Randomized maps for assessing the reliability of patients clusters in DNA microarray data analyses

Abstract

Objective: Clustering algorithms may be applied to the analysis of DNA microarray data to identify novel subgroups that may lead to new taxonomies of diseases defined at bio-molecular level. A major problem related to the identification of biologically meaningful clusters is the assessment of their reliability, since clustering algorithms may find clusters even if no structure is present. Methodology:Recently, methods based on random "perturbations" of the data, such as bootstrapping, noise injections techniques and random subspace methods have been applied to the problem of cluster validity estimation. In this framework, we propose stability measures that exploits the high dimensionality of DNA microarray data and the redundancy of information stored in microarray chips. To this end we randomly project the original gene expression data into lower dimensional subspaces, approximately preserving the distance between the examples according to the Johnson-Lindenstrauss (JL) theory. The stability of the clusters discovered in the original high dimensional space is estimated by comparing them with the clusters discovered in randomly projected lower dimensional subspaces. The proposed cluster-stability measures may be applied to validate and to quantitatively assess the reliability of the clusters obtained by a large class of clustering algorithms. Results and conclusion:We tested the effectiveness of our approach with high dimensional synthetic data, whose distribution is a priori known, showing that the stability measures based on randomized maps correctly predict the number of clusters and the reliability of each individual cluster. Then we showed how to apply the proposed measures to the analysis of DNA microarray data, whose underlying distribution is unknown. We evaluated the validity of clusters discovered by hierarchical clustering algorithms in diffuse large B-cell lymphoma (DLBCL) and malignant melanoma patients, showing that the proposed reliability measures can support bio-medical researchers in the identification of stable clusters of patients and in the discovery of new subtypes of diseases characterized at bio-molecular level.

A stability-based algorithm to validate hierarchical clusters of genes

“…and by averaging the similarity matrices obtained from the application of a clustering algorithm C to the resulting projected data, we can compute the following stability index s for a cluster A (Bertoni and Valentini, 2006):…”

“…For each cluster obtained through the hierarchical clustering of the original data (Step 1), a stability index (Bertoni and Valentini, 2006) is computed using the similarity matrix constructed at Step 4.…”

“…Despite their successful application in several bioinformatics domains, these methods are well-suited to unsupervised problems characterised by a relatively low number of clusters and/or examples (Smolkin and Gosh, 2003;Bertoni and Valentini, 2006). Indeed if we try to apply them to the analysis of a very high number of clusters, computational problems may arise.…”

“…The main idea of this work consists in the assessment of the reliability of the clusters discovered by a hierarchical clustering algorithm, using a stability based measure borrowed from our previous work (Bertoni and Valentini, 2006). Differently from our previous approach, we do not need to know in advance the correct or approximate number of clusters but we can directly apply a stability measure that estimates the reliability of each individual cluster of the dendrogram computed by a hierarchical algorithm, thus reducing the complexity to a linear number of clusters with respect to the number of available examples.…”

Model order selection for bio-molecular data clustering

“…Different procedures have been introduced to randomly perturb the data, ranging from bootstrapping techniques [9,12,13], to noise injection into the data [14] or random projections into lower dimensional subspaces [15,16]. …”