title: Speeding up the Consensus Clustering methodology for microarray data analysis.
abstract: BACKGROUND
  The inference of the number of clusters in a dataset, a fundamental problem in Statistics, Data Analysis and Classification, is usually addressed via internal validation measures. The stated problem is quite difficult, in particular for microarrays, since the inferred prediction must be sensible enough to capture the inherent biological structure in a dataset, e.g., functionally related genes. Despite the rich literature present in that area, the identification of an internal validation measure that is both fast and precise has proved to be elusive. In order to partially fill this gap, we propose a speed-up of Consensus (Consensus Clustering), a methodology whose purpose is the provision of a prediction of the number of clusters in a dataset, together with a dissimilarity matrix (the consensus matrix) that can be used by clustering algorithms. As detailed in the remainder of the paper, Consensus is a natural candidate for a speed-up.
  
  
  RESULTS
  Since the time-precision performance of Consensus depends on two parameters, our first task is to show that a simple adjustment of the parameters is not enough to obtain a good precision-time trade-off. Our second task is to provide a fast approximation algorithm for Consensus. That is, the closely related algorithm FC (Fast Consensus) that would have the same precision as Consensus with a substantially better time performance. The performance of FC has been assessed via extensive experiments on twelve benchmark datasets that summarize key features of microarray applications, such as cancer studies, gene expression with up and down patterns, and a full spectrum of dimensionality up to over a thousand. Based on their outcome, compared with previous benchmarking results available in the literature, FC turns out to be among the fastest internal validation methods, while retaining the same outstanding precision of Consensus. Moreover, it also provides a consensus matrix that can be used as a dissimilarity matrix, guaranteeing the same performance as the corresponding matrix produced by Consensus. We have also experimented with the use of Consensus and FC in conjunction with NMF (Nonnegative Matrix Factorization), in order to identify the correct number of clusters in a dataset. Although NMF is an increasingly popular technique for biological data mining, our results are somewhat disappointing and complement quite well the state of the art about NMF, shedding further light on its merits and limitations.
  
  
  CONCLUSIONS
  In summary, FC with a parameter setting that makes it robust with respect to small and medium-sized datasets, i.e, number of items to cluster in the hundreds and number of conditions up to a thousand, seems to be the internal validation measure of choice. Moreover, the technique we have developed here can be used in other contexts, in particular for the speed-up of stability-based validation measures.
authors.affiliation: Dipartimento di Matematica ed Informatica, Universitá di Palermo, Via Archirafi 34, 90123 Palermo, Italy. raffaele@math.unipa.it.
authors.forename: Raffaele
authors.initials: R
authors.lastname: Giancarlo
authors.forename: Filippo
authors.initials: F
authors.lastname: Utro