for a transcriptional regulator, with five genes responding in a correlated manner.