Generating random AR(p) and MA(q) Toeplitz correlation matrices

AbstractMethods are proposed for generating random (p+1)×(p+1) Toeplitz correlation matrices that are consistent with a causal AR(p) Gaussian time series model. The main idea is to first specify distributions for the partial autocorrelations that are algebraically independent and take values in (−1,1), and then map to the Toeplitz matrix. Similarly, starting with pseudo-partial autocorrelations, methods are proposed for generating (q+1)×(q+1) Toeplitz correlation matrices that are consistent with an invertible MA(q) Gaussian time series model. The density can be uniform or non-uniform over the space of autocorrelations up to lag p or q, or over the space of autoregressive or moving average coefficients, by making appropriate choices for the densities of the (pseudo)-partial autocorrelations. Important intermediate steps are the derivations of the Jacobians of the mappings between the (pseudo)-partial autocorrelations, autocorrelations and autoregressive/moving average coefficients. The random generating methods are useful for models with a structured Toeplitz matrix as a parameter

A method for generating realistic correlation matrices

Simulating sample correlation matrices is important in many areas of statistics. Approaches such as generating Gaussian data and finding their sample correlation matrix or generating random uniform $[-1,1]$ deviates as pairwise correlations both have drawbacks. We develop an algorithm for adding noise, in a highly controlled manner, to general correlation matrices. In many instances, our method yields results which are superior to those obtained by simply simulating Gaussian data. Moreover, we demonstrate how our general algorithm can be tailored to a number of different correlation models. Using our results with a few different applications, we show that simulating correlation matrices can help assess statistical methodology.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS638 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

A note on eigenvalues of random block Toeplitz matrices with slowly growing bandwidth

This paper can be thought of as a remark of \cite{llw}, where the authors studied the eigenvalue distribution $\mu_{X_N}$ of random block Toeplitz band matrices with given block order $m$. In this note we will give explicit density functions of $\lim\limits_{N\to\infty}\mu_{X_N}$ when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of $m\times m$ Gaussian unitary ensemble (GUE for short). The series $\{\lim\limits_{N\to\infty}\mu_{X_N}|m\in\mathbb{N}\}$ can be seen as a transition from the standard normal distribution to semicircle distribution. We also show a similar relationship between GOE and block Toeplitz band matrices with symmetric blocks.Comment: 6 page

Learning detectors quickly using structured covariance matrices

Computer vision is increasingly becoming interested in the rapid estimation of object detectors. Canonical hard negative mining strategies are slow as they require multiple passes of the large negative training set. Recent work has demonstrated that if the distribution of negative examples is assumed to be stationary, then Linear Discriminant Analysis (LDA) can learn comparable detectors without ever revisiting the negative set. Even with this insight, however, the time to learn a single object detector can still be on the order of tens of seconds on a modern desktop computer. This paper proposes to leverage the resulting structured covariance matrix to obtain detectors with identical performance in orders of magnitude less time and memory. We elucidate an important connection to the correlation filter literature, demonstrating that these can also be trained without ever revisiting the negative set

Joint Covariance Estimation with Mutual Linear Structure

We consider the problem of joint estimation of structured covariance matrices. Assuming the structure is unknown, estimation is achieved using heterogeneous training sets. Namely, given groups of measurements coming from centered populations with different covariances, our aim is to determine the mutual structure of these covariance matrices and estimate them. Supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices, we develop a new efficient algorithm discovering the structure and using it to improve the estimation. Our technique is based on the application of principal component analysis in the matrix space. We also derive an upper performance bound of the proposed algorithm in the Gaussian scenario and compare it with the Cramer-Rao lower bound. Numerical simulations are presented to illustrate the performance benefits of the proposed method

Central Limit Theorem for local linear statistics in classical compact groups and related combinatorial identities

We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn

Geometric methods for estimation of structured covariances

We consider problems of estimation of structured covariance matrices, and in particular of matrices with a Toeplitz structure. We follow a geometric viewpoint that is based on some suitable notion of distance. To this end, we overview and compare several alternatives metrics and divergence measures. We advocate a specific one which represents the Wasserstein distance between the corresponding Gaussians distributions and show that it coincides with the so-called Bures/Hellinger distance between covariance matrices as well. Most importantly, besides the physically appealing interpretation, computation of the metric requires solving a linear matrix inequality (LMI). As a consequence, computations scale nicely for problems involving large covariance matrices, and linear prior constraints on the covariance structure are easy to handle. We compare this transportation/Bures/Hellinger metric with the maximum likelihood and the Burg methods as to their performance with regard to estimation of power spectra with spectral lines on a representative case study from the literature.Comment: 12 pages, 3 figure

Fisher Hartwig determinants, conformal field theory and universality in generalised XX models

We discuss certain quadratic models of spinless fermions on a 1D lattice, and their corresponding spin chains. These were studied by Keating and Mezzadri in the context of their relation to the Haar measures of the classical compact groups. We show how these models correspond to translation invariant models on an infinite or semi-infinite chain, which in the simplest case reduce to the familiar XX model. We give physical context to mathematical results for the entanglement entropy, and calculate the spin-spin correlation functions using the Fisher-Hartwig conjecture. These calculations rigorously demonstrate universality in classes of these models. We show that these are in agreement with field theoretic and renormalization group arguments that we provide