Positive-definite regularized estimation for high-dimensional covariance on scalar regression
Covariance is an important measure of marginal dependence among variables. However, heterogeneity in subject covariances and regression models for high-dimensional covariance matrices is not well studied. Compared to regression analysis for conditional means, modeling high-dimensional covariances is...
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| Vydáno v: | Biometrics Ročník 81; číslo 1 |
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07.01.2025
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| ISSN: | 0006-341X, 1541-0420, 1541-0420 |
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| Abstract | Covariance is an important measure of marginal dependence among variables. However, heterogeneity in subject covariances and regression models for high-dimensional covariance matrices is not well studied. Compared to regression analysis for conditional means, modeling high-dimensional covariances is much more challenging due to the large set of free parameters and the intrinsic positive-definite property that puts constraints on the regression parameters. In this paper, we propose a regularized estimation method for the regression coefficients of covariances under sufficient and necessary constraints for the positive definiteness of the conditional average covariance matrices given covariates. The proposed estimator satisfies the sparsity and positive-definite properties simultaneously. An alternating direction method of multipliers (ADMM) algorithm is proposed to solve the constrained and regularized optimization problem. We show the convergence of the proposed ADMM algorithm and derive the convergence rates of the proposed estimators for the regression coefficients and the heterogeneous covariances. The proposed method is evaluated by simulation studies, and its practical application is demonstrated by a case study on brain connectivity. |
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| AbstractList | Covariance is an important measure of marginal dependence among variables. However, heterogeneity in subject covariances and regression models for high-dimensional covariance matrices is not well studied. Compared to regression analysis for conditional means, modeling high-dimensional covariances is much more challenging due to the large set of free parameters and the intrinsic positive-definite property that puts constraints on the regression parameters. In this paper, we propose a regularized estimation method for the regression coefficients of covariances under sufficient and necessary constraints for the positive definiteness of the conditional average covariance matrices given covariates. The proposed estimator satisfies the sparsity and positive-definite properties simultaneously. An alternating direction method of multipliers (ADMM) algorithm is proposed to solve the constrained and regularized optimization problem. We show the convergence of the proposed ADMM algorithm and derive the convergence rates of the proposed estimators for the regression coefficients and the heterogeneous covariances. The proposed method is evaluated by simulation studies, and its practical application is demonstrated by a case study on brain connectivity.Covariance is an important measure of marginal dependence among variables. However, heterogeneity in subject covariances and regression models for high-dimensional covariance matrices is not well studied. Compared to regression analysis for conditional means, modeling high-dimensional covariances is much more challenging due to the large set of free parameters and the intrinsic positive-definite property that puts constraints on the regression parameters. In this paper, we propose a regularized estimation method for the regression coefficients of covariances under sufficient and necessary constraints for the positive definiteness of the conditional average covariance matrices given covariates. The proposed estimator satisfies the sparsity and positive-definite properties simultaneously. An alternating direction method of multipliers (ADMM) algorithm is proposed to solve the constrained and regularized optimization problem. We show the convergence of the proposed ADMM algorithm and derive the convergence rates of the proposed estimators for the regression coefficients and the heterogeneous covariances. The proposed method is evaluated by simulation studies, and its practical application is demonstrated by a case study on brain connectivity. Covariance is an important measure of marginal dependence among variables. However, heterogeneity in subject covariances and regression models for high-dimensional covariance matrices is not well studied. Compared to regression analysis for conditional means, modeling high-dimensional covariances is much more challenging due to the large set of free parameters and the intrinsic positive-definite property that puts constraints on the regression parameters. In this paper, we propose a regularized estimation method for the regression coefficients of covariances under sufficient and necessary constraints for the positive definiteness of the conditional average covariance matrices given covariates. The proposed estimator satisfies the sparsity and positive-definite properties simultaneously. An alternating direction method of multipliers (ADMM) algorithm is proposed to solve the constrained and regularized optimization problem. We show the convergence of the proposed ADMM algorithm and derive the convergence rates of the proposed estimators for the regression coefficients and the heterogeneous covariances. The proposed method is evaluated by simulation studies, and its practical application is demonstrated by a case study on brain connectivity. |
| Author | He, Jie Qiu, Yumou Zhou, Xiao-Hua |
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| Cites_doi | 10.5705/ss.2010.051 10.1016/j.brainres.2010.10.102 10.1080/01621459.2022.2034632 10.1080/01621459.2021.1917417 10.1080/01621459.2020.1855183 10.2307/2109358 10.1093/biostatistics/kxz057 10.1198/jasa.2009.0101 10.1080/01621459.2014.950375 10.1214/154957805100000104 10.1214/08-AOS600 10.1080/01621459.2012.725386 10.1016/j.neuron.2011.09.006 10.1001/jamapsychiatry.2015.0101 10.1080/01621459.1996.10476677 10.1080/01621459.2021.1970570 10.1080/10618600.2013.858633 10.1214/009053607000000758 10.1080/01621459.2015.1131699 10.1561/2200000016 10.1137/1.9781611970838 10.1214/23-AOAS1785 10.1186/s11689-022-09460-y |
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| Keywords | fMRI data high dimensionality ADMM algorithm regularization conditional average covariance matrix positive-definiteness constraint |
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| Title | Positive-definite regularized estimation for high-dimensional covariance on scalar regression |
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