An integral method for solving nonlinear eigenvalue problems

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour.     

Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model

We derive the asymptotics of the OLS estimator for a purely autoregressive spatial model. Only low-level conditions are used. As the sample size increases, the spatial matrix is assumed to approach a square-integrable function on the square (0,1)². The asymptotic distribution is a ratio of two infinite linear combinations of χ² variables. The formula involves eigenvalues of an integral operator associated with the function approached by the spatial matrices. Under the conditions imposed identification conditions for the maximum likelihood method and method of moments fail. A corrective two-step procedure using the OLS estimator is proposed.     

Non-white Wishart ensembles

We consider non-white Wishart ensembles , where X is a p×N random matrix with i.i.d. complex standard Gaussian entries and Σ is a covariance matrix, with fixed eigenvalues, close to the identity matrix. We prove that the largest eigenvalue of such random matrix ensembles exhibits a universal behavior in the large-N limit, provided Σ is “close enough” to the identity matrix. If not, we identify the limiting distribution of the largest eigenvalues, focusing on the case where the largest eigenvalues almost surely exit the support of the limiting Marchenko-Pastur's distribution.     

The higher-order derivatives of spectral functions

We are interested in higher-order derivatives of functions of the eigenvalues of real symmetric matrices with respect to the matrix argument. We describe a formula for the k-th derivative of such functions in two general cases.The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues.The second case describes the derivatives of the composition of a k-times differentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula significantly simplifies when the separable symmetric function is k-times continuously differentiable.As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one.To make the exposition as self contained as possible, we have included the necessary background results and definitions. Proofs of the intermediate technical results are collected in the appendices.     

On a new multivariate two-sample test

In this paper we propose a new test for the multivariate two-sample problem. The test statistic is the difference of the sum of all the Euclidean interpoint distances between the random variables from the two different samples and one-half of the two corresponding sums of distances of the variables within the same sample. The asymptotic null distribution of the test statistic is derived using the projection method and shown to be the limit of the bootstrap distribution. A simulation study includes the comparison of univariate and multivariate normal distributions for location and dispersion alternatives. For normal location alternatives the new test is shown to have power similar to that of the t- and T²-Test.     

The central limit theorem for sums of trimmed variables with heavy tails

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators and tests. Trimming also provides a profound insight into the partial sum behavior of i.i.d. sequences. There is a wide and nearly complete asymptotic theory of trimming, with one remarkable gap: no satisfactory criteria for the central limit theorem for modulus trimmed sums have been found, except for symmetric random variables. In this paper we investigate this problem in the case when the variables are in the domain of attraction of a stable law. Our results show that for modulus trimmed sums the validity of the central limit theorem depends sensitively on the behavior of the tail ratio P(X>t)/P(|X|>t) of the underlying variable X as t→∞ and paradoxically, increasing the number of trimmed elements does not generally improve partial sum behavior.     

Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series

We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p _n converges to infinity when the sample size n increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavy-tailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly dependent. We provide limit results for the weak convergence of these point processes to Poisson or cluster Poisson processes. Based on this convergence we can also derive the limit laws of various function als of the ordered eigenvalues such as the joint convergence of a finite number of the largest order statistics, the joint limit law of the largest eigenvalue and the trace, limit laws for successive ratios of ordered eigenvalues, etc. We also develop some limit theory for the singular values of the sample autocovariance matrices and their sums of squares. The theory is illustrated for simulated data and for the components of the S&P 500 stock index.     

Approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations

This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace (relation) below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspace, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.     

Error bounds in the isometric Arnoldi process

Error bounds for the eigenvalues computed in the isometric Arnoldi method are derived. The Arnoldi method applied to a unitary matrix U successively computes a sequence of unitary upper Hessenberg matrices H_k, k = 1,2,… The eigenvalues of the H_k's are increasingly better approximations to eigenvalues of U. An upper bound for the distance of the spectrum of H_k from the spectrum of U, and an upper bound for the distance between each individual eigenvalue of H_k and one of U are given. Between two eigenvalues of H_k on the unit circle, there is guaranteed to lie an eigenvalue of U. The results are applied to a problem in signal processing.     

Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images

In this article we develop a nonparametric methodology for estimating the mean change for matched samples on a Lie group. We then notice that for k≥5, a manifold of projective shapes of k-ads in 3D has the structure of a 3k−15 dimensional Lie group that is equivariantly embedded in a Euclidean space, therefore testing for mean change amounts to a one sample test for extrinsic means on this Lie group. The Lie group technique leads to a large sample and a nonparametric bootstrap test for one population extrinsic mean on a projective shape space, as recently developed by Patrangenaru, Liu and Sughatadasa. On the other hand, in the absence of occlusions, the 3D projective shape of a spatial k-ad can be recovered from a stereo pair of images, thus allowing one to test for mean glaucomatous 3D projective shape change detection from standard stereo pair eye images.     

