Joint Mean and Covariance Estimation with Unreplicated Matrix-Variate Data

Abstract

It has been proposed that complex populations, such as those that arise in genomics studies, may exhibit dependencies among observations as well as among variables. This gives rise to the challenging problem of analyzing unreplicated high-dimensional data with unknown mean and dependence structures. Matrixvariate approaches that impose various forms of (inverse) covariance sparsity allow flexible dependence structures to be estimated, but cannot directly be applied when the mean and covariance matrices are estimated jointly. We present a practical method utilizing generalized least squares and penalized (inverse) covariance estimation to address this challenge. We establish consistency and obtain rates of convergence for estimating the mean parameters and covariance matrices. The advantages of our approaches are: (i) dependence graphs and covariance structures can be estimated in the presence of unknown mean structure, (ii) the mean structure becomes more efficiently estimated when accounting for the dependence structure among observations; and (iii) inferences about the mean parameters become correctly calibrated. We use simulation studies and analysis of genomic data from a twin study of ulcerative colitis to illustrate the statistical convergence and the performance of our methods in practical settings. Several lines of evidence show that the test statistics for differential gene expression produced by our methods are correctly calibrated and improve power over conventional methods.By the above decompositions, it follows that SpB, J 0 , J 1 q can be expressed as SpB, J 0 , J 1 q " S II`SIII`S T

Tensor Graphical Lasso (TeraLasso)

“…Similar properties hold for the relative Frobenius norm error in view of Lemma 3.4. Note that by (17) and Lemma 3.4, under the settings of Theorem 3.2,…”

“…See [44], where such regression model is introduced for subgaussian matrix variate data. See also [16,17] and references therein for more recent applications of matrix variate models.…”

High dimensional precision matrix estimation under weak sparsity

“…In particular, the special case q = 0 corresponds to Theorem 3.3 of Zhou ( 2014) and these two results are consistent due to the dual properties between Lasso and the Dantzig selector. Moreover, our result can be generalized to the sub-Gaussian condition of the matrix data (Hornstein et al, 2019) and we omit the details.…”

Non-separable covariance models for spatio-temporal data, with applications to neural encoding analysis

“…For matrix-variate data with two-way dependencies, prior work depended on a large number of replicated data to obtain certain convergence guarantees, even when the data is observed in full and free of measurement error; see for example Dutilleul (1999), Werner et al (2008), Leng and Tang (2012), and Tsiligkaridis et al (2013). A recent line of work on matrix variate models (Kalaitzis et al, 2013;Zhou, 2014;Rudelson and Zhou, 2017) have focused on the design of estimators and efficient algorithms while establishing theoretical properties by using the Kronecker sum and product covariance models when a single or a small number of replicates are available from such matrix-variate distributions; See also Efron (2009), Allen andTibshirani (2010), andHornstein et al (2016) for related models and applications. Among these models, the Kronecker sum provides a covariance or precision matrix which is sparser than the Kronecker product (inverse) covariance model.…”