Stochastic, Dynamic Modelling and Signal Processing: Time Variable and State Dependent Parameter Estimation

Previous publications (e.g. Young, 1978, 1983, 1993a,b, 1998a,b, 1999a,b; Young and Runkle, 1989; Young and Minchin, 1991; Young et al., 1991; Young and Lees, 1993; Young and Beven, 1994; Young and Pedregal, 1997, 1998, 1999) have discussed an approach to nonstationary and nonlinear signal processing based on the identification and estimation of stochastic models with time variable (TVP) or state dependent (SDP) parameters. Here the term ‘nonstationarity’ is assumed to mean that the statistical properties of the signal, as defined by the parameters in an associated stochastic model, are changing over time at a rate which is ‘slow’ in relation to the rates of change of the stochastic state variables in the system under study. Although such nonstationary systems exhibit nonlinear behaviour, this can often be approximated well by TVP (or piece-wise linear) models, the parameters of which can be estimated using recursive methods of estimation in which the parameters are assumed to evolve in a simple stochastic manner (e.g. Young, 1984, 1999a). On the other hand, if the changes in the parameters are functions of the state or input variables (i.e. they actually constitute stochastic state variables), then the system is truly nonlinear and likely to exhibit severe nonlinear behaviour. Normally, this cannot be approximated in a simple TVP manner; in which case, recourse must be made to the alternative, and more powerful SDP modelling methods that are the main topic of this chapter. The extension of the TVP estimation methods to allow for state dependency, as described here, involves two statistical stages.