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28
On the distribution of the largest eigenvalue in principal components analysis
 Ann. Statist
, 2001
"... Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribu ..."
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Cited by 194 (2 self)
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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The
Matrix models for betaensembles
 J. Math. Phys
, 2002
"... This paper constructs tridiagonal random matrix models for general (β> 0) βHermite (Gaussian) and βLaguerre (Wishart) ensembles. These generalize the wellknown Gaussian and Wishart models for β = 1,2,4. Furthermore, in the cases of the βLaguerre ensembles, we eliminate the exponent quantization ..."
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Cited by 86 (19 self)
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This paper constructs tridiagonal random matrix models for general (β> 0) βHermite (Gaussian) and βLaguerre (Wishart) ensembles. These generalize the wellknown Gaussian and Wishart models for β = 1,2,4. Furthermore, in the cases of the βLaguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.
On the Distribution of the Largest Principal Component
 ANN. STATIST
, 2000
"... Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degr ..."
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Cited by 48 (0 self)
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Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n=p = 1: When centered by p = ( p n 1+ p p) 2 and scaled by p = ( p n 1+ p p)(1= p n 1+1= p p) 1=3 the distribution of x (1) approaches the TracyWidom law of order 1, which is dened in terms of the Painleve II dierential equation, and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to ...
Estimation of highdimensional prior and posterior covariance matrices in Kalman filter variants
 Journal of Multivariate Analysis
, 2007
"... This work studies the effect of using Monte Carlo based methods to estimate highdimensional systems. Recent focus in the geosciences has been on representing the atmospheric state using a probability density function, and, for extremely highdimensional systems, various sample based Kalman filter t ..."
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Cited by 47 (5 self)
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This work studies the effect of using Monte Carlo based methods to estimate highdimensional systems. Recent focus in the geosciences has been on representing the atmospheric state using a probability density function, and, for extremely highdimensional systems, various sample based Kalman filter techniques have been developed to address the problem of realtime assimilation of system information and observations. As the employed sample sizes are typically several orders of magnitude smaller than the system dimension, such sampling techniques inevitably induces considerable variability into the state estimate, primarily through prior and posterior sample covariance matrices. In this article we quantify this variability with mean squared error measures for two MonteCarlo based Kalman filter variants, the ensemble Kalman filter and the squareroot filter. Under weak assumptions, we derive exact expressions of the error measures. In other cases, we rely on matrix expansions and provide approximations. We show that covarianceshrinking (tapering) based on the Schur product of the prior sample covariance matrix and a positive definite function is a simple, computationally feasible, and very effective technique to reduce sample variability and to address rankdeficient sample covariances. We propose practical rules for obtaining optimally tapered sample covariance matrices. The theoretical results are verified and illustrated with extensive simulations.
FROM RANDOM MATRICES TO STOCHASTIC OPERATORS
"... Abstract. We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator disp ..."
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Cited by 12 (3 self)
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Abstract. We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel. 1.
Eigenvalue based spectrum sensing algorithms for cognitive radio
 IEEE Trans. on Communications
, 2009
"... Spectrum sensing is a fundamental component is a cognitive radio. In this paper, we propose new sensing methods based on the eigenvalues of the covariance matrix of signals received at the secondary users. In particular, two sensing algorithms are suggested, one is based on the ratio of the maximum ..."
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Cited by 11 (0 self)
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Spectrum sensing is a fundamental component is a cognitive radio. In this paper, we propose new sensing methods based on the eigenvalues of the covariance matrix of signals received at the secondary users. In particular, two sensing algorithms are suggested, one is based on the ratio of the maximum eigenvalue to minimum eigenvalue; the other is based on the ratio of the average eigenvalue to minimum eigenvalue. Using some latest random matrix theories (RMT), we quantify the distributions of these ratios and derive the probabilities of false alarm and probabilities of detection for the proposed algorithms. We also find the thresholds of the methods for a given probability of false alarm. The proposed methods overcome the noise uncertainty problem, and can even perform better than the ideal energy detection when the signals to be detected are highly correlated. The methods can be used for various signal detection applications without requiring the knowledge of signal, channel and noise power. Simulations based on randomly generated signals, wireless microphone signals and captured ATSC DTV signals are presented to verify the effectiveness of the proposed methods.
Performance analysis of transmit beamforming
 IEEE Trans. Commun
, 2005
"... Abstract—Using the theory of random matrices, a performance analysis is given for uncoded binary transmission over multipleinput multipleoutput channels, under the assumption that transmitter beamforming is used. In particular, exact finite antenna expressions are found for the average bit error r ..."
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Cited by 10 (2 self)
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Abstract—Using the theory of random matrices, a performance analysis is given for uncoded binary transmission over multipleinput multipleoutput channels, under the assumption that transmitter beamforming is used. In particular, exact finite antenna expressions are found for the average bit error rate (in the case of ergodic channels) for both noncoherent and coherent detection. Expressions for the the outage probability (in the case of quasistatic channels) are also given. Index Terms—Beamforming, bit error rate, outage probability, Rayleigh fading, Wishart matrices. I.
Global spectrum fluctuations for the βHermite and βLaguerre ensembles via matrix models
 J. Math. Phys
, 2006
"... ensembles via matrix models ..."
Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics
 Ann. Inst. H. Poincaré Probab. Statist
, 2005
"... Abstract In this paper we examine the zero and first order eigenvalue fluctuations for the fiHermite and fiLaguerre ensembles, using the matrix models we described in [5], in the limit as fi! 1. We find that the fluctuations are described by Gaussians of variance O(1=fi), centered at the roots of ..."
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Cited by 10 (5 self)
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Abstract In this paper we examine the zero and first order eigenvalue fluctuations for the fiHermite and fiLaguerre ensembles, using the matrix models we described in [5], in the limit as fi! 1. We find that the fluctuations are described by Gaussians of variance O(1=fi), centered at the roots of a corresponding Hermite (Laguerre) polynomial. We also show that the approximation is very good, even for small values of fi, by plotting exact level densities versus sum of Gaussians approximations. 1 Introduction This paper provides insight into the shape of random matrix laws such as the finite semicircle law, the finite quartercircle law and its generalization. We begin with a simple example. Suppose A is a random k \Theta k complex matrix with real and imaginary parts all i.i.d. standard normals. Let S = (A + AH)=2 be the Hermitian part of A. The matrix S has a distribution commonly known as the Gaussian Unitary Ensemble; this matrix distribution and the joint distribution of its (real) eigenvalues have been well studied. For a good reference on the subject, see Mehta [10].