47,724 research outputs found
High-dimensional graphs and variable selection with the Lasso
The pattern of zero entries in the inverse covariance matrix of a
multivariate normal distribution corresponds to conditional independence
restrictions between variables. Covariance selection aims at estimating those
structural zeros from data. We show that neighborhood selection with the Lasso
is a computationally attractive alternative to standard covariance selection
for sparse high-dimensional graphs. Neighborhood selection estimates the
conditional independence restrictions separately for each node in the graph and
is hence equivalent to variable selection for Gaussian linear models. We show
that the proposed neighborhood selection scheme is consistent for sparse
high-dimensional graphs. Consistency hinges on the choice of the penalty
parameter. The oracle value for optimal prediction does not lead to a
consistent neighborhood estimate. Controlling instead the probability of
falsely joining some distinct connectivity components of the graph, consistent
estimation for sparse graphs is achieved (with exponential rates), even when
the number of variables grows as the number of observations raised to an
arbitrary power.Comment: Published at http://dx.doi.org/10.1214/009053606000000281 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Foundational principles for large scale inference: Illustrations through correlation mining
When can reliable inference be drawn in the "Big Data" context? This paper
presents a framework for answering this fundamental question in the context of
correlation mining, with implications for general large scale inference. In
large scale data applications like genomics, connectomics, and eco-informatics
the dataset is often variable-rich but sample-starved: a regime where the
number of acquired samples (statistical replicates) is far fewer than the
number of observed variables (genes, neurons, voxels, or chemical
constituents). Much of recent work has focused on understanding the
computational complexity of proposed methods for "Big Data." Sample complexity
however has received relatively less attention, especially in the setting when
the sample size is fixed, and the dimension grows without bound. To
address this gap, we develop a unified statistical framework that explicitly
quantifies the sample complexity of various inferential tasks. Sampling regimes
can be divided into several categories: 1) the classical asymptotic regime
where the variable dimension is fixed and the sample size goes to infinity; 2)
the mixed asymptotic regime where both variable dimension and sample size go to
infinity at comparable rates; 3) the purely high dimensional asymptotic regime
where the variable dimension goes to infinity and the sample size is fixed.
Each regime has its niche but only the latter regime applies to exa-scale data
dimension. We illustrate this high dimensional framework for the problem of
correlation mining, where it is the matrix of pairwise and partial correlations
among the variables that are of interest. We demonstrate various regimes of
correlation mining based on the unifying perspective of high dimensional
learning rates and sample complexity for different structured covariance models
and different inference tasks
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
iPACOSE: an iterative algorithm for the estimation of gene regulation networks
In the context of Gaussian Graphical Models (GGMs) with high-
dimensional small sample data, we present a simple procedure to esti-
mate partial correlations under the constraint that some of them are
strictly zero. This method can also be extended to covariance selection.
If the goal is to estimate a GGM, our new procedure can be applied
to re-estimate the partial correlations after a first graph has been esti-
mated in the hope to improve the estimation of non-zero coefficients. In
a simulation study, we compare our new covariance selection procedure
to existing methods and show that the re-estimated partial correlation
coefficients may be closer to the real values in important cases
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