Bayes Linear Variance Adjustment for Locally Linear DLMs

Abstract

This paper exhibits quadratic products of linear combinations of observables which identify the covariance structure underlying the univariate locally linear time series dynamic linear model. The ®rst-and second-order moments for the joint distribution over these observables are given, allowing Bayes linear learning for the underlying covariance structure for the time series model. An example is given which illustrates the methodology and highlights the practical implications of the theory. In Wilkinson and Goldstein (1996), new methodology is developed for the revision of covariance structures underlying two-step invertible dynamic linear models (DLMs). Two-step invertible DLMs are essentially models without a trend component. Here, the locally linear DLM will be discussed, which does have a trend component, and so is not two-step invertible. Interest will be focused on univariate DLMs, since it is simpler to explain the theory in the univariate context. However, a full covariance matrix approach may be taken to the multivariate counterpart, thus generalizing the work in Wilkinson and Goldstein (1996). It will be shown how one may learn about the three dierent kinds of variance associated with the locally linear dynamic linear model, using partial prior speci®cation for certain aspects of the model. The theory will be applied to the modelling of sales of a particular product from a wholesale depot, and the importance of good estimation of all of the variance components will be demonstrated in the context of this example.Ã Correspondence to:

Bayes linear variance structure learning for inspection of large scale physical systems

“…as given in equation (8). In principle it is possible to learn about e 2 act and e 2 Xct separately.…”

“…This is similar to approach that taken in Wilkinson. 8 Using expressions for e 2 Xct thus obtained, we adjust our beliefs about M(W X ) using observed data and assumed known values for S r as explained below. In 'Mahalanobis variance learning' we present a fitting procedure using a Mahalanobis distance criterion to select an optimal combination of error variances.…”

“…The expectation of the square of the former linear combination is found using the facts that residuals are mutually uncorrelated in time, together with the exchangeability assumptions from equation (8) and the assumption of a fixed ratio of population variances in equation (14). Then…”

“…Farrow and Goldstein (1993) discusses Bayes linear methods for grouped multivariate repeated measurement studies with application to cross-over trials. Wilkinson (1997) discusses variance learning for a univariate linear growth DLM, and Wilkinson and Goldstein (1997) describes Bayes linear covariance matrix adjustment for a multivariate constant DLM. Shaw and Goldstein (1999) discusses Bayes linear experimental design for grouped multivariate exchangeable systems.…”

Posterior mean and variance approximation for regression and time series problems

“…Such considerations are useful in practice because they allow inference to a range of problems that otherwise the modeller would need to resort to Monte Carlo estimation (Gamerman, 1997) or to other simulation based methods (Kitagawa and Gersch, 1996). West and Harrison (1997, Chapter 4) and Wilkinson (1997) discuss how the above mentioned regression estimation can be applied to a sequential estimation problem, which is necessary to consider in time series analysis.…”