Sample Correlation Matrices Research Articles

Large sample correlation matrices with unbounded spectrum

In this paper, we demonstrate that the diagonal of a high-dimensional sample covariance matrix stemming from n independent observations of a p-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix regardless of the spectral norm of the population covariance matrix. Our assumptions involve p and n tending to infinity, with p/n tending to a constant which might be positive or zero. Consequently, we investigate the asymptotic properties of the sample correlation matrix with a divergent spectrum, and we explore its applications by deriving the limiting spectral distribution for its eigenvalues and analyzing the convergence of divergent and non-divergent spiked eigenvalues under a generalized spiked correlation framework.

Factor retention in ordered categorical variables: Benefits and costs of polychoric correlations in eigenvalue-based testing

An essential step in exploratory factor analysis is to determine the optimal number of factors. The Next Eigenvalue Sufficiency Test (NEST; Achim, 2017) is a recent proposal to determine the number of factors based on significance tests of the statistical contributions of candidate factors indicated by eigenvalues of sample correlation matrices. Previous simulation studies have shown NEST to recover the optimal number of factors in simulated datasets with high accuracy. However, these studies have focused on continuous variables. The present work addresses the performance of NEST for ordinal data. It has been debated whether factor models – and thus also the optimal number of factors – for ordinal variables should be computed for Pearson correlation matrices, which are known to underestimate correlations for ordinal datasets, or for polychoric correlation matrices, which are known to be instable. The central research question is to what extent the problems associated with Pearson correlations and polychoric correlations deteriorate NEST for ordinal datasets. Implementations of NEST tailored to ordinal datasets by utilizing polychoric correlations are proposed. In a simulation, the proposed implementations were compared to the original implementation of NEST which computes Pearson correlations even for ordinal datasets. The simulation shows that substituting polychoric correlations for Pearson correlations improves the accuracy of NEST for binary variables and large sample sizes (N = 500). However, the simulation also shows that the original implementation using Pearson correlations was the most accurate implementation for Likert-type variables with four response categories when item difficulties were homogeneous.

Bayesian Structural Equation Models of Correlation Matrices

We present a method for Bayesian structural equation modeling of sample correlation matrices as correlation structures. The method transforms the sample correlation matrix to an unbounded vector using the matrix logarithm function. Bayesian inference about the unbounded vector is performed assuming a multivariate-normal likelihood, with a mean based on the transformed model-implied correlation matrix, and a covariance assumed to be of known form. Using Monte Carlo simulation, we examine the performance of the method with normal and ordinal indicators, as well as the capacity of the method to estimate models that account for misspecification. The performance of the approach is often adequate suggesting that the proposed method can be used for Bayesian analysis of correlation structures. We conclude with a discussion of potential applications of the approach, as well as future directions needed to further develop the method.

A Strong Limit Theorem of the Largest Entries of a Sample Correlation Matrices under a Strong Mixing Assumption

We are interested in an n by p matrix Xn where the n rows are strictly stationary α-mixing random vectors and each of the p columns is an independent and identically distributed random vector; p=pn goes to infinity as n→∞, satisfiying 0<c1≤pn/nτ≤c2<∞, where τ>0, c2≥c1>0. We obtain a logarithmic law of Ln=max1≤i<j≤pn|ρij| using the Chen–Stein Poisson approximation method, where ρij denotes the sample correlation coefficient between the ith column and the jth column of Xn.

Central limit theorem of linear spectral statistics of high-dimensional sample correlation matrices

A high-dimensional sample correlation matrix is an important random matrix in multivariate statistical analysis. Its central limit theory is one of the main theoretical bases for making statistical inferences on high-dimensional correlation matrices. Under the high-dimensional framework in which the data dimension tends to infinity proportionally with the sample size, we establish the central limit theorems (CLT) for the linear spectral statistics (LSS) of sample correlation matrices in two settings: (1) the population follows an independent component structure; (2) the population follows an elliptical structure, including some heavy-tailed distributions. The results show that the CLTs of the LSS of the sample correlation matrices are very different in the two settings. In particular, even if the population correlation matrix is an identity matrix, the CLTs are different in the two settings. An application of our two established CLTs is provided.

Properties of eigenvalues and eigenvectors of large-dimensional sample correlation matrices

