High-Dimensional Mahalanobis Distances of Complex Random Vectors

In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed ratio c=p/n→∞. We investigate the distributional properties of complex MD when the random samples are independent, but not necessarily identically distributed. Some results regarding the F-matrix F=S2−1S1—the product of a sample covariance matrix S1 (from the independent variable array (be(Zi)1×n) with the inverse of another covariance matrix S2 (from the independent variable array (Zj≠i)p×n)—are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components S1 and S2 of the F-matrix is not required.

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Asymptotic Distributions of Functions of the Eigenvalues of the Sample Covariance Matrix and Canonical Correlation Matrix in Multivariate Time Series

10.21236/ada170282 ◽

1986 ◽

Author(s):

M. Taniguchi ◽

P. R. Krishnaiah

Keyword(s):

Time Series ◽

Covariance Matrix ◽

Canonical Correlation ◽

Correlation Matrix ◽

Multivariate Time Series ◽

Asymptotic Distributions ◽

Sample Covariance Matrix ◽

Sample Covariance

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Score matching filters

10.5194/egusphere-egu21-16125 ◽

2021 ◽

Author(s):

Marie Turčičová ◽

Jan Mandel ◽

Kryštof Eben

Keyword(s):

Kalman Filter ◽

Covariance Matrix ◽

Markov Random Fields ◽

Computational Cost ◽

Time Step ◽

Sample Covariance ◽

Monte Carlo Approximation ◽

Gauss Markov Random Fields ◽

Ensemble Filtering ◽

Non Gaussian

A widely popular group of data assimilation methods in meteorological and geophysical sciences is formed by filters based on Monte-Carlo approximation of the traditional Kalman filter, e.g. Ensemble Kalman filter (EnKF), Ensemble square-root filter and others. Due to the computational cost, ensemble size is usually small compared to the dimension of the state vector. Traditional EnKF implicitly uses the sample covariance which is a poor estimate of the background covariance matrix - singular and contaminated by spurious correlations. We focus on modelling the background covariance matrix by means of a linear model for its inverse. This is particularly useful in Gauss-Markov random fields (GMRF), where the inverse covariance matrix has a banded structure. The parameters of the model are estimated by the score matching method which provides estimators in a closed form, cheap to compute. The resulting estimate is a key component of the proposed ensemble filtering algorithms. Under the assumption that the state vector is a GMRF in every time-step, the Score matching filter with Gaussian resampling (SMF-GR) gives in every time-step a consistent (in the large ensemble limit) estimator of mean and covariance matrix of the forecast and analysis distribution. Further, we propose a filtering method called Score matching ensemble filter (SMEF), based on regularization of the EnKF. This filter performs well even for non-Gaussian systems with non-linear dynamics. The performance of both filters is illustrated on a simple linear convection model and Lorenz-96.

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Asymptotic distributions of functions of a sample covariance matrix under the elliptical distribution

Canadian Journal of Statistics ◽

10.2307/3315589 ◽

1994 ◽

Vol 22 (2) ◽

pp. 273-283 ◽

Cited By ~ 12

Author(s):

Toshiya Iwashita ◽

Minoru Siotani

Keyword(s):

Covariance Matrix ◽

Asymptotic Distributions ◽

Sample Covariance Matrix ◽

Elliptical Distribution ◽

Sample Covariance

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First- and Second-Order Moments of the Normalized Sample Covariance Matrix of Spherically Invariant Random Vectors

IEEE Signal Processing Letters ◽

10.1109/lsp.2006.888400 ◽

2007 ◽

Vol 14 (6) ◽

pp. 425-428 ◽

Cited By ~ 16

Author(s):

Sbastien Bausson ◽

Frdric Pascal ◽

Philippe Forster ◽

Jean-Philippe Ovarlez ◽

Pascal Larzabal

Keyword(s):

Covariance Matrix ◽

Second Order ◽

Sample Covariance Matrix ◽

Random Vectors ◽

Sample Covariance

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Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Journal of Multivariate Analysis ◽

10.1016/0047-259x(87)90083-2 ◽

1987 ◽

Vol 22 (1) ◽

pp. 156-176 ◽

Cited By ~ 5

Author(s):

M Taniguchi ◽

P.R Krishnaiah

Keyword(s):

Time Series ◽

Covariance Matrix ◽

Canonical Correlation ◽

Correlation Matrix ◽

Multivariate Time Series ◽

Asymptotic Distributions ◽

Sample Covariance Matrix ◽

Sample Covariance

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Analysis of grid operation state based on improved maximum eigenvalue sample covariance matrix algorithm

Proceedings of the Institution of Mechanical Engineers Part I Journal of Systems and Control Engineering ◽

10.1177/09596518211001270 ◽

2021 ◽

pp. 095965182110012

Author(s):

Dinghui Wu ◽

Juan Zhang ◽

Bo Wang ◽

Tinglong Pan

Keyword(s):

Covariance Matrix ◽

Signal To Noise Ratio ◽

Poor Performance ◽

Sample Covariance Matrix ◽

Maximum Eigenvalue ◽

Signal To Noise ◽

Matrix Algorithm ◽

Sample Covariance ◽

Noise Ratio ◽

Application Fields

Traditional static threshold–based state analysis methods can be applied to specific signal-to-noise ratio situations but may present poor performance in the presence of large sizes and complexity of power system. In this article, an improved maximum eigenvalue sample covariance matrix algorithm is proposed, where a Marchenko–Pastur law–based dynamic threshold is introduced by taking all the eigenvalues exceeding the supremum into account for different signal-to-noise ratio situations, to improve the calculation efficiency and widen the application fields of existing methods. The comparison analysis based on IEEE 39-Bus system shows that the proposed algorithm outperforms the existing solutions in terms of calculation speed, anti-interference ability, and universality to different signal-to-noise ratio situations.

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