Review of spectral methods for spatial processes

Spectral methods are a powerful tool for studying the spatial structure of random fields and generally offer significant computational benefits. The objective of this talk is to introduce the audience to Fourier analysis and spectral methods for spatial temporal processes. I will review discrete and continuous Fourier analysis, Fourier representation of non-periodic functions, the aliasing phenomenon in the spectral domain, the Fast Fourier Transform (FFT) to efficiently approximate Fourier integrals, and some of the spectral methods to approximate a Gaussian likelihood.

A stationary process has a spectral representation in terms of sine and cosine waves, we will generalize this result to nonstationary and nonseparable spatial temporal processes.

I will present some of the commonly used classes of spectral densities for stationary processes, and some new one models for nonstationary and nonseparable space-time processes.

All these methods will be applied to air pollution data provided by the US EPA.