On the Distribution of the Ratio of the Largest Eigenvalue to the Trace of a Wishart Matrix

Written by Boaz Nadler (2011)

The Problem

Matlab Code + Demo

References

Let l1 denote the largest sample eigenvalue of a p by p Wishart matrix W with identity covariance, computed from n samples. In several problems, including testing for multiplicative components of interaction  and signal detection , there is an interest in the distribution of the following random variable

U = l1 / [ Trace(W) / p ].

In principle, as p,n → ∞, with p/n→c, after proper centering and scaling, the asymptotic distribution of the ratio is the same as that of the largest eigenvalue itself, namely a Tracy-Widom distribution

Pr [ (U - μn,p) / σn,p] → TWβ(s)
where β=1 for real valued data and β=2 for complex valued data.

However, as shown in , this may be a poor approximation to this distribution, in particular for small to modest values of p,n. Instead, in  a refined approximation was derived

Pr [ (U - μn,p) / σn,p] ~ TWβ(s) + 1/(β np) (μn,pn,p)2 TWβ''(s).

Matlab Code + Demo

We provide a Matlab function that inverts the modified distribution for the ratio, and for a given right tail probability α provides an approximate threshold t(α,p,n), such that Pr[U > t] ~ α.

function [t s ] = TW_trace_ratio_threshold (p, n, beta, alpha)

where

p          = dimension

n          = number of samples,

beta     = 1 or 2 for real or complex valued noise,

alpha    = right tail probability, a number between 0 and 1.

Additional required routines are:

function [mu_np sigma_np] = KN_mu_sigma(n,p,beta);    % computes the centering and scaling constants

TW_beta1.mat, TW_beta2.mat                     % pre-computed tables of the Tracy-Widom distributions and their derivatives, using Matlab code by Prof. Folkmar Bornemann and based on 

Demo: function demo_ratio.m compares the accuracy of the theoretical formula to empirical simulations, as a function of number of samples n, similar to the figure in the paper .

References

 B. Nadler, On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix, Journal of Multivariate Analysis, 2011.

 I.M. Johnstone, On the distribution of the largest eigenvalues in principal components analysis, Annals of Statistics, vol. 29, pp. 295-327, 2001.

 J.R. Schott, A note on the critical values used in stepwise tests for multiplicative components of interaction, Communications in Statistics - Theory and Methods, 1986.

 O. Besson, L. L. Scharf, CFAR matched direction detector, IEEE Transactions on Signal Processing, 2006.

 F. Bornemann, On the numerical evaluation of distributions in random matrix theory, Markov Processes and Related Fields, 2010.