Sample Correlation Matrix Research Articles

Meta-Analytical SEM: Equivalence Between Maximum Likelihood and Generalized Least Squares

Meta-analysis plays a key role in combining studies to obtain more reliable results. In social, behavioral, and health sciences, measurement units are typically not well defined. More meaningful results can be obtained by standardizing the variables and via the analysis of the correlation matrix. Structural equation modeling (SEM) with the combined correlations, called meta-analytical SEM (MASEM), is a powerful tool for examining the relationship among latent constructs as well as those between the latent constructs and the manifest variables. Three classes of methods have been proposed for MASEM: (1) generalized least squares (GLS) in combining correlations and in estimating the structural model, (2) normal-distribution-based maximum likelihood (ML) in combining the correlations and then GLS in estimating the structural model (ML-GLS), and (3) ML in combining correlations and in estimating the structural model (ML). The current article shows that these three methods are equivalent. In particular, (a) the GLS method for combining correlation matrices in meta-analysis is asymptotically equivalent to ML, (b) the three methods (GLS, ML-GLS, ML) for MASEM with correlation matrices are asymptotically equivalent, (c) they also perform equally well empirically, and (d) the GLS method for SEM with the sample correlation matrix in a single study is asymptotically equivalent to ML, which has being discussed extensively in the SEM literature regarding whether the analysis of a correlation matrix yields consistent standard errors and asymptotically valid test statistics. The results and analysis suggest that a sample-size weighted GLS method is preferred for combining correlations and for MASEM.

Estimating large covariance matrix with network topology for high-dimensional biomedical data

Interactions between features of high-dimensional biomedical data often exhibit complex and organized, yet latent, network topological structures. Estimating the non-sparse large covariance matrix of these high-dimensional biomedical data while preserving and recognizing the latent network topology are challenging. A two step procedure is proposed that first detects latent network topological structures from the sample correlation matrix by implementing new penalized optimization and then regularizes the covariance matrix by leveraging the detected network topological information. The network topology guided regularization can reduce false positive and false negative rates simultaneously because it allows edges to borrow strengths from each other precisely. Empirical data examples demonstrate that organized latent network topological structures widely exist in high-dimensional biomedical data across platforms and identifying these network structures can effectively improve estimating covariance matrix and understanding interactive relationships between biomedical features.

A new method for better portfolio investment: A case of the Korean stock market

In this study, a method is devised to estimate a correlation matrix capable of constructing a well-diversified portfolio by the Markowitz mean-variance (MV) optimization function (MVOF), after which evidence is presented to empirically prove that the proposed method effectively reduces the sensitivity of portfolio output caused by the error of input variables, such as the mean and standard deviation of stocks in a portfolio. The proposed method removes the property of a market factor included in the sample correlation matrix through random matrix theory. The results demonstrate the comparative advantage of the proposed method in effectively reducing the sensitivity on both the estimation error and the prediction error from the mean and standard deviation. In particular, this comparative advantage is dependent on the striking reduction of portfolio risk gained by constructing the well-diversified portfolio. The proposed method also achieves high investment performance in the risk-return domain, and is particularly stronger in the unstable situation of either a market crash or a higher-risk portfolio. Consequently, this study offers new insight into how to enhance the practical applicability of the MVOF by controlling the property of the market factor in the sample correlation matrix.

Correlation-Based Meta-Analytic Structural Equation Modeling: Effects of Parameter Covariance on Point and Interval Estimates

More and more researchers use meta-analysis to conduct multivariate analysis to summarize previous findings. In the correlation-based meta-analytic structural equation modeling (cMASEM), the average sample correlation matrix is used to estimate the average population model. Using a simple mediation model, we illustrated that random effects covariation in population parameters can theoretically bias the path coefficient estimates and lead to nonnormal random effects distribution of the correlations. We developed an R function for researchers to examine by simulation the impact of random effects in other models. We then reanalyzed two real data sets and conducted a simulation study to examine the magnitude of the impact on realistic situations. Simulation results suggest parameter bias is typically negligible (less than .02), parameter bias and root mean square error do not differ across methods, 95% confident intervals are sometimes more accurate for the two-stage structural equation modeling approach with a diagonal random effects model, and power is sometimes higher for the traditional Viswesvaran-Ones approach. Given the increasing popularity of cMASEM in organizational research, these simulation results form the basis for us to make several recommendations on its application.

