Modified Cholesky Decomposition Research Articles

Alternative modified Cholesky decomposition of the precision matrix of longitudinal data

The correlation matrix might be of scientific interest for longitudinal data. However, few studies have focused on both robust estimation of the correlation matrix against model misspecification and robustness to outliers in the data, when the precision matrix possesses a typical structure. In this paper, we propose an alternative modified Cholesky decomposition (AMCD) for the precision matrix of longitudinal data, which results in robust estimation of the correlation matrix against model misspecification of the innovation variances. A joint mean-covariance model with multivariate normal distribution and AMCD is established, the quasi-Fisher scoring algorithm is developed, and the maximum likelihood estimators are proven to be consistent and asymptotically normally distributed. Furthermore, a double-robust joint modeling approach with multivariate Laplace distribution and AMCD is established, and the quasi-Newton algorithm for maximum likelihood estimation is developed. The simulation studies and real data analysis demonstrate the effectiveness of the proposed AMCD method.

Regularized estimation of Kronecker structured covariance matrix using modified Cholesky decomposition

In this paper, we study a Kronecker structured model for covariance matrices when data are matrix-valued. Using the modified Cholesky decomposition for Kronecker structured covariance matrix, we propose a regularized covariance estimator by imposing shrinkage and smoothing penalties on the Cholesky factors. A regularized flip-flop (RFF) algorithm is developed to produce a statistically efficient estimator for a large covariance matrix of matrix-valued data. Asymptotic properties are investigated and the performance of the estimator is evaluated by simulations. The results presented are applied to real data example.

Bayesian adaptive Lasso estimation of large graphical model based on modified Cholesky decomposition

In this paper, based on the modified Cholesky decomposition of the precision matrix, we propose Bayesian adaptive Lasso estimation and maximum adaptive posterior estimation for graphical model. We also recover the graph by minimizing the decoupled shrinkage and selection loss function.

Fast DOA Estimation Algorithms via Positive Incremental Modified Cholesky Decomposition for Augmented Coprime Array Sensors.

This paper proposes a fast direction of arrival (DOA) estimation method based on positive incremental modified Cholesky decomposition atomic norm minimization (PI-CANM) for augmented coprime array sensors. The approach incorporates coprime sampling on the augmented array to generate a non-uniform, discontinuous virtual array. It then utilizes interpolation to convert this into a uniform, continuous virtual array. Based on this, the problem of DOA estimation is equivalently formulated as a gridless optimization problem, which is solved via atomic norm minimization to reconstruct a Hermitian Toeplitz covariance matrix. Furthermore, by positive incremental modified Cholesky decomposition, the covariance matrix is transformed from positive semi-definite to positive definite, which simplifies the constraint of optimization problem and reduces the complexity of the solution. Finally, the Multiple Signal Classification method is utilized to carry out statistical signal processing on the reconstructed covariance matrix, yielding initial DOA angle estimates. Experimental outcomes highlight that the PI-CANM algorithm surpasses other algorithms in estimation accuracy, demonstrating stability in difficult circumstances such as low signal-to-noise ratios and limited snapshots. Additionally, it boasts an impressive computational speed. This method enhances both the accuracy and computational efficiency of DOA estimation, showing potential for broad applicability.

Statistical inference of multi-state transition model for longitudinal data with measurement error and heterogeneity

. Multi-state transition model is typically used to analyze longitudinal data in medicine and sociology. Moreover, variables in longitudinal studies usually are error-prone, and random effects are heterogeneous, which will result in biased estimates of the interest parameters. This article is intended to estimate the parameters of the multi-state transition model for longitudinal data with measurement error and heterogeneous random effects and further consider the covariate related to the covariance matrix of random effects is also error-prone when the covariate in the transition model is error-prone. We model the covariance matrix of random effects through the modified Cholesky decomposition and propose a pseudo-likelihood method based on the Monte Carlo expectation-maximization algorithm and the Bayesian method based on Markov Chain Monte Carlo to infer and calculate the whole estimates. Meanwhile, we obtain the asymptotic properties and evaluate the finite sample performance of the proposed method by simulation, which is well in terms of Bias, RMSE, and coverage rate of confidence intervals. In addition, we apply the proposed method to the MFUS data.

