Large Dimensional Covariance Matrix Research Articles

Minority oversampling for imbalanced time series classification

Many vital real-world applications involve time-series data with skewed distribution. Compared to traditional imbalanced learning problems, the classification of imbalanced time-series data is more challenging due to the high dimensionality and high inter-variable correlation. This paper proposes a structure-preserving Oversampling method to resolve the High-dimensional Imbalanced Time-series classification (OHIT). OHIT leverages a density-ratio-based shared nearest neighbor clustering algorithm to capture the modes of minority class in high-dimensional space. It for each mode applies the shrinkage technique of large-dimensional covariance matrix to obtain an accurate and reliable covariance structure. The structure-preserving synthetic samples are eventually generated based on the multivariate Gaussian distribution with the estimated covariance matrix. In addition, to further promote the performance of classifying imbalanced time-series data, we integrate OHIT into boosting framework to obtain a new ensemble algorithm OHITBoost. Extensive experiments on several publicly available time-series datasets (including unimodal and multimodal) demonstrate the effectiveness of OHIT and OHITBoost in terms of F1, G-mean, and AUC.

How can an economic scenario generation model cope with abrupt changes in financial markets?

PurposeIt is quite possible that financial institutions including life insurance companies would encounter turbulent situations such as the COVID-19 pandemic before policies mature. Constructing models that can generate scenarios for major assets to cover abrupt changes in financial markets is thus essential for the financial institution's risk management.Design/methodology/approachThe key issues in such modeling include how to manage the large number of risk factors involved, how to model the dynamics of chosen or derived factors and how to incorporate relations among these factors. The authors propose the orthogonal ARMA–GARCH (autoregressive moving-average–generalized autoregressive conditional heteroskedasticity) approach to tackle these issues. The constructed economic scenario generation (ESG) models pass the backtests covering the period from the beginning of 2018 to the end of May 2020, which includes the turbulent situations caused by COVID-19.FindingsThe backtesting covering the turbulent period of COVID-19, along with fan charts and comparisons on simulated and historical statistics, validates our approach.Originality/valueThis paper is the first one that attempts to generate complex long-term economic scenarios for a large-scale portfolio from its large dimensional covariance matrix estimated by the orthogonal ARMA–GARCH model.

Improved Shrinkage Estimator of Large-Dimensional Covariance Matrix under the Complex Gaussian Distribution

Estimating the covariance matrix of a random vector is essential and challenging in large dimension and small sample size scenarios. The purpose of this paper is to produce an outperformed large-dimensional covariance matrix estimator in the complex domain via the linear shrinkage regularization. Firstly, we develop a necessary moment property of the complex Wishart distribution. Secondly, by minimizing the mean squared error between the real covariance matrix and its shrinkage estimator, we obtain the optimal shrinkage intensity in a closed form for the spherical target matrix under the complex Gaussian distribution. Thirdly, we propose a newly available shrinkage estimator by unbiasedly estimating the unknown scalars involved in the optimal shrinkage intensity. Both the numerical simulations and an example application to array signal processing reveal that the proposed covariance matrix estimator performs well in large dimension and small sample size scenarios.

Improvement of sphericity test for large-dimensional covariance matrix

This paper discusses the problem of testing the sphericity of population covariance matrix when the sample size n and the dimensionality p both tend to infinity with p / n → c ∈ (0,1). A new test statistic is presented by utilizing an inequality, and asymptotic properties of the proposed statistic are derived for generally distributed population under the null hypothesis. Numerical simulations demonstrate that the proposed statistic has a significant improvement on test powers under the alternative hypothesis by comparing with some congeneric statistics.

A Cholesky-based estimation for large-dimensional covariance matrices

ABSTRACTThis paper develops a new method to estimate a large-dimensional covariance matrix when the variables have no natural ordering among themselves. The modified Cholesky decomposition technique is used to provide a set of estimates of the covariance matrix under multiple orderings of variables. The proposed estimator is in the form of a linear combination of these available estimates and the identity matrix. It is positive definite and applicable in large dimensions. The merits of the proposed estimator are demonstrated through the numerical study and a real data example by comparison with several existing methods.

Testing covariance structure of large-dimensional data based on Wald’s score test

This paper considers testing the covariance matrices structure based on Wald’s score test in large-dimensional setting. The tests for identity and sphericity of large-dimensional covariance matrices are reviewed by the generalized CLT for the linear spectral statistics of large-dimensional sample covariance matrices from [D. D. Jiang, Tests for large-dimensional covariance structure based on Rao’s score test, J. Multivariate Anal. 152 (2016) 28–39]. The proposed test can be applicable for large-dimensional non-Gaussian variables in a wider range. Furthermore, the simulation study is conducted to compare the proposed test with other large-dimensional covariance matrix tests. As seen from the simulation results, our proposed test is feasible for large-dimensional data without restriction of population distribution and provides the accurate and steady empirical sizes, which are almost around the nominal size.

Optimal estimation of a large-dimensional covariance matrix under Stein’s loss

This paper introduces a new method for deriving covariance matrix estimators that are decision-theoretically optimal within a class of nonlinear shrinkage estimators. The key is to employ large-dimensional asymptotics: the matrix dimension and the sample size go to infinity together, with their ratio converging to a finite, nonzero limit. As the main focus, we apply this method to Stein’s loss. Compared to the estimator of Stein (Estimation of a covariance matrix (1975); J. Math. Sci. 34 (1986) 1373–1403), ours has five theoretical advantages: (1) it asymptotically minimizes the loss itself, instead of an estimator of the expected loss; (2) it does not necessitate post-processing via an ad hoc algorithm (called “isotonization”) to restore the positivity or the ordering of the covariance matrix eigenvalues; (3) it does not ignore any terms in the function to be minimized; (4) it does not require normality; and (5) it is not limited to applications where the sample size exceeds the dimension. In addition to these theoretical advantages, our estimator also improves upon Stein’s estimator in terms of finite-sample performance, as evidenced via extensive Monte Carlo simulations. To further demonstrate the effectiveness of our method, we show that some previously suggested estimators of the covariance matrix and its inverse are decision-theoretically optimal in the large-dimensional asymptotic limit with respect to the Frobenius loss function.

Identity tests for high dimensional data using RMT

In this work, we redefined two important statistics, the CLRT test [Z. Bai, D. Jiang, J. Yao, S. Zheng, Corrections to LRT on large-dimensional covariance matrix by RMT, The Annals of Statistics 37 (6B) (2009) 3822–3840] and the LW test [O. Ledoit, M. Wolf, Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size, The Annals of Statistics (2002) 1081–1102] on identity tests for high dimensional data using random matrix theories. Compared with existing CLRT and LW tests, the new tests can accommodate data which has unknown means and non-Gaussian distributions. Simulations demonstrate that the new tests have good properties in terms of size and power. What is more, even for Gaussian data, our new tests perform favorably in comparison to existing tests. Finally, we find the CLRT is more sensitive to eigenvalues less than 1 while the LW test has more advantages in relation to detecting eigenvalues larger than 1.

Corrections to LRT on large-dimensional covariance matrix by RMT

In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension p is large compared to the sample size n. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F-matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with χ2 approximation fails. Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.