A Note on Collinearity Diagnostics and Centering

ABSTRACTThe usual approach for diagnosing collinearity proceeds by centering and standardizing the regressors. The sample correlation matrix of the predictors is then the basic tool for describing approximate linear combinations that may distort the conclusions of a standard least-square analysis. However, as indicated by several authors, centering may eventually fail to detect the sources of ill-conditioning. In spite of this earlier claim, there does not seem to be in the literature a fully clear explanation of the reasons for this bad potential behavior of the traditional strategy for analyzing collinearity. This note studies this issue in some detail. Results derived are motivated by the analysis of a well-known real dataset. Practical conclusions are illustrated with several examples.

ANÁLISE FATORIAL MULTIVARIADA APLICADA NA CARACTERIZAÇÃO DE CONTAMINANTES DE UM DEPÓSITO DE MINÉRIO DE FERRO

Factorial analysis is the technique of multivariate statistics which aims at the reduction of the number of initial variables of analysis, in order to decrease information loss. The technique creates a small group of common factors not observed from the interpretation of the existing interrelations within the set of observed variables. In this article, the factor analysis was applied using the R program with the objective of finding factors that represent the relation between the contaminants of an iron ore. To this end, the variables Fe, SiO2, Al2O3, P, Mn, TiO2, CaO, MgO and PPC (loss by calcination) were analyzed. The factors were extracted using the Main Components method, using the sample correlation matrix. It was observed that 83.5% of the total variance could be explained by 3 factors. Using the orthogonal rotation of the Varimax type, it was observed that the PPC, CaO and MgO variables have higher loads in factor 1 (carbonate presence), Al and TiO2 variables have higher loads in factor 2 (chemical weathering and/or igneous intrusions) and factor 3 is related to the Manganese variable.

Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from n independent observations of a p-dimensional time series with iid components converge almost surely to (1+γ)2 and (1−γ)2, respectively, as n→∞, if p∕n→γ∈(0,1] and the truncated variance of the entry distribution is “almost slowly varying”, a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marčenko–Pastur law, which extends a result in Bai and Zhou (2008). We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study.In our proofs we use the method of moments combined with a Path-Shortening Algorithm, which efficiently uses the structure of sample correlation matrices, to calculate precise bounds for matrix norms. We believe that this new approach could be of further use in random matrix theory.

On the Performance of MUSIC With Toeplitz Rectification in the Context of Large Arrays

When using subspace methods for DoA estimation such as MUSIC, it is well known that a performance loss occurs when the number of available samples $N$ is not large compared to the number of sensors $M$ . This degradation is mainly due to the use of the sample correlation matrix (SCM), which is a poor estimator of the true correlation matrix of the observations in this situation. When the latter exhibits a Toeplitz structure, a standard trick consists in correcting the structure of the SCM by averaging its entries along the subdiagonals. This procedure, known as Toeplitz rectification, is widely known to improve the estimation of the true correlation matrix, hence the performance of the corresponding subspace methods. In this paper, we propose a statistical analysis of the MUSIC method using Toeplitz rectified SCM (refered to as R-MUSIC), in the context where $M,N$ are of the same order of magnitude. More precisely, considering the asymptotic regime in which $M,N$ converge to infinity at the same rate, we prove the consistency and asymptotic normality of the R-MUSIC DoA estimates. Numerical simulations show the accurate prediction provided by the proposed theoretical analysis.

Robust and sparse correlation matrix estimation for the analysis of high-dimensional genomics data.

