title: Statistical Significance of Clustering using Soft Thresholding.
abstract: Clustering methods have led to a number of important discoveries in bioinformatics and beyond. A major challenge in their use is determining which clusters represent important underlying structure, as opposed to spurious sampling artifacts. This challenge is especially serious, and very few methods are available, when the data are very high in dimension. Statistical Significance of Clustering (SigClust) is a recently developed cluster evaluation tool for high dimensional low sample size data. An important component of the SigClust approach is the very definition of a single cluster as a subset of data sampled from a multivariate Gaussian distribution. The implementation of SigClust requires the estimation of the eigenvalues of the covariance matrix for the null multivariate Gaussian distribution. We show that the original eigenvalue estimation can lead to a test that suffers from severe inflation of type-I error, in the important case where there are a few very large eigenvalues. This paper addresses this critical challenge using a novel likelihood based soft thresholding approach to estimate these eigenvalues, which leads to a much improved SigClust. Major improvements in SigClust performance are shown by both mathematical analysis, based on the new notion of Theoretical Cluster Index, and extensive simulation studies. Applications to some cancer genomic data further demonstrate the usefulness of these improvements.
authors.affiliation: Department of Epidemiology and Biostatistics, University of Georgia, Athens, GA 30605.
authors.forename: Hanwen
authors.initials: H
authors.lastname: Huang
authors.affiliation: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599; Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599; Lineberger Comprehensive Cancer Center, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599; Carolina Center for Genome Sciences, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599.
authors.forename: Yufeng
authors.initials: Y
authors.lastname: Liu
authors.affiliation: Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706.
authors.forename: Ming
authors.initials: M
authors.lastname: Yuan
authors.affiliation: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599; Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599; Lineberger Comprehensive Cancer Center, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599.
authors.forename: J S
authors.initials: JS
authors.lastname: Marron
keywords: Clustering; Covariance Estimation; High Dimension; Invariance Principles; Unsupervised Learning