Spatial Sign Covariance Matrix Research Articles

On the eigenvectors of large-dimensional sample spatial sign covariance matrices

To investigate the limiting behavior of eigenvectors of the sample spatial sign covariance matrix (SSCM), the eigenvector empirical spectral distribution (VESD) is defined with weights depending on the eigenvectors. In this paper, we first show that the VESD of a large-dimensional sample SSCM converges to a generalized Marčenko–Pastur distribution when both the dimension p of observations and the sample size n tend to infinity proportionally. Further, the central limit theorem of linear spectral statistics of VESD is established, which suggests that the eigenmatrix of sample SSCM and the classical sample covariance matrix are asymptotically the same.

On eigenvalues of a high-dimensional spatial-sign covariance matrix

This paper investigates limiting spectral properties of a high-dimensional sample spatial-sign covariance matrix when both the dimension of the observations and the sample size grow to infinity. The underlying population is general enough to include the popular independent components model and the family of elliptical distributions. The first result of the paper shows that the empirical spectral distribution of a high dimensional sample spatial-sign covariance matrix converges to a generalized Marčenko-Pastur distribution. Secondly, a new central limit theorem for a class of related linear spectral statistics is established.

Bias Adjusted Sign Covariance Matrix

The spatial sign covariance matrix (SSCM), also known as the normalized sample covariance matrix (NSCM), has been widely used in signal processing as a robust alternative to the sample covariance matrix (SCM). It is well-known that the SSCM does not provide consistent estimates of the eigenvalues of the shape matrix (normalized scatter matrix). To alleviate this problem, we propose BASIC (Bias Adjusted SIgn Covariance), which performs an approximate bias correction to the eigenvalues of the SSCM under the assumption that the samples are generated from zero mean unspecified complex elliptically symmetric distributions (the real-valued case is also addressed). We then use the bias correction in order to develop a robust regularized SSCM based estimator, BASIC Shrinkage estimator (BASICS), which is suitable for high dimensional problems, where the dimension can be larger than the sample size. We assess the proposed estimator with several numerical examples as well as in a linear discriminant analysis (LDA) classification problem with real data sets. The simulations show that the proposed estimator compares well to competing robust covariance matrix estimators but has the advantage of being significantly faster to compute.

Linear Pooling of Sample Covariance Matrices

We consider the problem of estimating high-dimensional covariance matrices of $K$-populations or classes in the setting where the sample sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an asymptotically unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.

A generalized spatial sign covariance matrix

The well-known spatial sign covariance matrix (SSCM) carries out a radial transform which moves all data points to a sphere, followed by computing the classical covariance matrix of the transformed data. Its popularity stems from its robustness to outliers, fast computation, and applications to correlation and principal component analysis. In this paper we study more general radial functions. It is shown that the eigenvectors of the generalized SSCM are still consistent and the ranks of the eigenvalues are preserved. The influence function of the resulting scatter matrix is derived, and it is shown that its asymptotic breakdown value is as high as that of the original SSCM. A simulation study indicates that the best results are obtained when the inner half of the data points are not transformed and points lying far away are moved to the center.

A multivariate sign chart for monitoring dependence among mixed-type data

Abstract Statistical process control (SPC) has been widely utilised for quality improvement and surveillance in industrial engineering. Modern industrial applications have witnessed more and more mixed-type quality characteristics such as those consisting of ordinal categorical and continuous ones. However, traditional charting techniques consider the dependence in either categorical or continuous data and hardly combine the two in quality control. Under the assumption that the ordinal attribute levels of a factor are determined by a latent continuous variable, there exists an order among categorical observations of this factor, which is similar to that among continuous observations. Then mixed-type observations can be transformed into a unified framework of standardized ranks, based on directions of which with respect to their centre parameter, the spatial-sign covariance matrix can be calculated for statistical surveillance of cross-dependence among mixed-type factors. The affine invariant property of consequent charting statistic helps improve the efficiency of detecting dependence shifts in mixed-type data. Simulation results demonstrate the superiority of proposed control chart and an additive manufacturing (3D printing) example shows that it can perform excellently well in practice.

A robust multivariate sign control chart for detecting shifts in covariance matrix under the elliptical directions distributions

Most existing control charts monitoring the covariance matrix of multiple variables were restricted to multivariate normal distribution. When the process distribution is non-normal, the performance of these control charts could potentially be (highly) affected, especially for heavy-tail distributions. To construct a robust multivariate control chart for monitoring the covariance matrix, we applied spatial sign covariance matrix and maximum norm to the exponentially weighted moving average (EWMA) scheme and proposed a Phase II control chart. The novel chart is distribution-free under the family of elliptical directions distributions. Comparison studies demonstrate that the novel method is very powerful in detecting various shifts, especially for heavy-tailed distributions. The implementation of the proposed control chart is demonstrated by a white wine data.

The Spatial Sign Covariance Matrix and Its Application for Robust Correlation Estimation

We summarize properties of the spatial sign covariance matrix and especially consider the relationship between its eigenvalues and those of the shape matrix of an elliptical distribution. The explicit relationship known in the bivariate case was used to construct the spatial sign correlation coefficient, which is a non-parametric and robust estimator for the correlation coefficient within the elliptical model. We consider a multivariate generalization, which we call the multivariate spatial sign correlation matrix. A small simulation study indicates that the new estimator is very efficient under various elliptical distributions if the dimension is large. We furthermore derive its influence function under certain conditions which indicates that the multivariate spatial sign correlation becomes more sensitive to outliers as the dimension increases.

