A Linear Time Biclustering Algorithm for Time Series Gene Expression Data

Non-supervised machine learning methods have been used in the analysis of gene expression data obtained from microarray experiments. Recently, biclustering, a non-supervised approach that performs simultaneous clustering on the row and column dimensions of the data matrix, has been shown to be remarkably effective in a variety of applications. The goal of biclustering [1] is to find subgroups of genes and subgroups of conditions, where the genes exhibit highly correlated behaviors. In the most common settings, biclustering is an NP-complete problem [3], and heuristic approaches are used to obtain sub-optimal solutions using reasonable computational resources [2]. There exists, however, one particular restriction of the problem that has not been considered before, and that leads to a tractable problem and, indeed, to a surprisingly efficient linear time algorithm for the problem of finding all maximal biclusters. This restriction is applicable when the gene expression data corresponds to snapshots in time of the expression level of the genes. Under this experimental setup, the researcher is, in many cases, particularly interested in biclusters with contiguous columns, that correspond to samples taken in consecutive instants in time where the genes exhibit coherent expression levels. Our algorithm is based on the use of suffix trees, built over a set of strings obtained by first discretizing the values in the gene expression matrix and then performing an alphabet transformation that associates with each symbol in the matrix the column where it belongs. The main result is that, if T is the generalized suffix tree built over these strings, then each internal node satisfying a particular condition, checkable in constant time, defines a contiguous column maximal bicluster. For example, consider (an hypothetical) discretized matrix shown on the left of the following figure: