2,901 research outputs found
Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models
This article provides a central limit theorem for a consistent estimator of
population eigenvalues with large multiplicities based on sample covariance
matrices. The focus is on limited sample size situations, whereby the number of
available observations is known and comparable in magnitude to the observation
dimension. An exact expression as well as an empirical, asymptotically
accurate, approximation of the limiting variance is derived. Simulations are
performed that corroborate the theoretical claims. A specific application to
wireless sensor networks is developed.Comment: 30 p
Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models
30 pp.International audienceThis article provides a central limit theorem for a consistent estimator of population eigenvalues with large multiplicities based on sample covariance matrices. The focus is on limited sample size situations, whereby the number of available observations is known and comparable in magnitude to the observation dimension. An exact expression as well as an empirical, asymptotically accurate, approximation of the limiting variance is derived. Simulations are performed that corroborate the theoretical claims. A specific application to wireless sensor networks is developed
Signal Processing in Large Systems: a New Paradigm
For a long time, detection and parameter estimation methods for signal
processing have relied on asymptotic statistics as the number of
observations of a population grows large comparatively to the population size
, i.e. . Modern technological and societal advances now
demand the study of sometimes extremely large populations and simultaneously
require fast signal processing due to accelerated system dynamics. This results
in not-so-large practical ratios , sometimes even smaller than one. A
disruptive change in classical signal processing methods has therefore been
initiated in the past ten years, mostly spurred by the field of large
dimensional random matrix theory. The early works in random matrix theory for
signal processing applications are however scarce and highly technical. This
tutorial provides an accessible methodological introduction to the modern tools
of random matrix theory and to the signal processing methods derived from them,
with an emphasis on simple illustrative examples
Robust spiked random matrices and a robust G-MUSIC estimator
A class of robust estimators of scatter applied to information-plus-impulsive
noise samples is studied, where the sample information matrix is assumed of low
rank; this generalizes the study of (Couillet et al., 2013b) to spiked random
matrix models. It is precisely shown that, as opposed to sample covariance
matrices which may have asymptotically unbounded (eigen-)spectrum due to the
sample impulsiveness, the robust estimator of scatter has bounded spectrum and
may contain isolated eigenvalues which we fully characterize. We show that, if
found beyond a certain detectability threshold, these eigenvalues allow one to
perform statistical inference on the eigenvalues and eigenvectors of the
information matrix. We use this result to derive new eigenvalue and eigenvector
estimation procedures, which we apply in practice to the popular array
processing problem of angle of arrival estimation. This gives birth to an
improved algorithm based on the MUSIC method, which we refer to as robust
G-MUSIC
Estimation of the Covariance Matrix of Large Dimensional Data
This paper deals with the problem of estimating the covariance matrix of a
series of independent multivariate observations, in the case where the
dimension of each observation is of the same order as the number of
observations. Although such a regime is of interest for many current
statistical signal processing and wireless communication issues, traditional
methods fail to produce consistent estimators and only recently results relying
on large random matrix theory have been unveiled. In this paper, we develop the
parametric framework proposed by Mestre, and consider a model where the
covariance matrix to be estimated has a (known) finite number of eigenvalues,
each of it with an unknown multiplicity. The main contributions of this work
are essentially threefold with respect to existing results, and in particular
to Mestre's work: To relax the (restrictive) separability assumption, to
provide joint consistent estimates for the eigenvalues and their
multiplicities, and to study the variance error by means of a Central Limit
theorem
On determining the number of spikes in a high-dimensional spiked population model
In a spiked population model, the population covariance matrix has all its
eigenvalues equal to units except for a few fixed eigenvalues (spikes).
Determining the number of spikes is a fundamental problem which appears in many
scientific fields, including signal processing (linear mixture model) or
economics (factor model). Several recent papers studied the asymptotic behavior
of the eigenvalues of the sample covariance matrix (sample eigenvalues) when
the dimension of the observations and the sample size both grow to infinity so
that their ratio converges to a positive constant. Using these results, we
propose a new estimator based on the difference between two consecutive sample
eigenvalues
Statistical eigen-inference from large Wishart matrices
We consider settings where the observations are drawn from a zero-mean
multivariate (real or complex) normal distribution with the population
covariance matrix having eigenvalues of arbitrary multiplicity. We assume that
the eigenvectors of the population covariance matrix are unknown and focus on
inferential procedures that are based on the sample eigenvalues alone (i.e.,
"eigen-inference"). Results found in the literature establish the asymptotic
normality of the fluctuation in the trace of powers of the sample covariance
matrix. We develop concrete algorithms for analytically computing the limiting
quantities and the covariance of the fluctuations. We exploit the asymptotic
normality of the trace of powers of the sample covariance matrix to develop
eigenvalue-based procedures for testing and estimation. Specifically, we
formulate a simple test of hypotheses for the population eigenvalues and a
technique for estimating the population eigenvalues in settings where the
cumulative distribution function of the (nonrandom) population eigenvalues has
a staircase structure. Monte Carlo simulations are used to demonstrate the
superiority of the proposed methodologies over classical techniques and the
robustness of the proposed techniques in high-dimensional, (relatively) small
sample size settings. The improved performance results from the fact that the
proposed inference procedures are "global" (in a sense that we describe) and
exploit "global" information thereby overcoming the inherent biases that
cripple classical inference procedures which are "local" and rely on "local"
information.Comment: Published in at http://dx.doi.org/10.1214/07-AOS583 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Cleaning large correlation matrices: tools from random matrix theory
This review covers recent results concerning the estimation of large
covariance matrices using tools from Random Matrix Theory (RMT). We introduce
several RMT methods and analytical techniques, such as the Replica formalism
and Free Probability, with an emphasis on the Marchenko-Pastur equation that
provides information on the resolvent of multiplicatively corrupted noisy
matrices. Special care is devoted to the statistics of the eigenvectors of the
empirical correlation matrix, which turn out to be crucial for many
applications. We show in particular how these results can be used to build
consistent "Rotationally Invariant" estimators (RIE) for large correlation
matrices when there is no prior on the structure of the underlying process. The
last part of this review is dedicated to some real-world applications within
financial markets as a case in point. We establish empirically the efficacy of
the RIE framework, which is found to be superior in this case to all previously
proposed methods. The case of additively (rather than multiplicatively)
corrupted noisy matrices is also dealt with in a special Appendix. Several open
problems and interesting technical developments are discussed throughout the
paper.Comment: 165 pages, article submitted to Physics Report
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