FOURIER INFERENCE: SOME METHODS FOR THE ANALYSIS OF ARRAY AND NONGAUSSIAN SERIES DATA¹

Abstract

Fourier inference is a collection of analytic techniques and philosophic attitudes, for the analysis of data, wherein essential use is made of empirical Fourier transforms. This paper sets down some basic results concerning the finite Fourier transforms of stationary process data and then, to illustrate the approach, uses those results to develop procedures for: 1) estimating cloud and storm motion, 2) passive sonar and 3) fitting finite parameter models to nonGaussian time series via bispectral fitting. This last procedure is illustrated by an analysis of a stretch of Mississippi River runoff data. Examples l), 2) refer to data having the form Y(xj,yj,t) for j = 1, . . . , J and t = 0, . . . , T-1 say, and view that data as part of a realization of a spatial-temporal process. Such data has become common in geophysics generally and in hydrology particularly. The goal of this paper is to present some new statistical procedures pertinent to problems in the water sciences, equally it is to iUustrate the genesis of those procedures and how their properties may be approximated.

Sea level characteristics in the Bay of Fundy/Gulf of Maine

“…The form of the likelihood is motivated by the asymptotic normality of the discrete Fourier transform, and minimization of (6) leads to an estimate of . See Brillinger (1985) for details on this and other statistics based on the discrete Fourier transform.…”

On spectral and bispectral estimator of the parameter of nongaussian data

“…We are given functions A and B defined by (12) and (13) such that Α(ω, OQ), 0 < ω < 1 is symmetric about 1/2 and Β(ωι, ω·ι, θ$) is symmetric according to (7) and (8).…”

“…The handicap of the nonGaussian parameter estimation is that the exact spectral and bispectral densities are supposed to be known up to some parameters of the model. The models have been successfully considered are the linear (nonGaussian) Brillinger [7], [8] and the bilinear one Terdik [24].…”

An Application of Statistics to Meteorology: Estimation of Motion

“…If that extreme point is (v u ;ŵ u ), the mean velocity of the storm is estimated by the average ofv u andŵ u . (Marshall 1980) Brillinger 1985) indicated extensions of the results of (Hannan & Thomson 1973) to provide large sample distributions for maximum cross-correlation estimates in the case of two time slices. (Brillinger 1993) is concerned with estimating the joint distributions of several successive motions given consecutive locations of moving particles.…”