Estimation of banded time-varying precision matrix based on SCAD and group lasso

A new banded time-varying precision matrix estimator is proposed for high-dimensional time series. The estimator utilizes the modified Cholesky decomposition, and the two factors in the decomposition are dynamically estimated by applying the GARCH model to the innovation variance and the Kalman filter on the Cholesky factor. The SCAD penalty and group lasso penalty are imposed on the Cholesky factor to estimate the banded structure. An efficient algorithm based on the alternating direction method of multipliers (ADMM), local linear approximation (LLA), and blockwise coordinate descent (BCD) algorithms is developed. The convergence of the algorithm is proven theoretically, and the estimator is guaranteed to be banded. Simulation and real-data analysis demonstrate the favorable performance of the proposed algorithm compared to other methods.

A new approach for ultrahigh-dimensional covariance matrix estimation

We propose a method for estimating ultrahigh dimensional covariance matrix without the Gaussian distribution and the order of variables assumptions. More specifically, by combining the modified Cholesky decomposition (MCD) and refitted cross validation (RCV), a permutation invariant method called CovPRCV is developed to estimate covariance matrix under the “Permutation-Average” framework. The employment of RCV procedure attenuates significantly spurious correlation in the ultrahigh dimensional data. We derive the consistency of proposed estimator under the Frobenius norm without requiring banded structure and normal distribution. The finite-sample performance is assessed via simulation studies, which indicate that the proposed method is promising compared with its competitors in many interesting scenarios. We also apply the proposed method to analyze a prostate dataset.

Ensemble LDA via the modified Cholesky decomposition

A binary classification problem in the high-dimensional settings is studied via the ensemble learning with each base classifier constructed from the linear discriminant analysis (LDA), and these base classifiers are integrated by the weighted voting. The precision matrix in the LDA rule is estimated by the modified Cholesky decomposition (MCD), which is able to provide us with a set of precision estimates by considering multiple variable orderings, and hence yield a group of different LDA classifiers. Such available LDA classifiers are then integrated to improve the classification performance. The simulation and the application studies are conducted to demonstrate the merits of the proposed method.

A novel sparse linear mixed model for multi-source mixed-frequency data fusion in telemedicine

Digital health and telemonitoring have resulted in a wealth of information to be collected to monitor, manage, and improve human health. The multi-source mixed-frequency health data overwhelm the modeling capacity of existing statistical and machine learning models, due to many challenging properties. Although predictive analytics for big health data plays an important role in telemonitoring, there is a lack of rigorous prediction model that can automatically predicts patients’ health conditions, e.g., Disease Severity Indicators (DSIs), from multi-source mixed-frequency data. Sleep disorder is a prevalent cardiac syndrome that is characterized by abnormal respiratory patterns during sleep. Although wearable devices are available to administrate sleep studies at home, the manual scoring process to generate the DSI remains a bottleneck in automated monitoring and diagnosis of sleep disorder. To address the multi-fold challenges for precise prediction of the DSI from high-dimensional multi-source mixed-frequency data in sleep disorder, we propose a sparse linear mixed model that combines the modified Cholesky decomposition with group lasso penalties to enable joint group selection of fixed effects and random effects. A novel Expectation Maximization (EM) algorithm integrated with an efficient Majorization Maximization (MM) algorithm is developed for model estimation of the proposed sparse linear mixed model with group variable selection. The proposed method was applied to the SHHS data for telemonitoring and diagnosis of sleep disorder and found that a few significant feature groups that are consistent with prior medical studies on sleep disorder. The proposed method also outperformed a few benchmark methods with the highest prediction accuracy.

Scalable Bayesian High-dimensional Local Dependence Learning

In this work, we propose a scalable Bayesian procedure for learning the local dependence structure in a high-dimensional model where the variables possess a natural ordering. The ordering of variables can be indexed by time, the vicinities of spatial locations, and so on, with the natural assumption that variables far apart tend to have weak correlations. Applications of such models abound in a variety of fields such as finance, genome associations analysis and spatial modeling. We adopt a flexible framework under which each variable is dependent on its neighbors or predecessors, and the neighborhood size can vary for each variable. It is of great interest to reveal this local dependence structure by estimating the covariance or precision matrix while yielding a consistent estimate of the varying neighborhood size for each variable. The existing literature on banded covariance matrix estimation, which assumes a fixed bandwidth cannot be adapted for this general setup. We employ the modified Cholesky decomposition for the precision matrix and design a flexible prior for this model through appropriate priors on the neighborhood sizes and Cholesky factors. The posterior contraction rates of the Cholesky factor are derived which are nearly or exactly minimax optimal, and our procedure leads to consistent estimates of the neighborhood size for all the variables. Another appealing feature of our procedure is its scalability to models with large numbers of variables due to efficient posterior inference without resorting to MCMC algorithms. Numerical comparisons are carried out with competitive methods, and applications are considered for some real datasets.

