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

A mathematical model of a large open fire

A mathematical model capable of predicting the detailed characteristics of large, liquid fuel, axisymmetric, pool fires is described. The predicted characteristics include spatial distributions of flame gas velocity, soot concentration and chemical specie concentrations including carbon monoxide, carbon dioxide, water, unreacted oxygen, unreacted fuel and nitrogen. Comparisons of the predictions with experimental values are also given

Free Probability, Sample Covariance Matrices and Stochastic Eigen-Inference

Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This talk is a survey specifically oriented towards the needs and interests of a computationally inclined audience. We include the important mathematics (free probability) that permit the characterization of a large class of random matrices. We discuss how computational software is transforming this theory into practice by highlighting its use in the context of a stochastic eigen-inference application.Singapore-MIT Alliance (SMA

Performance Evaluation of Hierarchical Simulation Systems

Simulation needs for design analysis, verification, and testing have become increasingly important as integrated circuit size and complexity have grown. One technique for dealing with this problem is to utilize hierarchical modeling and simulation methods. This paper presents an analysis of hierarchical simulation systems in terms of two performance measures; the number of statements required for describing a system, and the simulation system execution time associated with a given hierarchical system representation. A model of hierarchical simulation system performance is developed. The performance of the hierarchical simulator, lsim2, is examined through its use on the set of benchmark circuits and the results discussed in light of model predictions

LSIM2 User\u27s Manual

Lsim2 is gate/switch-level digital logic simulator. It enables users to model digital circuits both at the gate and switch level and incorporates features the support investigation of the simulation task itself. Lsim2 is an augmented version of the original lsim* with the addition of several new MSI-type components models. This user\u27s manual describes procedures for specifying a circuit in lsim2, mechanisms for controlling the simulation, and approaches to modeling systems

Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d)>0 is the exponent associated to the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n^{-2(\theta(d) + \theta(2))}. For d=1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0,1] has an unusual scaling form given by n^{-\tilde \phi(k/\log n)} where \tilde \phi(x) is a universal large deviation function.Comment: 4 pages, 3 figures. Minor changes. Accepted version in Phys. Rev. Let

Lattice congruences of the weak order

We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let \eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show that the fibers of \eta_K constitute the smallest lattice congruence with 1\equiv s for every s\in(S-K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order.Comment: 26 pages, 4 figure