Continuous Time Brownian Motion Models for Analysis of Sequential Data

Abstract

SummaryThe paper discusses techniques for analysis of sequential data from variable processes, particularly techniques that can be used even when the data are not equally spaced in time. The techniques are based on models which use continuous time Brownian motion and its integrals to describe the pattern of variation in an underlying physical process and use white noise to describe variation due to measurement processes. The paper concentrates on making statements about what has been happening in the region that is covered by the data rather than on making predictions about regions that are far from the data. It is argued that the continuous time integrated Brownian motion plus white noise model is always preferable to its discrete time analogue: the local linear trend model. Further integrals of Brownian motion can be fitted as an alternative to using splines. All of these continuous time Brownian motion models can be fitted to data that are associated with a time interval (interval data) as well as to data that are associated with a single time (spot data). They can be fitted by using mixed model methodology as well as by using Kalman filtering and smoothing.