A novel robust estimation for high-dimensional precision matrices.

In this paper we propose a new robust estimation of precision matrices for high-dimensional data when the number of variables is larger than the sample size. Different from the existing methods in literature, the proposed model combines the technique of modified Cholesky decomposition (MCD) with the robust generalized M-estimators. The MCD reparameterizes a precision matrix and transforms its estimation into solving a series of linear regressions, in which the commonly used robust techniques can be conveniently incorporated. Additionally, the proposed method adopts the model averaging idea to address the ordering issue in the MCD approach, resulting in an accurate estimation for precision matrices. Simulations and real data analysis are conducted to illustrate the merits of the proposed estimator.

Sparse covariance matrix estimation for ultrahigh dimensional data

We introduce a Covariance matrix Refitted Cross Validation (CovRCV) estimation procedure without requiring the Gaussian assumption. Specifically, we first use the modified Cholesky decomposition(MCD) to transform the problem of covariance matrix estimation to that of coefficient estimation in the regression setting. Then, we use an estimation method based on RCV to mitigate the prevalent spurious correlation in the ultrahigh dimensional regression. The proposed procedure is easy to implement. Furthermore, we establish the consistency of the proposed CovRCV estimation and show the rate of convergence without assuming the banded structure in Bickel and Levina (2008). The finite sample performance of our method is assessed via an intensive numerical study. Our numerical comparison shows that the CovRCV estimation outperforms the existing ones in various interesting scenarios. The advantages of our method are further illustrated by a real case study.

Modifying Covariance Localization to Mitigate Sampling Errors from the Ensemble Data Assimilation

The ensemble-based Kalman filter requires at least a considerable ensemble (e.g., 10,000 members) to identify relevant error covariance at great distances for multidimensional geophysical systems. However, increasing numerous ensemble sizes will enlarge sampling errors. This study proposes a modified Cholesky decomposition based on the covariance localization (CL) scheme, namely a covariance localization scheme with modified Cholesky decomposition (CL-MC). Our main idea utilizes a modified Cholesky (MC) decomposition technique for estimating the background error covariance matrix; meanwhile, we employ the tunable singular value decomposition method on the background error covariance to improve the ensemble increment and avoid the imbalance of the system. To verify if the proposed method can effectively mitigate the sampling errors, numerical experiments are conducted on the Lorenz-96 model and large-scale model (SPEEDY model). The results show that the CL-MC method outperforms the CL method for different data assimilation parameters (ensemble sizes and localizations). Furthermore, by performing one year of assimilation experiments on the SPEEDY model, it is found that the 1-day forecast RMSEs obtained by the CL are approximately equal to the 5-day forecast RMSEs of CL-MC. So, the CL-MC method has potential advantages for long-term forecasting. Maybe the proposed CL-MC method achieves good prospects for widespread application in atmospheric general circulation models.

Regularized estimation of the Mahalanobis distance based on modified Cholesky decomposition

Estimating inverse covariance matrix is an essential part of many statistical methods. This paper proposes a regularized estimator for the inverse covariance matrix. Modified Cholesky decomposition (MCD) is utilized to construct positive definite estimators. Instead of directly regularizing the inverse covariance matrix itself, we impose regularization on the Cholesky factor. The estimated inverse covariance matrix is used to build Mahalanobis distance (MD). The proposed method is evaluated by detecting outliers through simulations and empirical studies.

Robust estimation in multivariate heteroscedastic regression models with autoregressive covariance structures using EM algorithm

In the analysis of repeated or clustered measurements, it is crucial to determine the dynamics that affect the mean, variance, and correlations of the data, which will be possible using appropriate models. One of these models is the joint mean–covariance model, which is a multivariate heteroscedastic regression model with autoregressive covariance structures. In these models, parameter estimation is usually carried on under normality assumption, but the resulting estimators will be very sensitive to the outliers or non-normality of data. In this study, we propose a robust alternative method and an EM algorithm for estimating the parameters of joint mean–covariance models. Robustification is achieved using a multivariate heavy-tailed distribution with the same number of parameters as the multivariate normal distribution. To simplify the estimation procedure, a modified Cholesky decomposition is adopted to factorize the dependence structure in terms of unconstrained autoregressive and scale innovation parameters. Also, the technique for the prediction of future responses is given. An intensive simulation study and a real data example are provided to demonstrate the performance of the proposed method.

