Browse
Submit Data
Databases
API
Help

Dataset Information

0 Views

0 Connections

0 Citations

0 Reanalyses

0 Downloads

Omics score: 0

Asymptotics of eigenstructure of sample correlation matrices for high-dimensional spiked models.

ABSTRACT: Sample correlation matrices are widely used, but for high-dimensional data little is known about their spectral properties beyond "null models", which assume the data have independent coordinates. In the class of spiked models, we apply random matrix theory to derive asymptotic first-order and distributional results for both leading eigenvalues and eigenvectors of sample correlation matrices, assuming a high-dimensional regime in which the ratio p/n, of number of variables p to sample size n, converges to a positive constant. While the first-order spectral properties of sample correlation matrices match those of sample covariance matrices, their asymptotic distributions can differ significantly. Indeed, the correlation-based fluctuations of both sample eigenvalues and eigenvectors are often remarkably smaller than those of their sample covariance counterparts.

SUBMITTER: Morales-Jimenez D

PROVIDER: S-EPMC8026145 | biostudies-literature | 2021 Apr

REPOSITORIES: biostudies-literature

ACCESS DATA

Json Xml

Publications

Asymptotics of eigenstructure of sample correlation matrices for high-dimensional spiked models.

Morales-Jimenez David D Johnstone Iain M IM McKay Matthew R MR Yang Jeha J

Statistica Sinica 20210401 2

Sample correlation matrices are widely used, but for high-dimensional data little is known about their spectral properties beyond "null models", which assume the data have independent coordinates. In the class of spiked models, we apply random matrix theory to derive asymptotic first-order and distributional results for both leading eigenvalues and eigenvectors of sample correlation matrices, assuming a high-dimensional regime in which the ratio <i>p/n</i>, of number of variables <i>p</i> to sam ...[more]

PMID: 33833489

Similar Datasets

Asymptotics of empirical eigenstructure for high dimensional spiked covariance.

Project description:We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al. (2013). They also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies.

| S-EPMC5563862 | biostudies-literature

TEST FOR HIGH DIMENSIONAL CORRELATION MATRICES.

Project description:Testing correlation structures has attracted extensive attention in the literature due to both its importance in real applications and several major theoretical challenges. The aim of this paper is to develop a general framework of testing correlation structures for the one-, two-, and multiple sample testing problems under a high-dimensional setting when both the sample size and data dimension go to infinity. Our test statistics are designed to deal with both the dense and sparse alternatives. We systematically investigate the asymptotic null distribution, power function, and unbiasedness of each test statistic. Theoretically, we make great efforts to deal with the non-independency of all random matrices of the sample correlation matrices. We use simulation studies and real data analysis to illustrate the versatility and practicability of our test statistics.

| S-EPMC6709985 | biostudies-literature

The Statistics and Mathematics of High Dimension Low Sample Size Asymptotics.

Project description:The aim of this paper is to establish several deep theoretical properties of principal component analysis for multiple-component spike covariance models. Our new results reveal an asymptotic conical structure in critical sample eigendirections under the spike models with distinguishable (or indistinguishable) eigenvalues, when the sample size and/or the number of variables (or dimension) tend to infinity. The consistency of the sample eigenvectors relative to their population counterparts is determined by the ratio between the dimension and the product of the sample size with the spike size. When this ratio converges to a nonzero constant, the sample eigenvector converges to a cone, with a certain angle to its corresponding population eigenvector. In the High Dimension, Low Sample Size case, the angle between the sample eigenvector and its population counterpart converges to a limiting distribution. Several generalizations of the multi-spike covariance models are also explored, and additional theoretical results are presented.

| S-EPMC5173295 | biostudies-literature

Joint Estimation of Multiple High-dimensional Precision Matrices.

