Dataset Information

Penalized model-based clustering with unconstrained covariance matrices.

ABSTRACT: Clustering is one of the most useful tools for high-dimensional analysis, e.g., for microarray data. It becomes challenging in presence of a large number of noise variables, which may mask underlying clustering structures. Therefore, noise removal through variable selection is necessary. One effective way is regularization for simultaneous parameter estimation and variable selection in model-based clustering. However, existing methods focus on regularizing the mean parameters representing centers of clusters, ignoring dependencies among variables within clusters, leading to incorrect orientations or shapes of the resulting clusters. In this article, we propose a regularized Gaussian mixture model permitting a treatment of general covariance matrices, taking various dependencies into account. At the same time, this approach shrinks the means and covariance matrices, achieving better clustering and variable selection. To overcome one technical challenge in estimating possibly large covariance matrices, we derive an E-M algorithm utilizing the graphical lasso (Friedman et al 2007) for parameter estimation. Numerical examples, including applications to microarray gene expression data, demonstrate the utility of the proposed method.

SUBMITTER: Zhou H

PROVIDER: S-EPMC2867492 | biostudies-literature | 2009 Jan

REPOSITORIES: biostudies-literature

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Publications

Penalized model-based clustering with unconstrained covariance matrices.

Zhou Hui H Pan Wei W Shen Xiaotong X

Electronic journal of statistics 20090101

Clustering is one of the most useful tools for high-dimensional analysis, e.g., for microarray data. It becomes challenging in presence of a large number of noise variables, which may mask underlying clustering structures. Therefore, noise removal through variable selection is necessary. One effective way is regularization for simultaneous parameter estimation and variable selection in model-based clustering. However, existing methods focus on regularizing the mean parameters representing center ...[more]

PMID: 20463857

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Project description:This article investigates a form of rank deficiency in phenotypic covariance matrices derived from geometric morphometric data, and its impact on measures of phenotypic integration. We first define a type of rank deficiency based on information theory then demonstrate that this deficiency impairs the performance of phenotypic integration metrics in a model system. Lastly, we propose methods to treat for this information rank deficiency. Our first goal is to establish how the rank of a typical geometric morphometric covariance matrix relates to the information entropy of its eigenvalue spectrum. This requires clear definitions of matrix rank, of which we define three: the full matrix rank (equal to the number of input variables), the mathematical rank (the number of nonzero eigenvalues), and the information rank or "effective rank" (equal to the number of nonredundant eigenvalues). We demonstrate that effective rank deficiency arises from a combination of methodological factors-Generalized Procrustes analysis, use of the correlation matrix, and insufficient sample size-as well as phenotypic covariance. Secondly, we use dire wolf jaws to document how differences in effective rank deficiency bias two metrics used to measure phenotypic integration. The eigenvalue variance characterizes the integration change incorrectly, and the standardized generalized variance lacks the sensitivity needed to detect subtle changes in integration. Both metrics are impacted by the inclusion of many small, but nonzero, eigenvalues arising from a lack of information in the covariance matrix, a problem that usually becomes more pronounced as the number of landmarks increases. We propose a new metric for phenotypic integration that combines the standardized generalized variance with information entropy. This metric is equivalent to the standardized generalized variance but calculated only from those eigenvalues that carry nonredundant information. It is the standardized generalized variance scaled to the effective rank of the eigenvalue spectrum. We demonstrate that this metric successfully detects the shift of integration in our dire wolf sample. Our third goal is to generalize the new metric to compare data sets with different sample sizes and numbers of variables. We develop a standardization for matrix information based on data permutation then demonstrate that Smilodon jaws are more integrated than dire wolf jaws. Finally, we describe how our information entropy-based measure allows phenotypic integration to be compared in dense semilandmark data sets without bias, allowing characterization of the information content of any given shape, a quantity we term "latent dispersion". [Canis dirus; Dire wolf; effective dispersion; effective rank; geometric morphometrics; information entropy; latent dispersion; modularity and integration; phenotypic integration; relative dispersion.].

Dataset Information

Penalized model-based clustering with unconstrained covariance matrices.

Publications

Penalized model-based clustering with unconstrained covariance matrices.

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