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DETERMINANT MAXIMIZATION WITH LINEAR MATRIX INEQUALITY CONSTRAINTS
"... The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the s ..."
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Cited by 169 (18 self)
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The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the semidefinite programming problem. We give an overview of the applications of the determinant maximization problem, pointing out simple cases where specialized algorithms or analytical solutions are known. We then describe an interiorpoint method, with a simplified analysis of the worstcase complexity and numerical results that indicate that the method is very efficient, both in theory and in practice. Compared to existing specialized algorithms (where they are available), the interiorpoint method will generally be slower; the advantage is that it handles a much wider variety of problems.
Fast And Exact Simulation Of Stationary Gaussian Processes Through Circulant Embedding Of The Covariance Matrix
, 1997
"... . Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid## This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over m +1 equispaced point ..."
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Cited by 41 (1 self)
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. Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid## This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over m +1 equispaced points on a line can be produced at the cost of an initial FFT of length 2m with each new realization requiring an additional FFT of the same length. In particular, the paper first notes that if an (m+1)×(m+1) Toeplitz correlation matrix R can be embedded in a nonnegative definite 2M×2M circulant matrix S, exact realizations of the normal multivariate y #N(0,R) can be generated via FFTs of length 2M . Theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which M = m is nonnegative definite. Extensions to simulations of stationary fields in higher dimensions are also provided and illustrated. Key words. geostatistics, ...
Nonparametric Modelling of Hierarchically Exchangeable Data
, 2003
"... Hierarchically exchangeable data are characterized by the exchangeability of a population of units and the exchangeability of observations from each individual unit. A flexible model for such data is the hierarchical logisticnormal model, which provides unconstrained sampling distributions at the w ..."
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Cited by 9 (0 self)
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Hierarchically exchangeable data are characterized by the exchangeability of a population of units and the exchangeability of observations from each individual unit. A flexible model for such data is the hierarchical logisticnormal model, which provides unconstrained sampling distributions at the withinunit level and an unconstrained covariance structure at the betweenunit level. Also, the sampling distribution at the betweenunit level is unimodal in a weak sense. Parameter estimation and inference for the hierarchical logisticnormal model is relatively straightforward via Markov chain Monte Carlo or an approximate EM algorithm. These and other features of the hierarchical logistic normal model are explored, and the model is applied to the analysis of tumor locations in a mammalian population. A comparison is made to a similar data analysis based on Dirichlet distributions.
Simulation of Stationary Gaussian Vector Fields
 Statistics and Computing
"... In earlier work we described a circulant embedding approach for simulating scalarvalued stationary Gaussian random fields on a finite rectangular grid, with the covariance function prescribed. In this sequel, we describe how the circulant embedding approach can be used to simulate stationary Gaussi ..."
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Cited by 7 (1 self)
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In earlier work we described a circulant embedding approach for simulating scalarvalued stationary Gaussian random fields on a finite rectangular grid, with the covariance function prescribed. In this sequel, we describe how the circulant embedding approach can be used to simulate stationary Gaussian vector fields. As in the scalar case, the simulation procedure is theoretically exact if a certain nonnegativity condition is satisfied. In the vector setting, this exactness condition takes the form of a nonnegative definiteness condition on a certain set of Hermitian matrices. The main computational tool used is the Fast Fourier Transform. Consequently, when implemented appropriately, the procedure is highly efficient, in terms of both CPU time and storage. Key Words and Phrases: AMS (1991) Subject Classification. 1 Grace Chan, Department of Statistics, School of Mathematics, University of New South Wales, Sydney 2052, Australia. 2 A.T.A. Wood, School of Mathematical Sciences, Universi...
Localization Of The Eigenvalues Of Toeplitz Matrices Using Additive Decomposition, Embedding In Circulants, And The Fourier Transform
 Proc. 10th IFAC Symposium on System Identification SYSID’94, Copenhagen
, 1992
"... . This paper explores the relationship between Toeplitz and circulant matrices. Upper and lower bounds for all eigenvalues of hermitian Toeplitz matrices are given, capitalizing on the possibility of embedding a Toeplitz matrix in a circulant, and of expressing any n \Theta n Toeplitz matrix as a ..."
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Cited by 4 (0 self)
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. This paper explores the relationship between Toeplitz and circulant matrices. Upper and lower bounds for all eigenvalues of hermitian Toeplitz matrices are given, capitalizing on the possibility of embedding a Toeplitz matrix in a circulant, and of expressing any n \Theta n Toeplitz matrix as a sum of two matrices with known eigenvalues. The bounds can be simultaneously found using a single discrete Fourier transform evaluation. Thus, the total computation time is O(log 2 n) per bound. Simulation results indicate that the bounds are sharper than a few other known bounds. Key Words. Computational methods; digital signal processing; eigenvalues; fast Fourier transforms; matrix algebra; numerical methods. 1. INTRODUCTION The class of Toeplitz matrices is extremely important, for a number of theoretical and practical reasons. These matrices arise naturally in a variety of problems, including trigonometric moment problems, optimum filtering, linear prediction and spectral estimat...
Fast, Exact Synthesis of Gaussian and nonGaussian LongRangeDependent Processes
, 1999
"... 1=f noise and statistically selfsimilar processes such as fractional Brownian motion (fBm) are vital for modeling numerous realworld phenomena, from network traffic to DNA to the stock market. Although several algorithms exist for synthesizing discretetime samples of a 1=f process, these algorith ..."
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Cited by 2 (0 self)
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1=f noise and statistically selfsimilar processes such as fractional Brownian motion (fBm) are vital for modeling numerous realworld phenomena, from network traffic to DNA to the stock market. Although several algorithms exist for synthesizing discretetime samples of a 1=f process, these algorithms are inexact, meaning that the covariance of the synthesized processes can deviate significantly from that of a true 1=f process. However, the Fast Fourier Transform (FFT) can be used to exactly and efficiently synthesize such processes in O(N log N) operations for a lengthN signal. Strangely enough, the key is to apply the FFT to match the target process's covariance structure, not its frequency spectrum. In this paper, we prove that this FFTbased synthesis is exact not only for 1=f processes such as fBm, but also for a wide class of longrange dependent processes. Leveraging the flexibility of the FFT approach, we develop new models for processes that exhibit one type of fBm scaling be...
Realisability conditions for second order marginals of biphased media
, 2013
"... Thispaperconcernsthesecondordermarginals ofbiphasedrandommedia. We give discriminating necessary conditions for a bivariate function to besuch a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they a ..."
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Thispaperconcernsthesecondordermarginals ofbiphasedrandommedia. We give discriminating necessary conditions for a bivariate function to besuch a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper. 1
A Maximum Entropy Approach to the Realizability of Spin Correlation Matrices
 ENTROPY
, 2013
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