Project description:Motivated by analysis of gene expression data measured in different tissues or disease states, we consider joint estimation of multiple precision matrices to effectively utilize the partially shared graphical structures of the corresponding graphs. The procedure is based on a weighted constrained ℓ∞/ℓ1 minimization, which can be effectively implemented by a second-order cone programming. Compared to separate estimation methods, the proposed joint estimation method leads to estimators converging to the true precision matrices faster. Under certain regularity conditions, the proposed procedure leads to an exact graph structure recovery with a probability tending to 1. Simulation studies show that the proposed joint estimation methods outperform other methods in graph structure recovery. The method is illustrated through an analysis of an ovarian cancer gene expression data. The results indicate that the patients with poor prognostic subtype lack some important links among the genes in the apoptosis pathway.

| S-EPMC5351783 | biostudies-literature

Robust estimation of high-dimensional covariance and precision matrices.

Project description:High-dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components. Covariance and precision matrices provide a useful summary of such structure, yet the performance of popular matrix estimators typically hinges upon a sub-Gaussianity assumption. This paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions. The proposed estimators, under a bounded fourth moment assumption, achieve the same minimax convergence rates as do existing methods under a sub-Gaussianity assumption. Consistency of the proposed estimators is also established under the weak assumption of bounded 2 + ε moments for ε ∈ (0, 2). The associated convergence rates depend on ε.

| S-EPMC6188670 | biostudies-literature

Power and Sample Size Calculations for Generalized Estimating Equations via Local Asymptotics.

Project description:We consider the problem of calculating power and sample size for tests based on generalized estimating equations (GEE), that arise in studies involving clustered or correlated data (e.g., longitudinal studies and sibling studies). Previous approaches approximate the power of such tests using the asymptotic behavior of the test statistics under fixed alternatives. We develop a more accurate approach in which the asymptotic behavior is studied under a sequence of local alternatives that converge to the null hypothesis at root-m rate, where m is the number of clusters. Based on this approach, explicit sample size formulae are derived for Wald and quasi-score test statistics in a variety of GEE settings. Simulation results show that in the important special case of logistic regression with exchangeable correlation structure, previous approaches can inflate the projected sample size (to obtain nominal 90% power using the Wald statistic) by over 10%, whereas the proposed approach provides an accuracy of around 2%.

| S-EPMC3903421 | biostudies-literature

Performance of binary prediction models in high-correlation low-dimensional settings: a comparison of methods.

Project description:BackgroundClinical prediction models are developed widely across medical disciplines. When predictors in such models are highly collinear, unexpected or spurious predictor-outcome associations may occur, thereby potentially reducing face-validity of the prediction model. Collinearity can be dealt with by exclusion of collinear predictors, but when there is no a priori motivation (besides collinearity) to include or exclude specific predictors, such an approach is arbitrary and possibly inappropriate.MethodsWe compare different methods to address collinearity, including shrinkage, dimensionality reduction, and constrained optimization. The effectiveness of these methods is illustrated via simulations.ResultsIn the conducted simulations, no effect of collinearity was observed on predictive outcomes (AUC, R2, Intercept, Slope) across methods. However, a negative effect of collinearity on the stability of predictor selection was found, affecting all compared methods, but in particular methods that perform strong predictor selection (e.g., Lasso). Methods for which the included set of predictors remained most stable under increased collinearity were Ridge, PCLR, LAELR, and Dropout.ConclusionsBased on the results, we would recommend refraining from data-driven predictor selection approaches in the presence of high collinearity, because of the increased instability of predictor selection, even in relatively high events-per-variable settings. The selection of certain predictors over others may disproportionally give the impression that included predictors have a stronger association with the outcome than excluded predictors.

| S-EPMC8751246 | biostudies-literature

Homogeneity tests of covariance matrices with high-dimensional longitudinal data.