Microarray technology can be used to study the expression of thousands of genes across a number of different experimental conditions, usually hundreds. The underlying principle is that genes sharing similar expression patterns, across different samples, can be part of the same co-expression system, or they may share the same biological functions. Groups of genes are usually identified based on cluster analysis. Clustering methods rely on the similarity matrix between genes. A common choice to measure similarity is to compute the sample correlation matrix. Dimensionality reduction is another popular data analysis task which is also based on covariance/correlation matrix estimates. Unfortunately, covariance/correlation matrix estimation suffers from the intrinsic noise present in high-dimensional data. Sources of noise are: sampling variations, presents of outlying sample units, and the fact that in most cases the number of units is much larger than the number of genes. In this paper, we propose a robust correlation matrix estimator that is regularized based on adaptive thresholding. The resulting method jointly tames the effects of the high-dimensionality, and data contamination. Computations are easy to implement and do not require hand tunings. Both simulated and real data are analyzed. A Monte Carlo experiment shows that the proposed method is capable of remarkable performances. Our correlation metric is more robust to outliers compared with the existing alternatives in two gene expression datasets. It is also shown how the regularization allows to automatically detect and filter spurious correlations. The same regularization is also extended to other less robust correlation measures. Finally, we apply the ARACNE algorithm on the SyNTreN gene expression data. Sensitivity and specificity of the reconstructed network is compared with the gold standard. We show that ARACNE performs better when it takes the proposed correlation matrix estimator as input. The R software is available at https://github.com/angy89/RobustSparseCorrelation. aserra@unisa.it or robtag@unisa.it. Supplementary data are available at Bioinformatics online.

Estimating the number of sources in white Gaussian noise: simple eigenvalues based approaches

Estimating the number of sources is a key task in many array signal processing applications. Conventional algorithms such as Akaike's information criterion (AIC) and minimum description length (MDL) suffer from underestimation and overestimation errors. In this study, the authors propose four algorithms to estimate the number of sources in white Gaussian noise. The authors’ proposed algorithms are categorised into two main categories; namely, sample correlation matrix (CorrM) based and correlation coefficient matrix (CoefM) based. Their proposed algorithms are applied on the CorrM and CoefM eigenvalues. They propose to use two decision statistics, which are the moving increment and the moving standard deviation of the estimated eigenvalues as metrics to estimate the number of sources. For their two CorrM based algorithms, the decision statistics are compared to thresholds to decide on the number of sources. They show that the conventional process to estimate the threshold is mathematically tedious with high computational complexity. Alternatively, they define two threshold formulas through linear regression fitting. For their two CoefM based algorithms, they re-define the problem as a simple maximum value search problem. Results show that the proposed algorithms perform on par or better than AIC and MDL as well as recently modified algorithms at medium and high signal-to-noise ratio (SNR) levels and better at low SNR levels and low number of samples, while using a lower complexity criterion function.

Relative Accuracy of Two Modified Parallel Analysis Methods that Use the Proper Reference Distribution.

Parallel analysis (PA) assesses the number of factors in exploratory factor analysis. Traditionally PA compares the eigenvalues for a sample correlation matrix with the eigenvalues for correlation matrices for 100 comparison datasets generated such that the variables are independent, but this approach uses the wrong reference distribution. The proper reference distribution of eigenvalues assesses the kth factor based on comparison datasets with k-1 underlying factors. Two methods that use the proper reference distribution are revised PA (R-PA) and the comparison data method (CDM). We compare the accuracies of these methods using Monte Carlo methods by manipulating the factor structure, factor loadings, factor correlations, and number of observations. In the 17 conditions in which CDM was more accurate than R-PA, both methods evidenced high accuracies (i.e.,>94.5%). In these conditions, CDM had slightly higher accuracies (mean difference of 1.6%). In contrast, in the remaining 25 conditions, R-PA evidenced higher accuracies (mean difference of 12.1%, and considerably higher for some conditions). We consider these findings in conjunction with previous research investigating PA methods and concluded that R-PA tends to offer somewhat stronger results. Nevertheless, further research is required. Given that both CDM and R-PA involve hypothesis testing, we argue that future research should explore effect size statistics to augment these methods.

