A bootstrap test for testing the equality of two ultra-high dimensional covariance matrices

High dimensional covariance matrices have attracted much attention of statisticians and econometricians during the past decades. Vast literature is devoted to the research in high dimensional covariance matrices. However, most of them are for constant covariance matrices. In many applications, constant covariance matrices are not appropriate, e.g. in portfolio allocation, dynamic covariance matrices would make much more sense. Simply assuming each entry of a covariance matrix is a function of time to introduce a dynamic structure would not work. In this paper, we are going to introduce a class of high dimensional dynamic covariance matrices in which a kind of additive structure is embedded. We will show the proposed high dimensional dynamic covariance matrices have many advantages in applications. An estimation procedure is also proposed to estimate the proposed high dimensional dynamic covariance matrices. Asymptotic properties are built to justify the proposed estimation procedure. Intensive simulation studies show the proposed estimation procedure works very well when sample size is finite. Finally, we apply the proposed high dimensional dynamic covariance matrices, together with the proposed estimation procedure, to portfolio allocation. The results look very interesting.

This paper considers the estimation of structured clutter-plus-noise covariance matrix (CNCM) in space-time adaptive processing (STAP) for airborne radar systems. Specially, the CNCM is modeled as a sum of Kronecker products involving two lower dimensional temporal and spatial covariance matrices, with persymmetric structure. Then, resorting to the Kronecker Product principal component analysis (KronPCA) based algorithm, a novel estimator of the high dimensional and persymmetric CNCM is proposed. Furthermore, the proposed method explores the sparse factors of the CNCM and recovers low-rank persymmetric covariance matrices. At analysis stage, we assess the performance of the proposed algorithm through simulations.

Computationally efficient banding of large covariance matrices for ordered data and connections to banding the inverse Cholesky factor.

Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely used techniques, in particular in principal component analysis (PCA). In many modern data analysis problems, statisticians are faced with large datasets where the sample size, n, is of the same order of magnitude as the number of variables p. Random matrix theory predicts that in this context, the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance. We propose to use a fundamental result in random matrix theory, the Marcenko–Pastur equation, to better estimate the eigenvalues of large dimensional covariance matrices. The Marcenko–Pastur equation holds in very wide generality and under weak assumptions. The estimator we obtain can be thought of as “shrinking” in a nonlinear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues. Inspired by ideas of random matrix theory, we also suggest a change of point of view when thinking about estimation of high-dimensional vectors: we do not try to estimate directly the vectors but rather a probability measure that describes them. We think this is a theoretically more fruitful way to think about these problems. Our estimator gives fast and good or very good results in extended simulations. Our algorithmic approach is based on convex optimization. We also show that the proposed estimator is consistent.

This manuscript presents an approach to perform generalized linear regression with multiple high dimensional covariance matrices as the outcome. In many areas of study, such as resting-state functional magnetic resonance imaging (fMRI) studies, this type of regression can be utilized to characterize variation in the covariance matrices across units. Model parameters are estimated by maximizing a likelihood formulation of a generalized linear model, conditioning on a well-conditioned linear shrinkage estimator for multiple covariance matrices, where the shrinkage coefficients are proposed to be shared across matrices. Theoretical studies demonstrate that the proposed covariance matrix estimator is optimal achieving the uniformly minimum quadratic loss asymptotically among all linear combinations of the identity matrix and the sample covariance matrix. Under certain regularity conditions, the proposed estimator of the model parameters is consistent. The superior performance of the proposed approach over existing methods is illustrated through simulation studies. Implemented to a resting-state fMRI study acquired from the Alzheimer's Disease Neuroimaging Initiative, the proposed approach identified a brain network within which functional connectivity is significantly associated with Apolipoprotein E ε4, a strong genetic marker for Alzheimer's disease.

Estimating covariance matrices is an important part of portfolio selection, risk management, and asset pricing. This paper reviews the recent development in estimating high dimensional covariance matrices, where the number of variables can be greater than the number of observations. The limitations of the sample covariance matrix are discussed. Several new approaches are presented, including the shrinkage method, the observable and latent factor method, the Bayesian approach, and the random matrix theory approach. For each method, the construction of covariance matrices is given. The relationships among these methods are discussed.

