constraints

. 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, ...

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 logistic-normal model, which provides unconstrained sampling distributions at the within-unit level and an unconstrained covariance structure at the betweenunit level. Also, the sampling distribution at the between-unit level is unimodal in a weak sense. Parameter estimation and inference for the hierarchical logistic-normal 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.

In earlier work we described a circulant embedding approach for simulating scalar-valued 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 non-negativity 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...

. 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...

1=f noise and statistically self-similar processes such as fractional Brownian motion (fBm) are vital for modeling numerous real-world phenomena, from network traffic to DNA to the stock market. Although several algorithms exist for synthesizing discrete-time 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 length-N 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 FFT-based synthesis is exact not only for 1=f processes such as fBm, but also for a wide class of long-range dependent processes. Leveraging the flexibility of the FFT approach, we develop new models for processes that exhibit one type of fBm scaling be...