2,331 research outputs found
An Introduction to Wishart Matrix Moments
These lecture notes provide a comprehensive, self-contained introduction to
the analysis of Wishart matrix moments. This study may act as an introduction
to some particular aspects of random matrix theory, or as a self-contained
exposition of Wishart matrix moments. Random matrix theory plays a central role
in statistical physics, computational mathematics and engineering sciences,
including data assimilation, signal processing, combinatorial optimization,
compressed sensing, econometrics and mathematical finance, among numerous
others. The mathematical foundations of the theory of random matrices lies at
the intersection of combinatorics, non-commutative algebra, geometry,
multivariate functional and spectral analysis, and of course statistics and
probability theory. As a result, most of the classical topics in random matrix
theory are technical, and mathematically difficult to penetrate for non-experts
and regular users and practitioners. The technical aim of these notes is to
review and extend some important results in random matrix theory in the
specific context of real random Wishart matrices. This special class of
Gaussian-type sample covariance matrix plays an important role in multivariate
analysis and in statistical theory. We derive non-asymptotic formulae for the
full matrix moments of real valued Wishart random matrices. As a corollary, we
derive and extend a number of spectral and trace-type results for the case of
non-isotropic Wishart random matrices. We also derive the full matrix moment
analogues of some classic spectral and trace-type moment results. For example,
we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and
full matrix cases. Laplace matrix transforms and matrix moment estimates are
also studied, along with new spectral and trace concentration-type
inequalities
Partially Adaptive Estimation via Maximum Entropy Densities
We propose a partially adaptive estimator based on information theoretic maximum entropy estimates of the error distribution. The maximum entropy (maxent) densities have simple yet flexible functional forms to nest most of the mathematical distributions. Unlike the nonparametric fully adaptive estimators, our parametric estimators do not involve choosing a bandwidth or trimming, and only require estimating a small number of nuisance parameters, which is desirable when the sample size is small. Monte Carlo simulations suggest that the proposed estimators fare well with non-normal error distributions. When the errors are normal, the efficiency loss due to redundant nuisance parameters is negligible as the proposed error densities nest the normal. The proposed partially adaptive estimator compares favorably with existing methods, especially when the sample size is small. We apply the estimator to a bio-pharmaceutical example and a stochastic frontier model.
Approximations of Shannon Mutual Information for Discrete Variables with Applications to Neural Population Coding
Although Shannon mutual information has been widely used, its effective
calculation is often difficult for many practical problems, including those in
neural population coding. Asymptotic formulas based on Fisher information
sometimes provide accurate approximations to the mutual information but this
approach is restricted to continuous variables because the calculation of
Fisher information requires derivatives with respect to the encoded variables.
In this paper, we consider information-theoretic bounds and approximations of
the mutual information based on Kullback--Leibler divergence and R\'{e}nyi
divergence. We propose several information metrics to approximate Shannon
mutual information in the context of neural population coding. While our
asymptotic formulas all work for discrete variables, one of them has consistent
performance and high accuracy regardless of whether the encoded variables are
discrete or continuous. We performed numerical simulations and confirmed that
our approximation formulas were highly accurate for approximating the mutual
information between the stimuli and the responses of a large neural population.
These approximation formulas may potentially bring convenience to the
applications of information theory to many practical and theoretical problems.Comment: 31 pages, 6 figure
On the Analytic Wavelet Transform
An exact and general expression for the analytic wavelet transform of a
real-valued signal is constructed, resolving the time-dependent effects of
non-negligible amplitude and frequency modulation. The analytic signal is first
locally represented as a modulated oscillation, demodulated by its own
instantaneous frequency, and then Taylor-expanded at each point in time. The
terms in this expansion, called the instantaneous modulation functions, are
time-varying functions which quantify, at increasingly higher orders, the local
departures of the signal from a uniform sinusoidal oscillation. Closed-form
expressions for these functions are found in terms of Bell polynomials and
derivatives of the signal's instantaneous frequency and bandwidth. The analytic
wavelet transform is shown to depend upon the interaction between the signal's
instantaneous modulation functions and frequency-domain derivatives of the
wavelet, inducing a hierarchy of departures of the transform away from a
perfect representation of the signal. The form of these deviation terms
suggests a set of conditions for matching the wavelet properties to suit the
variability of the signal, in which case our expressions simplify considerably.
One may then quantify the time-varying bias associated with signal estimation
via wavelet ridge analysis, and choose wavelets to minimize this bias
- …