Correlation Tests and Linear Spectral Statistics of the Sample Correlation Matrix

Testing the independence of the entries of multidimensional Gaussian observations is a very important problem in statistics, with a number of applications in signal processing, radar, cognitive radio, seismography, and multiple other fields. Typically, the problem is formulated as a binary hypothesis test, whereby the presence of correlation is declared when the value of a certain statistic is higher than a certain predetermined threshold. Most of the statistics for correlation tests are constructed from the sample correlation matrix (also known as sample coherence matrix in signal processing), which is defined as a power-normalized version of the sample covariance matrix. In this paper, correlation tests constructed from linear spectral statistics (LSS) of the sample correlation matrix are analyzed under the asymptotic framework where both sample size and observation dimension become large but comparable in magnitude. A central limit theorem (CLT) is established on this class of statistics, which is valid for generally correlated Gaussian observations. Results show that LSS asymptotically fluctuate as Gaussian random variables under both the hypotheses, with an asymptotic mean and variance that can be established for each particular test. In particular, this general CLT can be used to establish the asymptotic behavior of two of the most important correlation test statistics, namely the generalized likelihood ratio test and the Frobenius norm test, under both null and alternative hypotheses. As a by-product, it is established that LSS of sample covariance and sample correlation matrices have exactly the same first order behavior, but quite different asymptotic fluctuations in the second-order regime. In both the cases, the LSS asymptotically behave as Gaussian random variables, although with quite different asymptotic means and variances.

Blind CFO Estimations Based on Subspace-Based Approaches Under Single Data Block for Interleaved OFDMA Uplink

This letter presents computationally efficient subspace-based estimators to estimate carrier frequency offset (CFO) for interleaved orthogonal frequency division multiple access (OFDMA) uplink systems. Conventional subspace-based estimators rely on eigenvalue decomposition (EVD) of sample correlation matrix to compute signal subspace or noise subspace. For interleaved OFDMA with large subchannel, conventional subspace-based CFO estimators inevitably lead to intensive computational complexity due to calculating sample correlation matrix and its EVD. To circumvent this problem, in conjugation with center-symmetric trimmed correlation matrix and Nystrom method, the required signal subspace or orthogonal projection matrix only needs to calculate two submatrices of center-symmetric trimmed correlation matrix. Therefore, the proposed signal subspace-based and noise subspace-based estimators not only can estimate CFO, but also save computational cost, especially in high SNR and large subchannel scenarios. Computer simulation results are also provided for illustrating the effectiveness of the presented blind CFO estimators under single OFDMA data block.

Improving Portfolio Allocation Through Covariance Matrix Filtering

The sample covariance matrix is known to contain substantial statistical noise, making it inappropriate for use in financial decision making. Leading researchers have proposed various filtering methods that attempt to reduce the level of noise in the covariance matrix estimator. In most cases, these methods can be interpreted by analysing how they adjust the eigen-structure of the sample correlation matrix. This paper compares the filtering methods using a theoretical eigen-framework as well as a practical South African experiment. By focussing on the eigen-structure, the sources of statistical noise are identified. The sample correlation matrix suffers from excess dispersion in its eigenvalues and excess dispersion in its pairwise correlations. Bayesian shrinkage estimators that effectively remove the excess dispersion provide superior performance in terms of out-of-sample portfolio risk and turnover. Specifically, the optimal filtering method is a blend between the sample covariance matrix, its diagonal elements and the covariance matrix based on the constant correlation model.

Test of independence for high-dimensional random vectors based on freeness in block correlation matrices

In this paper, we are concerned with the independence test for $k$ high-dimensional sub-vectors of a normal vector, with fixed positive integer $k$. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is proposed for this purpose. We then construct the so-called Schott type statistic as our test statistic, which turns out to be a particular linear spectral statistic of the block correlation matrix. Interestingly, the limiting behavior of the Schott type statistic can be figured out with the aid of the Free Probability Theory and the Random Matrix Theory. Specifically, we will bring the so-called real second order freeness for Haar distributed orthogonal matrices, derived in Mingo and Popa (2013)[10], into the framework of this high-dimensional testing problem. Our test does not require the sample size to be larger than the total or any partial sum of the dimensions of the $k$ sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those statistics proposed in Jiang and Yang (2013)[8] and Jiang, Bai and Zheng (2013)[7], is satisfactory. Real data analysis is also used to illustrate our method.

