Estimating covariance and precision matrices along subspaces

In many engineering applications, estimation of covariance and precision matrices is of great importance, helping researchers understand the dependency and conditional dependency between variables of interest. Among various matrix estimation methods, the modified Cholesky decomposition is a commonly used technique. It has the advantage of transforming the matrix estimation task into solving a sequence of regression models. Moreover, the sparsity on the regression coefficients implies certain sparse structure on the covariance and precision matrices. In this chapter, we first overview the Cholesky-based covariance and precision matrices estimation. It is known that the Cholesky-based matrix estimation depends on a prespecified ordering of variables, which is often not available in practice. To address this issue, we then introduce several techniques to enhance the Cholesky-based estimation of covariance and precision matrices. These approaches are able to ensure the positive definiteness of the matrix estimate and applicable in general situations without specifying the ordering of variables. The advantage of Cholesky-based estimation is illustrated by numerical studies and several real-case applications.

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.

Accurate estimation of large covariance and precision matrices is an essential prerequisite of portfolio selection and financial risk management. Among factor-based covariance estimators, high-frequency data provides additional information but also introduces noise, which could degrade the estimation of betas. In this paper, we propose a new Mixed-frequency and FActor-based (MFACE) combining high-frequency (intraday) data and low-frequency (daily) betas. We apply approximate factor structure to construct the high-dimensional minimum variance portfolio. We establish the consistency and obtain the convergence rate of the covariance estimator and the corresponding precision matrix estimator. A comprehensive simulation study investigates the estimation accuracy under different combinations of intrady sampling frequency, intraday sample size and daily sample size. Out-of-sample forecasts demonstrate that our estimator explains the daily volatility of the equally-weighted portfolio pretty well. The minimum variance portfolio that embodies low-frequency betas also achieves the minimum risk at relatively low cost among several estimators.

Estimation of sparse covariance matrices and their inverse subject to positive definiteness constraints has drawn a lot of attention in recent years. The abundance of high-dimensional data, where the sample size (n) is less than the dimension (d), requires shrinkage estimation methods since the maximum likelihood estimator is not positive definite in this case. Furthermore, when n is larger than d but not sufficiently larger, shrinkage estimation is more stable than maximum likelihood as it reduces the condition number of the precision matrix. Frequentist methods have utilized penalized likelihood methods, whereas Bayesian approaches rely on matrix decompositions or Wishart priors for shrinkage. In this paper we propose a new method, called the Bayesian Covariance Lasso (BCLASSO), for the shrinkage estimation of a precision (covariance) matrix. We consider a class of priors for the precision matrix that leads to the popular frequentist penalties as special cases, develop a Bayes estimator for the precision matrix, and propose an efficient sampling scheme that does not precalculate boundaries for positive definiteness. The proposed method is permutation invariant and performs shrinkage and estimation simultaneously for non-full rank data. Simulations show that the proposed BCLASSO performs similarly as frequentist methods for non-full rank data.

Structure preserving non-negative matrix factorization for dimensionality reduction

The problem of dimensionality reduction has recently receiveda lot of attention in areas such as machine learning, pattern recognition and computer vision. Dimensionality reduction is an important pre-processing step which aims to obtain a low-rank approximation problem with limited loss of vital information. It involves a mapping of high dimensional data samples into a low dimensional space such that the most important features in the original data are preserved. Locality preserving partial least squares (LPPLS) is a feature extraction method developed to preserve the local structure of data. The original LPPLS algorithm does not make use of class information of data when available. We propose in this paper a modification of LPPLS such that the class information of the data points istaken into consideration in the construction of the neighborhood graph. By doing so, our proposed method is allowed to have more discriminating power than LPPLS. Experimental results on a real-world dataset confirm the effectiveness of our proposed method.

The determination of the covariance matrix and its inverse, the precision matrix, is critical in the statistical analysis of cosmological measurements. The covariance matrix is typically estimated with a limited number of simulations at great computational cost before inversion into the precision matrix; therefore, it can be ill-conditioned and overly noisy when the sample size n used for estimation is not much larger than the data vector dimension. In this work, we consider a class of methods known as shrinkage estimation for the precision matrix, which combines an empirical estimate with a target that is either analytical or stochastic. These methods include linear and non-linear shrinkage applied to the covariance matrix (the latter represented by the so-called NERCOME estimator), and the direct linear shrinkage estimation of the precision matrix which we introduce in a cosmological setting. By performing Bayesian parameter inference and using metrics like matrix loss functions, the Kullback–Leibler divergence and the eigenvalue spectrum, we compare their performance against the standard sample estimator with varying sample size n. We have found the shrinkage estimators to significantly improve the posterior distribution at low n, especially for the linear shrinkage estimators either inverted from the covariance matrix or applied directly to the precision matrix, with an empirical target constructed from the sample estimate. Our results are particularly relevant to the analyses of Stage-IV spectroscopic galaxy surveys such as the Dark Energy Spectroscopic Instrument and Euclid, whose statistical power can be limited by the computational cost of obtaining an accurate precision matrix estimate.

