Rank Covariance Matrix for a Partially Known Covariance Matrix
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Classical multivariate methods are often based on the sample covariance matrix, which is very sensitive to outlying observations. One alternative to the covariance matrix is the a ne equivariant rank covariance matrix (RCM) that has been studied for example in Visuri et al. (2003). In this article we assume that the covariance matrix is partially known and study how to estimate the corresponding RCM. We use the properties that the RCM is a ne equivariant and that the RCM is proportional to the inverse of the regular covariance matrix, and reduce the problem of estimating the RCM to estimating marginal rank covariance matrices. This is a great advantage when the dimension of the original data vectors is large.Keywords:
Matérn covariance function
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Matérn covariance function
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The problem of estimating a normal covariance matrix is considered from a decision-theoretic point of view, where the dimension of the covariance matrix is larger than the sample size. This paper addresses not only the nonsingular case but also the singular case in terms of the covariance matrix. Based on James and Stein's minimax estimator and on an orthogonally invariant estimator, some classes of estimators are unifiedly defined for any possible ordering on the dimension, the sample size and the rank of the covariance matrix. Unified dominance results on such classes are provided under a Stein-type entropy loss. The unified dominance results are applied to improving on an empirical Bayes estimator of a high-dimensional covariance matrix.
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공분산 행렬은 다변량 통계분석에서 중요한 역할을 하고 있으며 전통적인 다변량 분석의 경우 표본 공분산 행렬이 참공분산 행렬의 추정량으로 주로 사용되었다. 하지만 변수의 수가 표본의 크기보다 훨씬 큰 고차원 데이터와 같은 경우에는 표본 공분산 행렬은 비정칙행렬이 되어 기존의 다변량 기법을 사용하는 데 적절하지 않을 수가 있다. 최근 이러한 문제점을 해결하기 위해 축소추정, 경계추정, 수정 콜레스키 분해 추정 등의 새로운 공분산 행렬의 추정량들이 제안되었다. 본 논문에서는 추정량들의 성능에 영향을 미칠 수 있는 여러 현실적인 상황들을 가정하여 모의실험을 통해 참공분산 행렬의 추정량들의 성능을 비교하였다. The covariance matrix is important in multivariate statistical analysis and a sample covariance matrix is used as an estimator of the covariance matrix. High dimensional data has a larger dimension than the sample size; therefore, the sample covariance matrix may not be suitable since it is known to perform poorly and event not invertible. A number of covariance matrix estimators have been recently proposed with three different approaches of shrinkage, thresholding, and modified Cholesky decomposition. We compare the performance of these newly proposed estimators in various situations.
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Matérn covariance function
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We develop a method for estimating well-conditioned and sparse covariance and inverse covariance matrices from a sample of vectors drawn from a sub-gaussian distribution in high dimensional setting. The proposed estimators are obtained by minimizing the quadratic loss function and joint penalty of `1 norm and variance of its eigenvalues. In contrast to some of the existing methods of covariance and inverse covariance matrix estimation, where often the interest is to estimate a sparse matrix, the proposed method is flexible in estimating both a sparse and well-conditioned covariance matrix simultaneously. The proposed estimators are optimal in the sense that they achieve the minimax rate of estimation in operator norm for the underlying class of covariance and inverse covariance matrices. We give a very fast algorithm for computation of these covariance and inverse covariance matrices which is easily scalable to large scale data analysis problems. The simulation study for varying sample sizes and variables shows that the proposed estimators performs better than several other estimators for various choices of structured covariance and inverse covariance matrices. We also use our proposed estimator for tumor tissues classification using gene expression data and compare its performance with some other classification methods.
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In this article, we employ a regression formulation to estimate the high-dimensional covariance matrix for a given network structure. Using prior information contained in the network relationships, we model the covariance as a polynomial function of the symmetric adjacency matrix. Accordingly, the problem of estimating a high-dimensional covariance matrix is converted to one of estimating low dimensional coefficients of the polynomial regression function, which we can accomplish using ordinary least squares or maximum likelihood. The resulting covariance matrix estimator based on the maximum likelihood approach is guaranteed to be positive definite even in finite samples. Under mild conditions, we obtain the theoretical properties of the resulting estimators. A Bayesian information criterion is also developed to select the order of the polynomial function. Simulation studies and empirical examples illustrate the usefulness of the proposed methods.
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The condition of PINCUS (1974) for the estimability of covariance components in normal models is extended to the case of singular covariance matrices
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Estimating large covariance and precision (inverse covariance) matrices has become increasingly important in high dimensional statistics because of its wide applications. The estimation problem is challenging not only theoretically due to the constraint of its positive definiteness, but also computationally because of the curse of dimensionality. Many types of estimators have been proposed such as thresholding under the sparsity assumption of the target matrix, banding and tapering the sample covariance matrix. However, these estimators are not always guaranteed to be positive-definite, especially, for finite samples, and the sparsity assumption is rather restrictive. We propose a novel two-stage adaptive method based on the Cholesky decomposition of a general covariance matrix. By banding the precision matrix in the first stage and adapting the estimates to the second stage estimation, we develop a computationally efficient and statistically accurate method for estimating high dimensional precision matrices. We demonstrate the finite-sample performance of the proposed method by simulations from autoregressive, moving average, and long-range dependent processes. We illustrate its wide applicability by analyzing financial data such S&P 500 index and IBM stock returns, and electric power consumption of individual households. The theoretical properties of the proposed method are also investigated within a large class of covariance matrices.
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Matérn covariance function
Covariance mapping
Scatter matrix
Matrix (chemical analysis)
CMA-ES
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For statistical inferences that involve covariance matrices, it is desirable to obtain an accurate covariance matrix estimate with a well-structured eigen-system. We propose to estimate the covariance matrix through its matrix logarithm based on an approximate log-likelihood function. We develop a generalization of the Leonard and Hsu log-likelihood approximation that no longer requires a nonsingular sample covariance matrix. The matrix log-transformation provides the ability to impose a convex penalty on the transformed likelihood such that the largest and smallest eigenvalues of the covariance matrix estimate can be regularized simultaneously. The proposed method transforms the problem of estimating the covariance matrix into the problem of estimating a symmetric matrix, which can be solved efficiently by an iterative quadratic programming algorithm. The merits of the proposed method are illustrated by a simulation study and two real applications in classification and portfolio optimization. Supplementary materials for this article are available online.
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Matrix (chemical analysis)
Scatter matrix
Matérn covariance function
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Matrix (chemical analysis)
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