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OCIAM**

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Oxford Centre for Industrial and Applied Mathematics
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Dr Leonard A Smith's Publications**

Indistinguishable states II :
Imperfect model scenarios

K. Judd and L.A. Smith. In Review (July 2001)

ABSTRACT
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.

## Contact Information

Tel: 01865-2-70517

E-mail: lenny@maths.ox.ac.uk
Last updated: 14 Feb 2001

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