Pfaffian point process for the Gaussian real generalised eigenvalue problem

The generalised eigenvalues for a pair of N?× N matrices (X ₁, X ₂) are defined as the solutions of the equation det (X ₁ ? λX ₂)?=?0, or equivalently, for X ₂ invertible, as the eigenvalues of ${X_{2}^{-1}X_{1}}$ . We consider Gaussian real matrices X ₁, X ₂, for which the generalised eigenvalues have the rotational invariance of the half-sphere, or after a fractional linear transformation, the rotational invariance of the unit disk. In these latter variables we calculate the joint eigenvalue probability density function, the probability p _N,k of finding k real eigenvalues, the densities of real and complex eigenvalues (the latter being related to an average over characteristic polynomials), and give an explicit Pfaffian formula for the higher correlation functions ${\rho_{(k_1,k_2)}}$ . A limit theorem for p _N,k is proved, and the scaled form of ${\rho_{(k_1,k_2)}}$ is shown to be identical to the analogous limit for the correlations of the eigenvalues of real Gaussian matrices. We show that these correlations satisfy sum rules characteristic of the underlying two-component Coulomb gas.     

On the sum of the Laplacian eigenvalues of a tree

Given an n-vertex graph G=(V,E), the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L=D-A, where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k∈{1,…,n}. This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenkovi? and Gutman [10].     

On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors

We consider Bayesian analysis of data from multivariate linear regression models whose errors have a distribution that is a scale mixture of normals. Such models are used to analyze data on financial returns, which are notoriously heavy-tailed. Let π denote the intractable posterior density that results when this regression model is combined with the standard non-informative prior on the unknown regression coefficients and scale matrix of the errors. Roughly speaking, the posterior is proper if and only if n≥d+k, where n is the sample size, d is the dimension of the response, and k is number of covariates. We provide a method of making exact draws from π in the special case where n=d+k, and we study Markov chain Monte Carlo (MCMC) algorithms that can be used to explore π when n>d+k. In particular, we show how the Haar PX-DA technology studied in Hobert and Marchev (2008) [11] can be used to improve upon Liu’s (1996) [7] data augmentation (DA) algorithm. Indeed, the new algorithm that we introduce is theoretically superior to the DA algorithm, yet equivalent to DA in terms of computational complexity. Moreover, we analyze the convergence rates of these MCMC algorithms in the important special case where the regression errors have a Student’s t distribution. We prove that, under conditions on n, d, k, and the degrees of freedom of the t distribution, both algorithms converge at a geometric rate. These convergence rate results are important from a practical standpoint because geometric ergodicity guarantees the existence of central limit theorems which are essential for the calculation of valid asymptotic standard errors for MCMC based estimates.     

Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix

In this paper, we propose a new methodology to deal with PCA in high-dimension, low-sample-size (HDLSS) data situations. We give an idea of estimating eigenvalues via singular values of a cross data matrix. We provide consistency properties of the eigenvalue estimation as well as its limiting distribution when the dimension d and the sample size n both grow to infinity in such a way that n is much lower than d. We apply the new methodology to estimating PC directions and PC scores in HDLSS data situations. We give an application of the findings in this paper to a mixture model to classify a dataset into two clusters. We demonstrate how the new methodology performs by using HDLSS data from a microarray study of prostate cancer.     

On the sum of Laplacian eigenvalues of graphs

Let k be a natural number and let G be a graph with at least k vertices. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most , where e(G) is the number of edges of G. We prove this conjecture for k=2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G)+2k-1.     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

Method of solving the generalized problem on eigenvalues of multidimensional circuits, I

This work is devoted to the development of a new method for solving the complete generalized problem of large-scale electrical circuit eigenvalues. We prove that the determinant of the conductance node matrix of an arbitrary k-loop LC-circuit is the Weinstein function for the loop impendance matrix of this circuit, and vice versa. A recurrent method of imposing constraints on the basic problem was defined, which permitted us to separate roots of characteristic polynomials in the recurrent process. Also a condition for conservativity of multiple eigenvalues was defined. A solution algorithm for the problem of defining a full range of eigenvalues of an oscillatory system with a finite number of degrees of freedom was suggested.     

Wishart-Laplace distributions associated with matrix quadratic forms

For a normal random matrix Y with mean zero, necessary and sufficient conditions are obtained for Y^′W_kY to be Wishart-Laplace distributed and {Y^′W_kY} to be independent, where each W_k is assumed to be symmetric rather than nonnegative definite.     

Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components X_k, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (namely 2,3, and 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n→∞). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, and the isolation of nonlinearities. We also show that the previously known densities (for n?4) are in fact the only solutions in a natural and very large class of potential RBS densities. This finding clarifies the need for new methods, such as those presented here.     

Computable eigenvalue bounds for rank-k perturbations

We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given.