Linear Smoothing of Noisy Spatial Temporal Series

Jumat Jumat

doi:10.3844/jmssp.2005.309.321

Abstract

The main objective of the study is the development of a linear filter to extract the signal from a spatio-temporal series affected by measureme nt error. We assume that the evolution of the unobservable signal can be modelled by a space time autoregressive process. In its vectorial form, the model admits a state space representation allowing the direct application of the Kalman filter machinery to predict the unobservable state vector on the basis of the sample information. Having introduced the model, referred to as a STARG+Noise model, the study discusses Maximum Likelihood (ML) parameter estimation assuming knowledge of the variance of the noise process. Consistent method of moments estimators of the autoregressive coefficients and noise variance are also derived, primarily to be used as inputs in the ML estimation procedure. Finally, we consider some simulation studies and an investigation involving sulphur diox ide level monitoring.

Highlights

The study is concerned with parameter estimation and smoothing of a spatio-temporal series corrupted by noise
We assume that the evolution of the unobservable signal can be modelled by a space time autoregressive process
Following Dryden et al.[4], we propose the SpaceTime Adjusted Maximum Likelihood Estimator (STAMLE), which is an approximation to the ML estimators, provided the noise variance is known or can be consistently estimated

Summary

INTRODUCTION

The study is concerned with parameter estimation and smoothing of a spatio-temporal series corrupted by noise. As can be immediately verified, when the process is assumed to evolve over time according to a p-th order autoregressive model, assuming a separable covariance is equivalent to postulating the following linear model for the observed spatial time series α(B)ξt = εt where εt is a zero mean multivariate white noise process with autocovariance function E(εtε′t+h) = ΣS if h = 0 and E(εtε′t+h) = 0 elsewhere and α(B) is the scalar autoregressive polynomial α(B) = 1-α1B-α2B2-...-αpBp, where α1, α2,..., αp are scalar parameters and B is the usual backward shift operator. Kalman filter to derive a recursive procedure for estimating the parameters of a STARMAX model as well dealing with missing data The implementation of such algorithms is not a difficult task on its own; for huge spatial temporal datasets, the filter dimensionality as well as matrix inversions may suggest the adoption of some “tricks” which can reduce the computational burden. We will briefly deal with some relevant computational simplifications

Inversion of A0

Method

CONCLUSION