1,568 research outputs found
Minimax Rate-Optimal Estimation of High-Dimensional Covariance Matrices with Incomplete Data
Missing data occur frequently in a wide range of applications. In this paper, we consider estimation of high-dimensional covariance matrices in the presence of missing observations under a general missing completely at random model in the sense that the missingness is not dependent on the values of the data. Based on incomplete data, estimators for bandable and sparse covariance matrices are proposed and their theoretical and numerical properties are investigated.
Minimax rates of convergence are established under the spectral norm loss and the proposed estimators are shown to be rate-optimal under mild regularity conditions. Simulation studies demonstrate that the estimators perform well numerically. The methods are also illustrated through an application to data from four ovarian cancer studies. The key technical tools developed in this paper are of independent interest and potentially useful for a range of related problems in high-dimensional statistical inference with missing data
Posterior contraction in sparse Bayesian factor models for massive covariance matrices
Sparse Bayesian factor models are routinely implemented for parsimonious
dependence modeling and dimensionality reduction in high-dimensional
applications. We provide theoretical understanding of such Bayesian procedures
in terms of posterior convergence rates in inferring high-dimensional
covariance matrices where the dimension can be larger than the sample size.
Under relevant sparsity assumptions on the true covariance matrix, we show that
commonly-used point mass mixture priors on the factor loadings lead to
consistent estimation in the operator norm even when . One of our major
contributions is to develop a new class of continuous shrinkage priors and
provide insights into their concentration around sparse vectors. Using such
priors for the factor loadings, we obtain similar rate of convergence as
obtained with point mass mixture priors. To obtain the convergence rates, we
construct test functions to separate points in the space of high-dimensional
covariance matrices using insights from random matrix theory; the tools
developed may be of independent interest. We also derive minimax rates and show
that the Bayesian posterior rates of convergence coincide with the minimax
rates upto a term.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1215 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Exponentially convergent data assimilation algorithm for Navier-Stokes equations
The paper presents a new state estimation algorithm for a bilinear equation
representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS)
equations on a torus in R2. This state equation is subject to uncertain but
bounded noise in the input (Kolmogorov forcing) and initial conditions, and its
output is incomplete and contains bounded noise. The algorithm designs a
time-dependent gain such that the estimation error converges to zero
exponentially. The sufficient condition for the existence of the gain are
formulated in the form of algebraic Riccati equations. To demonstrate the
results we apply the proposed algorithm to the reconstruction a chaotic fluid
flow from incomplete and noisy data
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