On the eigenvalues of the spatial sign covariance matrix in more than two dimensions

We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.

Asymptotics of the two-stage spatial sign correlation

The spatial sign correlation (D\"urre, Vogel and Fried, 2015) is a highly robust and easy-to-compute, bivariate correlation estimator based on the spatial sign covariance matrix. Since the estimator is inefficient when the marginal scales strongly differ, a two-stage version was proposed. In the first step, the observations are marginally standardized by means of a robust scale estimator, and in the second step, the spatial sign correlation of the thus transformed data set is computed. D\"urre et al. (2015) give some evidence that the asymptotic distribution of the two-stage estimator equals that of the spatial sign correlation at equal marginal scales by comparing their influence functions and presenting simulation results, but give no formal proof. In the present paper, we close this gap and establish the asymptotic normality of the two-stage spatial sign correlation and compute its asymptotic variance for elliptical population distributions. We further derive a variance-stabilizing transformation, similar to Fisher's z-transform, and numerically compare the small-sample coverage probabilities of several confidence intervals.

Semiblind Channel Estimation for OFDM/OQAM Systems

In this letter, we propose a semiblind channel estimation technique for OFDM/OQAM wireless systems. The proposed technique exploits the real property of transmitted symbols to blindly identify the channel-induced rotation in the received signal, thereby reducing the pilot overhead for estimation purpose. Specifically, the channel phase over each subcarrier can be obtained from the spatial-sign covariance matrix of the received signal, while the channel amplitude can be expressed in terms of the subcarrier power. In effect, the frequency-domain noise can be reduced effectively by block averaging over multiple symbols. Since the channel delay spread is usually much smaller than the symbol duration, channel coefficients obtained from the estimated phase and amplitude can be further refined through low-rank filtering. Simulation results validate the efficacy of the proposed technique in time-dispersive fading channels, showing its robustness under different signal-to-noise ratio (SNR) conditions.

Corrigendum to “The spatial sign covariance matrix with unknown location” [J. Multivariate Anal. 130 (2014) 107–117

Corrigendum Corrigendum to ‘‘The spatial sign covariance matrix with unknown location’’ [J. Multivariate Anal. 130 (2014) 107–117] Alexander Durre a, Daniel Vogel b,∗, David E. Tyler c a Fakultat Statistik, Technische Universitat Dortmund, 44221 Dortmund, Germany b Fakultat fur Mathematik, Ruhr-Universitat Bochum, 44780 Bochum, Germany c Department of Statistics, Rutgers University, Piscataway, NJ 08854, USA

Spatial sign correlation

A new robust correlation estimator based on the spatial sign covariance matrix (SSCM) is proposed. We derive its asymptotic distribution and influence function at elliptical distributions. Finite sample and robustness properties are studied and compared to other robust correlation estimators by means of numerical simulations.

The asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions

The asymptotic efficiency of the spatial sign covariance matrix relative to affine equivariant estimators of scatter is studied. In particular, the spatial sign covariance matrix is shown to be asymptotically inadmissible, i.e., the asymptotic covariance matrix of the consistency-corrected spatial sign covariance matrix is uniformly larger than that of its affine equivariant counterpart, namely Tyler’s scatter matrix. Although the spatial sign covariance matrix has often been recommended when one is interested in principal components analysis, its inefficiency is shown to be most severe in situations where principal components are of greatest interest. Simulation shows that the inefficiency of the spatial sign covariance matrix also holds for small sample sizes, and that the asymptotic relative efficiency is a good approximation to the finite-sample efficiency for relatively modest sample sizes.

The spatial sign covariance matrix with unknown location

The consistency and asymptotic normality of the spatial sign covariance matrix with unknown location are shown. Simulations illustrate the different asymptotic behaviors when using the mean and the spatial median as a location estimator.

Tests and estimates of shape based on spatial signs and ranks

Nonparametric procedures for testing and estimation of the shape matrix in the case of multivariate elliptic distribution are considered. Testing for sphericity is an important special case. The tests and estimates are based on the spatial sign and rank covariance matrices. The estimates based on the spatial sign covariance matrix and symmetrized spatial sign covariance matrix are Tyler's [A distribution-free M-estimator of multivariate scatter, Ann. Statist. 15 (1987), pp. 234–251] shape matrix and and Dümbgen's [On Tyler's M-functional of scatter in high dimension, Ann. Inst. Statist. Math. 50 (1998), pp. 471–491] shape matrix, respectively. The test based on the spatial sign covariance matrix is the sign test statistic in the class of nonparametric tests proposed by Hallin and Paindaveine [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707–2756]. New tests and estimates based on the spatial rank covariance matrix are proposed. The shape estimates introduced in the paper play an important role in the inner standardisation of the spatial sign and rank tests for multivariate location. Limiting distributions of the tests and estimates are reviewed and derived, and asymptotic efficiencies as well as finite-sample efficiencies of the proposed tests are compared with those of the classical modified John's [Some optimal multivariate tests, Biometrika 58 (1971), pp. 123–127; The distribution of a statistic used for testing sphericity of normal distributions, Biometrika 59 (1972), pp. 169–173] test and the van der Waerden test (Hallin and Paindaveine, [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707–2756]). The symmetrised spatial sign- and rank-based estimates and tests seem to have a very high efficiency in the multivariate normal case, and they are much better than the classical estimate (shape matrix based on the regular covariance matrix) and test (John's test) for distributions with heavy tails.