Conditional generalized estimating equations of mean-variance-correlation for clustered data

Joint modelling of mean-covariance structure is an important topic in clustered data analysis. Existing methods, such as those based on modified Cholesky decomposition (MCD), alternative Cholesky decomposition (ACD) and hyperspherical coordinates decomposition (HPC), have two main restrictions. First, they often assume that responses in the same cluster are naturally ordered, for example, by time in longitudinal studies. Second, the existing methods model transformed parameters, for instance, the generalized autoregressive parameters and innovation variances in MCD/ACD, and the hyperspherical coordinates in HPC, making the dependence of correlation coefficients or variances on covariates hardly understandable. As an alternative, a data-driven method that models directly the mean, variances and correlation coefficients for clustered data is proposed. Comparing to the existing methods, the proposed approach not only has no need of natural order in responses but also works on original correlation coefficients and variances. The proposed models are flexible and interpretable, and the parameter estimators in joint generalized estimating equations (GEE) are shown to be consistent and asymptotically normally distributed. Consistent model selection criteria in spirit of quasi-likelihood under independence model criterion (QIC) are considered. The use of the proposed approach is demonstrated by intensive simulation studies and real data analysis.

A robust linear mixed-effects model for longitudinal data using an innovative multivariate skew-Huber distribution

Conventional linear mixed-effects modeling is routinely challenging when the validity of necessary assumptions is suspicious. In particular, robustifying model fitting is appealing in the presence of potential outlying points. This paper introduces a robust regression methodology in a parametric setting by constructing a novel multivariate skew-Huber distribution for longitudinal data with heavy-tails and skewed structures. Unlike preceding studies, our model allows for jointly estimating the tuning parameter, which controls the impact of outliers, with all other parameters using an undemanding computational algorithm. Moreover, by promoting an unconstrained parameterization through the modified Cholesky decomposition, the estimate of variance–covariance components can be merely accessible. We also present a spline mixed model to account for the covariate effect. To highlight the usefulness of our methodology, we conducted a simulation study and analyzed a data set collected on type 2 diabetic patients with microalbuminuria over a 6-year prospective cohort study. Findings show that our proposed robust model leads to convincing conclusions in empirical studies.

Robust estimation via modified Cholesky decomposition for modal partially nonlinear models with longitudinal data

This article proposes robust estimation via modified Cholesky decomposition for modal partially nonlinear models with longitudinal data. A new minorization-maximization algorithm is developed to calculate the modal regression estimates, and the balance between the robustness and efficiency can be achieved by choosing appropriate data adaptive bandwidth. Simulation studies and real data analysis are conducted to illustrate the finite sample performance of the proposed approach.

Dynamic covariance modeling with artificial neural networks

This article proposes a novel multivariate generalized autoregressive conditionally heteroscedastic (GARCH) model that incorporates the modified Cholesky decomposition for a covariance matrix in order to reduce the number of covariance parameters and increase the interpretation power of the model. The modified Cholesky decomposition for covariance matrix reduces the number of covariance parameters to , where p is the dimension of the stocks in the data, and enables us to obtain a regression equation. To account for the nonlinearity in the GARCH model, the parameters in our model are modeled using long short-term memory. The proposed model is compared with DCC model with respect to portfolio optimization and the distances between the actual covariance matrices and predicted covariance matrices. It is found that although the distances may or may not be reduced by our proposed model in different cases presented in this article, our proposed model outperforms the DCC model in terms of mean portfolio returns.

Constrained Bayesian doubly elastic net Lasso for linear quantile mixed models

In this article, we propose a novel constrained Bayesian elastic net approach for linear quantile mixed model shrinkage. A partially collapsed Gibbs sampling algorithm is developed for efficient posterior computation based on a modified Cholesky decomposition for the covariance matrix of random effects and an asymmetric Laplace distribution for the error distribution. We demonstrate the proposed method based on simulated data and an experimental dataset from a longitudinal study of age-related macular degeneration trial. Both simulation studies and real data analysis indicate that the proposed constrained Bayesian elastic net approach is competitive with the existing methods under a variety of scenarios, such as presence of a large number of covariates and collinearity.