Estimating high dimensional covariance matrices: A new look at the Gaussian conjugate framework

Spectrum sensing is a key component in cognitive radio networks. Recently there has been intense research interest in eigenvalue based sensing. The results presented so far rely on the distributions of infinite dimensional covariance matrices, therefore these analyses are not accurate for a small sample size. In this paper, we propose a new spectrum sensing method based on the distribution of the largest eigenvalue of the covariance matrices. Using distribution functions for finite dimensional matrices, we conduct exact analysis on the performance of the proposed detector. Essentially, the detection problem requires characterizing the decision threshold as a function of various parameters. The threshold optimization problem is characterized by a weighted sum of false alarm and miss detection probabilities. This detector outperforms the cooperative energy detector with all sample sizes and in the whole SNR range considered. Our proposed detection scheme has direct application in OFDM systems.

Modern tactical surveillance systems benefit from a network of distributed sensors that fuse multiplatform-multisensor data into a single integrated picture. Data fusion is complicated due to inconsistent dimensionality between sensors. For example, some radar systems provide range, bearing, and elevation measurements, while other systems provide two-dimensional measurements in range and bearing only. This paper presents a method for generating three dimensional track states and error covariance matrices from two dimensional tracks from two or more surveillance radars geographically separated in WGS-84 coordinates. Equations are developed for estimating the state and error covariance for the single sensor and multiplatform-multisensor cases. For surveillance radars with multiple tracks, track-to-track assignment is performed using the likelihood of the three dimensional track state for each candidate track-to-track association. Results of Monte Carlo simulations show that the new technique is a practical and efficient method that improves track accuracy, covariance consistency, and hence, the value of netting surveillance radars.

This paper analyzes the output signal-to-interference-plus-noise ratio (SINR) for a multiple-input multiple-output (MIMO) multicarrier code division multiple access (MC-CDMA) system with minimum mean square error receivers. Previous work on a single antenna MC-CDMA system is based on convergence of the limiting spectral distribution of large dimensional sample covariance matrices and the asymptotic freeness between random matrices. These concepts cannot directly be applied to a MIMO MC-CDMA system because some assumptions for single antenna do not match the case of multiple antenna. Therefore, this paper expands the concept of convergence and freeness to MIMO system by using extended Haar unitary random matrix and the Marcěnko-Pastur law. The analysis shows that the output SINR asymptotically converges to a deterministic value with a closed form expression. The analysis is used to obtain improved bit error rate calculation as verified by simulations.

Symmetric Positive Definite (SPD) matrices have been widely used in many computer vision tasks. Recently, there are growing interests in applying covariance matrices to image set classification due to their benefit of encoding image features as a data descriptor. Since SPD matrices follow a non-linear Riemannian geometry, exploiting an appropriate Riemannian metric is the key to successful classification. Adopting Riemannian metrics to classify covariance matrices of image sets is nontrivial, since such matrices are usually singular matrices. Besides, the computational complexity is intolerable while dealing with high dimensional covariance matrices. This paper proposes to use bidirectional covariance matrices instead of covariance matrices as a data descriptor. We model image sets both from the row and column directions of images and these bidirectional covariance matrices are proved to be compact and efficient. Improved accuracy and efficiency are obtained through experiments on standard datasets for comparing bidirectional covariance matrices with covariance matrices.

This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. The convergence rate captures a fundamental tradeoff between estimation error and approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, similar to low rank approximation of covariance matrices. The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm for Kronecker product covariance estimation. We show that a class of block Toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank $r$ that ensures a given level of bias. Simulations are presented to validate the theoretical bounds. As a real world application, we illustrate the utility of the proposed Kronecker covariance estimator for spatio-temporal linear least squares prediction of multivariate wind speed measurements.

This paper presents a new method for estimating high dimensional covariance matrices. Our method, permuted rank-penalized least-squares (PRLS), is based on Kronecker product series expansions of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. In addition, this framework allows one to tradeoff estimation error for approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, an analog to low rank approximation of covariance matrices [1]. The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm [2], [3].

Regularized Semiparametric Estimation of High Dimensional Dynamic Conditional Covariance Matrices

Testing the equality of two covariance matrices is a fundamental problem in statistics, and especially challenging when the data are high-dimensional. Through a novel use of random integration, we can test the equality of high-dimensional covariance matrices without assuming parametric distributions for the two underlying populations, even if the dimension is much larger than the sample size. The asymptotic properties of our test for arbitrary number of covariates and sample size are studied in depth under a general multivariate model. The finite-sample performance of our test is evaluated through numerical studies. The empirical results demonstrate that our test is highly competitive with existing tests in a wide range of settings. In particular, our proposed test is distinctly powerful under different settings when there exist a few large or many small diagonal disturbances between the two covariance matrices.