11,481 research outputs found
A sparse conditional Gaussian graphical model for analysis of genetical genomics data
Genetical genomics experiments have now been routinely conducted to measure
both the genetic markers and gene expression data on the same subjects. The
gene expression levels are often treated as quantitative traits and are subject
to standard genetic analysis in order to identify the gene expression
quantitative loci (eQTL). However, the genetic architecture for many gene
expressions may be complex, and poorly estimated genetic architecture may
compromise the inferences of the dependency structures of the genes at the
transcriptional level. In this paper we introduce a sparse conditional Gaussian
graphical model for studying the conditional independent relationships among a
set of gene expressions adjusting for possible genetic effects where the gene
expressions are modeled with seemingly unrelated regressions. We present an
efficient coordinate descent algorithm to obtain the penalized estimation of
both the regression coefficients and the sparse concentration matrix. The
corresponding graph can be used to determine the conditional independence among
a group of genes while adjusting for shared genetic effects. Simulation
experiments and asymptotic convergence rates and sparsistency are used to
justify our proposed methods. By sparsistency, we mean the property that all
parameters that are zero are actually estimated as zero with probability
tending to one. We apply our methods to the analysis of a yeast eQTL data set
and demonstrate that the conditional Gaussian graphical model leads to a more
interpretable gene network than a standard Gaussian graphical model based on
gene expression data alone.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS494 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Simultaneous Variable and Covariance Selection with the Multivariate Spike-and-Slab Lasso
We propose a Bayesian procedure for simultaneous variable and covariance
selection using continuous spike-and-slab priors in multivariate linear
regression models where q possibly correlated responses are regressed onto p
predictors. Rather than relying on a stochastic search through the
high-dimensional model space, we develop an ECM algorithm similar to the EMVS
procedure of Rockova & George (2014) targeting modal estimates of the matrix of
regression coefficients and residual precision matrix. Varying the scale of the
continuous spike densities facilitates dynamic posterior exploration and allows
us to filter out negligible regression coefficients and partial covariances
gradually. Our method is seen to substantially outperform regularization
competitors on simulated data. We demonstrate our method with a re-examination
of data from a recent observational study of the effect of playing high school
football on several later-life cognition, psychological, and socio-economic
outcomes
Covariance Estimation: The GLM and Regularization Perspectives
Finding an unconstrained and statistically interpretable reparameterization
of a covariance matrix is still an open problem in statistics. Its solution is
of central importance in covariance estimation, particularly in the recent
high-dimensional data environment where enforcing the positive-definiteness
constraint could be computationally expensive. We provide a survey of the
progress made in modeling covariance matrices from two relatively complementary
perspectives: (1) generalized linear models (GLM) or parsimony and use of
covariates in low dimensions, and (2) regularization or sparsity for
high-dimensional data. An emerging, unifying and powerful trend in both
perspectives is that of reducing a covariance estimation problem to that of
estimating a sequence of regression problems. We point out several instances of
the regression-based formulation. A notable case is in sparse estimation of a
precision matrix or a Gaussian graphical model leading to the fast graphical
LASSO algorithm. Some advantages and limitations of the regression-based
Cholesky decomposition relative to the classical spectral (eigenvalue) and
variance-correlation decompositions are highlighted. The former provides an
unconstrained and statistically interpretable reparameterization, and
guarantees the positive-definiteness of the estimated covariance matrix. It
reduces the unintuitive task of covariance estimation to that of modeling a
sequence of regressions at the cost of imposing an a priori order among the
variables. Elementwise regularization of the sample covariance matrix such as
banding, tapering and thresholding has desirable asymptotic properties and the
sparse estimated covariance matrix is positive definite with probability
tending to one for large samples and dimensions.Comment: Published in at http://dx.doi.org/10.1214/11-STS358 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
- …