Large-scale precision matrix estimation is of fundamental importance yet challenging in many contemporary applications for recovering Gaussian graphical models. In this paper, we suggest a new approach of innovated scalable efficient estimation (ISEE) for estimating large precision matrix. Motivated by the innovated transformation, we convert the original problem into that of large covariance matrix estimation. The suggested method combines the strengths of recent advances in high-dimensional sparse modeling and large covariance matrix estimation. Compared to existing approaches, our method is scalable and can deal with much larger precision matrices with simple tuning. Under mild regularity conditions, we establish that this procedure can recover the underlying graphical structure with significant probability and provide efficient estimation of link strengths. Both computational and theoretical advantages of the procedure are evidenced through simulation and real data examples.

We propose Bayesian methods for Gaussian graphical models that lead to sparse and adaptively shrunk estimators of the precision (inverse covariance) matrix. Our methods are based on lasso-type regularization priors leading to parsimonious parameterization of the precision matrix, which is essential in several applications involving learning relationships among the variables. In this context, we introduce a novel type of selection prior that develops a sparse structure on the precision matrix by making most of the elements exactly zero, in addition to ensuring positive definiteness - thus conducting model selection and estimation simultaneously. More importantly, we extend these methods to analyze clustered data using finite mixtures of Gaussian graphical model and infinite mixtures of Gaussian graphical models. We discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalizing constants that are functions of parameters of interest, which result from the restriction of positive definiteness on the correlation matrix. We evaluate the operating characteristics of our method via several simulations and demonstrate the application to real data examples in genomics.

Recent advances in shrinkage-based high-dimensional inference

Journal Article Nonparametric estimation of large covariance matrices of longitudinal data Get access Wei Biao Wu, Wei Biao Wu Search for other works by this author on: Oxford Academic Google Scholar Mohsen Pourahmadi Mohsen Pourahmadi Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 90, Issue 4, December 2003, Pages 831–844, https://doi.org/10.1093/biomet/90.4.831 Published: 01 December 2003

ABSTRACTWe propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrix estimators to have equal entries and also encourage zeros in the precision matrix estimator. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze three real datasets to demonstrate our method’s applicability. Supplementary materials, including an R package implementing our method, are available online.

Estimation of the covariance matrix and its inverse, the precision matrix, in high-dimensional situations is of great interest in many applications. In this paper, we focus on the estimation of a class of sparse precision matrices which are assumed to be approximately inversely closed for the case that the dimensionality p can be much larger than the sample size n, which is fundamentally different from the classical case that p <; n. Different in nature from state-of-the-art methods that are based on penalized likelihood maximization or constrained error minimization, based on the truncated Neumann series representation, we propose a computationally efficient precision matrix estimator that has a computational complexity of O (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ). We prove that the proposed estimator is consistent in probability and in L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> under the spectral norm. Moreover, its convergence is shown to be rate-optimal in the sense of minimax risk. We further prove that the proposed estimator is model selection consistent by establishing a convergence result under the entry-wise ∞-norm. Simulations demonstrate the encouraging finite sample size performance and computational advantage of the proposed estimator. The proposed estimator is also applied to a real breast cancer data and shown to outperform existing precision matrix estimators.

A variety of computationally efficient Bayesian models for the covariance matrix of a multivariate Gaussian distribution are available. However, all produce a relatively dense estimate of the precision matrix, and are therefore unsatisfactory when one wishes to use the precision matrix to consider the conditional independence structure of the data. This paper considers the posterior predictive distribution of model fit for these covariance models. We then undertake post-processing of the Bayes point estimate for the precision matrix to produce a sparse model whose expected fit lies within the upper 95% of the posterior predictive distribution of fit. The impact of the method for selecting the zero elements of the precision matrix is evaluated. Good results were obtained using models that encouraged a sparse posterior (G-Wishart, Bayesian adaptive graphical lasso) and selection using credible intervals. We also find that this approach is easily extended to the problem of finding a sparse set of elements that differ across a set of precision matrices, a natural summary when a common set of variables is observed under multiple conditions. We illustrate our findings with moderate dimensional data examples from finance and metabolomics.

The estimation of a covariance matrix from an insufficient amount of data is one of the most common problems in fields as diverse as multivariate statistics, wireless communications, signal processing, biology, learning theory, and finance. In a joint work of Marzetta, Tucci and Simon, a new approach to handle singular covariance matrices was suggested. The main idea was to use dimensionality reduction in conjunction with an average over the Stiefel manifold. In this paper, we continue with this research and we consider some new approaches to handle this problem. One of the methods is called the mean conjugate estimator under Ewens measure and uses a randomization of the sample covariance matrix over all the permutation matrices with respect to the Ewens measure. The techniques used to attack this problem are broad and run from random matrix theory to combinatorics.