Two-faced property of a market factor in asset pricing and diversification effect

This study empirically investigates the test hypothesis that a market factor acting as a representative common factor in the pricing models has a negative influence on constructing a well-diversified portfolio from the Markowitz mean–variance optimization function (MVOF). We use the comparative correlation matrix (C-CM) method to control a single eigenvalue among all eigenvalues included in the sample correlation matrix (S-CM), through the random matrix theory (RMT). In particular, this study observes the effect of the largest eigenvalue that has the property of the market factor. According to the results, the largest eigenvalue has the highest explanatory power on the stock return changes. The C-CM without the largest eigenvalue in the S-CM constructs a more diversified portfolio capable of improving the practical applicability of the MVOF. Moreover, the more diversified portfolio constructed from this C-CM has better out-of-sample performance in the future period. These results support the test hypothesis for the two-faced property of the market factor, defined by the largest eigenvalue.

Effects of common factors on stock correlation networks and portfolio diversification

This study empirically investigates the effects of common factors on the connectivity of the network among stocks and on the distribution of the investment weights for stocks. The network is defined as a stock correlation network from the minimal spanning tree (MST), and portfolio is defined as an efficient portfolio from the Markowitz mean-variance (MV) optimization function (MVOF). For these research goals, we devise a method using the comparative correlation matrix (C-CM), which does not have the property of a single common factor included in the sample correlation matrix (S-CM). The results reveal that common factors clearly affect the changes of connectivity among stocks in the networks, and that their influence is much greater on stocks with many links to other stocks in the network. Further, common factors significantly affect the determination of the investment weight's distribution for stocks from the MVOF. In particular, among the common factors, a market factor plays a dominant role in both structuring the network among stocks and in constructing the well-diversified portfolio. In addition, the devised method of the C-CM without the property of the market factor in the S-CM plays a crucial role in constructing a more diversified portfolio with better out-of-sample performance in the future period. These results are robust in both the Korean and the U.S. stocks markets.

Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution.

Correlation matrix plays a key role in many multivariate methods (e.g., graphical model estimation and factor analysis). The current state-of-the-art in estimating large correlation matrices focuses on the use of Pearson's sample correlation matrix. Although Pearson's sample correlation matrix enjoys various good properties under Gaussian models, its not an effective estimator when facing heavy-tail distributions with possible outliers. As a robust alternative, Han and Liu (2013b) advocated the use of a transformed version of the Kendall's tau sample correlation matrix in estimating high dimensional latent generalized correlation matrix under the transelliptical distribution family (or elliptical copula). The transelliptical family assumes that after unspecified marginal monotone transformations, the data follow an elliptical distribution. In this paper, we study the theoretical properties of the Kendall's tau sample correlation matrix and its transformed version proposed in Han and Liu (2013b) for estimating the population Kendall's tau correlation matrix and the latent Pearson's correlation matrix under both spectral and restricted spectral norms. With regard to the spectral norm, we highlight the role of "effective rank" in quantifying the rate of convergence. With regard to the restricted spectral norm, we for the first time present a "sign subgaussian condition" which is sufficient to guarantee that the rank-based correlation matrix estimator attains the optimal rate of convergence. In both cases, we do not need any moment condition.

Hypothesis testing for differentially correlated features.

In a multivariate setting, we consider the task of identifying features whose correlations with the other features differ across conditions. Such correlation shifts may occur independently of mean shifts, or differences in the means of the individual features across conditions. Previous approaches for detecting correlation shifts consider features simultaneously, by computing a correlation-based test statistic for each feature. However, since correlations involve two features, such approaches do not lend themselves to identifying which feature is the culprit. In this article, we instead consider a serial testing approach, by comparing columns of the sample correlation matrix across two conditions, and removing one feature at a time. Our method provides a novel perspective and favorable empirical results compared with competing approaches.

A test for the complete independence of high-dimensional random vectors

ABSTRACTThis paper discusses the problem of testing the complete independence of random variables when the dimension of observations can be much larger than the sample size. It is reported that two typical tests based on, respectively, the biggest off-diagonal entry and the largest eigenvalue of the sample correlation matrix lose their control of type I error in such high-dimensional scenarios, and exhibit distinct behaviours in type II error under different types of alternative hypothesis. Given these facts, we propose a permutation test procedure by synthesizing these two extreme statistics. Simulation results show that for finite dimension and sample size the proposed test outperforms the existing methods in various cases.