This paper is to study the properties of eigenvalues and eigenvectors of high-dimensional sample correlation matrices. We first improve the result of Jiang (Sankhyā 66 (2004) 35–48), Xiao and Zhou (J. Theoret. Probab. 23 (2010) 1–20) and the Theorem 1 of El Karoui (Ann. Appl. Probab. 19 (2009) 2362–2405), both concerning the limiting spectral distribution and the extreme eigenvalues of sample correlation matrices, by allowing a more general fourth moment condition. Then, we establish a central limit theorem (CLT) for the linear statistics of the eigenvectors of large sample correlation matrices. We discover that the difference between the functional CLT of the sample covariance matrix and the sample correlation matrix is fundamentally influenced by the direction of a nonrandom projection vector. In the special case where the square root of the correlation matrix is identity, the difference will be determined by the sum of the fourth powers of the entries of the projection vector. These results also indicate that the eigenmatrix of sample correlation matrices is not asymptotically Haar if the underlying distribution is Gaussian. In other words, the normalization based on the sample variances affects the asymptotic properties of the eigenmatrix of the Wishart matrix. Furthermore, we establish a theorem concerning CLT for the linear statistics of the eigenvectors of large sample covariance matrices. This theorem improves the main results in Bai, Miao and Pan (Ann. Probab. 35 (2007) 1532–1572), which requires the assumption that the fourth moment of the underlying variable matches the one of Gaussian distribution, as well as Theorem 1.3 in Pan and Zhou (Ann. Appl. Probab. 18 (2008) 1232–1270), which relaxed the Gaussian like fourth moment requirement but assumes the maximum entries of the projection vector converge to 0 uniformly. We illustrate the usefulness of the theoretical results through an application in communications.

Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations

Consider a p-dimensional population x∈Rp with i.i.d. coordinates that are regularly varying with index α∈(0,2). Since the variance of x is infinite, the diagonal elements of the sample covariance matrix Sn=n−1 ∑ i=1nxixi′ based on a sample x1,…,xn from the population tend to infinity as n increases and it is of interest to use instead the sample correlation matrix Rn={diag(Sn)}−1/2Sn{diag(Sn)}−1/2. This paper finds the limiting distributions of the eigenvalues of Rn when both the dimension p and the sample size n grow to infinity such that p/n→γ∈(0,∞). The family of limiting distributions {Hα,γ} is new and depends on the two parameters α and γ. The moments of Hα,γ are fully identified as sum of two contributions: the first from the classical Marčenko–Pastur law and a second due to heavy tails. Moreover, the family {Hα,γ} has continuous extensions at the boundaries α=2 and α=0 leading to the Marčenko–Pastur law and a modified Poisson distribution, respectively. Our proofs use the method of moments, the path-shortening algorithm developed in [18] (Stochastic Process. Appl. 128 (2018) 2779–2815) and some novel graph counting combinatorics. As a consequence, the moments of Hα,γ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions Hα,γ is also provided for comparison with the Marčenko–Pastur law.

The Asymptotic Distributions of the Largest Entries of Sample Correlation Matrices under an α-mixing Assumption

The Asymptotic Distributions of the Largest Entries of Sample Correlation Matrices under an α-mixing Assumption

Large sample correlation matrices: a comparison theorem and its applications

In this paper, we show that the diagonal of a high-dimensional sample covariance matrix stemming from n independent observations of a p-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix. We assume that n,p→∞ with p∕n tending to a constant which might be positive or zero. As applications, we provide an approximation of the sample correlation matrix R and derive a variety of results for its eigenvalues. We identify the limiting spectral distribution of R and construct an estimator for the population correlation matrix and its eigenvalues. Finally, the almost sure limits of the extreme eigenvalues of R in a generalized spiked correlation model are analyzed.

Limit Properties of the Largest Entries of High-Dimensional Sample Covariance and Correlation Matrices

In this paper, we consider the limit properties of the largest entries of sample covariance matrices and the sample correlation matrices. In order to make the statistics based on the largest entries of the sample covariance matrices and the sample correlation matrices more applicable in high-dimensional tests, the identically distributed assumption of population is removed. Under some moment’s assumption of the underlying distribution, we obtain that the almost surely limit and asymptotical distribution of the extreme statistics as both the dimension p and sample size n tend to infinity.

Point process convergence for the off-diagonal entries of sample covariance matrices

We study point process convergence for sequences of i.i.d. random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a (2+δ)th moment. We make heavy use of precise large deviation results for sums of i.i.d. random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the asymptotic Gumbel property of the largest entry.

Reconsidering the RBANS Factor Structure: a Systematic Literature Review and Meta-Analytic Factor Analysis.

The primary aim was toperform a systematic literature review and extract data necessary for a meta-analytic factor analysis of the RBANS. Secondary aims were to examine the potential validity and utility of the resulting factor structure. Literature was identified through a review of PsycINFO, PubMed, MEDLINE, Academic Search Complete, Psychology & Behavioral Sciences Collection, CINAHL Complete, Health Source: Nursing/Academic Edition, and SocINDEX. A two-stage meta-analytic structural equation modeling method was implemented to pool correlation matrices from primary studies and perform confirmatory factor analyses. Following model selection, factor scores were computed for two datasets and subjected to correlation and diagnostic accuracy analyses. A pooled correlation matrix was computed from 24 sample correlation matrices (N = 5299). Confirmatory factor analysis revealed that the theoretical five-factor model produced the best fit but only when error terms between Story Memory and Story Recall as well as between Figure Copy and Figure Recall were included. Regression-based factor scores showed mixed relationships with the manual-defined indices, and the overall diagnostic accuracy of the factor scores was adequate in both samples examined (AUC = 0.71 and 0.87). The five-factor model was an unexpected result given the failure of multiple previous studies to find support for that model. The five-factor model demonstrates several areas of potential improvement, including better representation of the factors by the indicators. The factor scores implied by this model also require further validation.

