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.

Keywords: Clustering; Covariance Estimation; High Dimension; Invariance Principles; Unsupervised Learning.

Figure 1

True and estimated covariance matrix eigenvalues based on the hard- and soft-thresholding methods for a simulated data set with d = 1000 and n = 50. This shows that some eigenvalues are highly over-estimated by the hard thresholding method. The soft thresholding method gives major improvement for this example.

Figure 2

Relationships between TCI and tuning parameter τ for three different settings (solid line). Dotted lines represent the TCI calculated from true eigenvalues. Shows situations where hard thresholding is anti-conservative (left panel), where energy preserving soft thresholding is anti-conservative (right panel), and where τ^⋆ is strictly between the two (central panel).

Figure 3

Empirical distributions of SigClust p-values for Simulation 3.2. This shows sample is too conservative, hard is anti-conservative, while soft strikes a nice balance in overall performance.

Figure 4

Empirical distributions of SigClust p-values for Simulation 3.3. The results indicate that hard is strongly anti-conservative, while sample is too conservative. Overall best is the soft method.

Figure 5

PCA projection scatter plot view of the BRCA data, showing 1D (diagonal) and 2D projections of the data onto PC directions. Groupings of colors and symbols indicate biological subtypes. Shows Basals are quite distinct from the others, and there is no strong evidence showing that LumA and LumB do not come from a single Gaussian distribution.