K. Judd and L.A. Smith. In Review (July 2001)
A previous paper ( Judd-Smith Indistinguishable States I) considered the problem of estimating the true state of a system given a perfect model. The perfect model scenario is potentially misleading because in practice all models are imperfect. This paper considers imperfect models. With an imperfect model it is often the case that the system state space and model state space are not equivalent, and so one must consider the projection of system state into model state space. Furthermore, for imperfect models it is almost certain that no trajectory of the model is consistent with an infinite series of observations, and consequently, there is no consistent way to estimate the projection of system state using trajectories. There are pseudo-orbits, however, that are consistent with observations and these can be used to estimate the projection of the system state. One then finds, just as in the perfect model scenario, that there is a set of states that are indistinguishable from the projection of the system state. The paper includes a discussion of how to estimate the set of indistinguishable states and the probability density on these states. There are two main conclusions drawn from this study. The first conclusion is that there is no state of the model that can be identified with the state of the system. The second conclusion is that one must be careful when using an imperfect model to forecast the system, because the initialization of the model state from noisy observations can give a model state that is a poor analogue for the system, and the method of forecast may not shadow the future behaviour of the system for very long. The latter conclusion holds even if one were able to obtain the true projection of the system state.
Last updated: 14 Feb 2001