Project description:This paper deals with the detection and identification of changepoints among covariances of high-dimensional longitudinal data, where the number of features is greater than both the sample size and the number of repeated measurements. The proposed methods are applicable under general temporal-spatial dependence. A new test statistic is introduced for changepoint detection, and its asymptotic distribution is established. If a changepoint is detected, an estimate of the location is provided. The rate of convergence of the estimator is shown to depend on the data dimension, sample size, and signal-to-noise ratio. Binary segmentation is used to estimate the locations of possibly multiple changepoints, and the corresponding estimator is shown to be consistent under mild conditions. Simulation studies provide the empirical size and power of the proposed test and the accuracy of the changepoint estimator. An application to a time-course microarray dataset identifies gene sets with significant gene interaction changes over time.

| S-EPMC6690172 | biostudies-literature

ASYMPTOTICS FOR CHANGE-POINT MODELS UNDER VARYING DEGREES OF MIS-SPECIFICATION.

Project description:Change-point models are widely used by statisticians to model drastic changes in the pattern of observed data. Least squares/maximum likelihood based estimation of change-points leads to curious asymptotic phenomena. When the change-point model is correctly specified, such estimates generally converge at a fast rate (n) and are asymptotically described by minimizers of a jump process. Under complete mis-specification by a smooth curve, i.e. when a change-point model is fitted to data described by a smooth curve, the rate of convergence slows down to n1/3 and the limit distribution changes to that of the minimizer of a continuous Gaussian process. In this paper we provide a bridge between these two extreme scenarios by studying the limit behavior of change-point estimates under varying degrees of model mis-specification by smooth curves, which can be viewed as local alternatives. We find that the limiting regime depends on how quickly the alternatives approach a change-point model. We unravel a family of 'intermediate' limits that can transition, at least qualitatively, to the limits in the two extreme scenarios. The theoretical results are illustrated via a set of carefully designed simulations. We also demonstrate how inference for the change-point parameter can be performed in absence of knowledge of the underlying scenario by resorting to subsampling techniques that involve estimation of the convergence rate.

| S-EPMC4678008 | biostudies-literature

TESTING HIGH-DIMENSIONAL COVARIANCE MATRICES, WITH APPLICATION TO DETECTING SCHIZOPHRENIA RISK GENES.

Project description:Scientists routinely compare gene expression levels in cases versus controls in part to determine genes associated with a disease. Similarly, detecting case-control differences in co-expression among genes can be critical to understanding complex human diseases; however statistical methods have been limited by the high dimensional nature of this problem. In this paper, we construct a sparse-Leading-Eigenvalue-Driven (sLED) test for comparing two high-dimensional covariance matrices. By focusing on the spectrum of the differential matrix, sLED provides a novel perspective that accommodates what we assume to be common, namely sparse and weak signals in gene expression data, and it is closely related with Sparse Principal Component Analysis. We prove that sLED achieves full power asymptotically under mild assumptions, and simulation studies verify that it outperforms other existing procedures under many biologically plausible scenarios. Applying sLED to the largest gene-expression dataset obtained from post-mortem brain tissue from Schizophrenia patients and controls, we provide a novel list of genes implicated in Schizophrenia and reveal intriguing patterns in gene co-expression change for Schizophrenia subjects. We also illustrate that sLED can be generalized to compare other gene-gene "relationship" matrices that are of practical interest, such as the weighted adjacency matrices.

| S-EPMC5655846 | biostudies-literature

OmicsDI is part of the ELIXIR infrastructure

OmicsDI is an Elixir interoperability service. Learn more ›

Tweets

OmicsDI Databases

PRIDE
PeptideAtlas
MassIVE
JPOST Repository
Physiome Model Repository

EGA
EVA
ENA
LINCS
PAXDB
Cell Collective

MetaboLights
Metabolomics Workbench
MetabolomeExpress
GNPS
BioModels
FAIRDOMHub

ArrayExpress
dbGaP
ExpressionAtlas
GEO
NODE

Information

Databases
Help
API
Contact us
Code on GitHub
Terms of use
Submit Data