Estimation of Parameters in Random Dynamical Systems

In random dynamical systems, e.g. described by stochastic differential equations, it is often difficult to infer the parameters. The main difficulty is that one hand no likelihoods can be computed, excluding Maximum Likelihood, EM or Bayesian Methods and on the other hand the system is random, excluding simple least squares comparison of the observations with expected trajectories of the dynamical system. We have developed a new R-package ”SysStat” containing general approaches for estimation in stochastic systems based on simulation rather than likelihood computation using approximate Bayes and approximate quasi-likelihood methods finding good approximations based on informed user choices and simulations of the models with varying parameters. The user choice especially includes finding functions of the data with high information contend in the sense of high quasi-likelihood. Although both methods are not directly applicable to dynamic systems, there are systematic ways of constructing such informative statistics for stochastic differential equation models, allowing to construct informative functions for the local dynamic and translating these functions to informative statistics of the global dynamic by time averaging. This allows to estimate parameters of dynamic stochastic models efficiently. Our main aim is the modeling of bioleaching processes, but the method has a more general applicability for various types of processes in the geosciences including stationary and transient, spatial, temporal and spatiotemporal processes and will thus be demonstrated with simple to understand toy examples.