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

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 [3] and signal detection [4], 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 [1], this may be a poor approximation to this distribution, in particular for small to modest values of p,n. Instead, in [1] 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.

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 [5]

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 [1].

References

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

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

[3] 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.

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

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