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

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.

Constructing networks by filtering correlation matrices: a null model approach.

Network analysis has been applied to various correlation matrix data. Thresholding on the value of the pairwise correlation is probably the most straightforward and common method to create a network from a correlation matrix. However, there have been criticisms on this thresholding approach such as an inability to filter out spurious correlations, which have led to proposals of alternative methods to overcome some of the problems. We propose a method to create networks from correlation matrices based on optimization with regularization, where we lay an edge between each pair of nodes if and only if the edge is unexpected from a null model. The proposed algorithm is advantageous in that it can be combined with different types of null models. Moreover, the algorithm can select the most plausible null model from a set of candidate null models using a model selection criterion. For three economic datasets, we find that the configuration model for correlation matrices is often preferred to standard null models. For country-level product export data, the present method better predicts main products exported from countries than sample correlation matrices do.

Be careful with your principal components.

Principal components analysis (PCA) is a common method to summarize a larger set of correlated variables into a smaller and more easily interpretable axes of variation. However, the different components need to be distinct from each other to be interpretable otherwise they only represent random directions. This is a fundamental assumption of PCA and, thus, needs to be tested every time. Sample correlation matrices will always result in a pattern of decreasing eigenvalues even if there is no structure. Tests are, therefore, needed to discern real patterns from illusionary ones. Furthermore, the loadings of the vectors need to be larger than expected by random data to be useful in the calculation of PC-scores. PC-scores calculated from nondistinct PC's have very large standard errors and cannot be used for biological interpretations. I give a number of examples to illustrate the potential problems with PCA. Robustness of the PC's increases with increasing sample size but not with the number of traits. I review a few simple test statistics appropriate for testing PC's and use a real-world example to illustrate how this can be done using randomization tests. PCA can be very useful but great care is needed to avoid spurious results.

Largest entries of sample correlation matrices from equi-correlated normal populations

The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient $\rho >0$ and both the population dimension $p$ and the sample size $n$ tend to infinity with $\log p=o(n^{\frac{1}{3}})$. As $0<\rho <1$, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussian distribution, and the same is true for the sample covariance matrix as $0<\rho <1/2$. This differs substantially from a well-known result for the independent case where $\rho =0$, in which the above limiting distribution is an extreme-value distribution. We then study the phase transition between these two limiting distributions and identify the regime of $\rho $ where the transition occurs. If $\rho $ is less than, larger than or is equal to the threshold, the corresponding limiting distribution is the extreme-value distribution, the Gaussian distribution and a convolution of the two distributions, respectively. The proofs rely on a subtle use of the Chen–Stein Poisson approximation method, conditioning, a coupling to create independence and a special property of sample correlation matrices. An application is given for a statistical testing problem.

TEST FOR HIGH DIMENSIONAL CORRELATION MATRICES.

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.

Strong convergence of ESD for large quaternion sample covariance matrices and correlation matrices when p/n → 0

In this paper, we study the strong convergence of empirical spectral distribution (ESD) of the large quaternion sample covariance matrices and correlation matrices when the ratio of the population dimension [Formula: see text] to sample size [Formula: see text] tends to zero. We prove that the ESD of renormalized quaternion sample covariance matrices converges almost surely to the semicircle law.

Random Matrix Theory for Heavy-Tailed Time Series

This paper is a review of recent results for large random matrices with heavy-tailed entries. First, we outline the development of and some classical results in random matrix theory. We focus on large sample covariance matrices, their limiting spectral distributions, and the asymptotic behavior of their largest and smallest eigenvalues and their eigenvectors. The limits significantly depend on the finite or infiniteness of the fourth moment of the entries of the random matrix. We compare the results for these two regimes which give rise to completely different asymptotic theories. Finally, the limits of the extreme eigenvalues of sample correlation matrices are examined.

A multiple testing approach to the regularisation of large sample correlation matrices

This paper proposes a regularisation method for the estimation of large covariance matrices that uses insights from the multiple testing (MT) literature. The approach tests the statistical significance of individual pair-wise correlations and sets to zero those elements that are not statistically significant, taking account of the multiple testing nature of the problem. The effective p-values of the tests are set as a decreasing function of N (the cross section dimension), the rate of which is governed by the nature of dependence of the underlying observations, and the relative expansion rates of N and T (the time dimension). In this respect, the method specifies the appropriate thresholding parameter to be used under Gaussian and non-Gaussian settings. The MT estimator of the sample correlation matrix is shown to be consistent in the spectral and Frobenius norms, and in terms of support recovery, so long as the true covariance matrix is sparse. The performance of the proposed MT estimator is compared to a number of other estimators in the literature using Monte Carlo experiments. It is shown that the MT estimator performs well and tends to outperform the other estimators, particularly when N is larger than T.