Friday 17 December 2010
Henry Abarbanel (California) Nonlinear State and Parameter Estimation in a Statistical Physics Setting

Models of complex dynamical systems often have unknown physical parameters and dynamical variables one cannot measure. Estimating these parameters and the unobserved state variables is necessary to allow prediction using the model. In the case where observations of the system of interest are noisy, the model of the system has errors, and the state of the system is uncertain at the beginning of observations, one can formulate the estimation problem as an exact path integral in discrete time and space. Within this framework we show how one can estimate the number of independent measurements required to make the state and parameter estimations and how one can evaluate the amount of information in bits associated with any measurement. The latter can be used to design observation programs focused on the most informative measurements. Several examples are given of simple physical systems and model systems from a geophysical setting. Systems with hundreds of degrees of freedom are discussed, and ideas on extending this to the very large systems of interest in studies of the atmosphere and ocean are examined. The path integral is evaluated numerically using Monte Carlo methods. We also discuss the "saddle path" approximation, and its corrections, to the path integral. It is the familiar 4D variational principle for this problem, and we show it has numerous local minima associated with instabilities on the synchronization manifold where observations equal model output. These instabilities require regularization before the state and parameters estimation can be carried out; we show how to accomplish this.