Measure of nonlinearity for stochastic systems

Knowledge of how nonlinear a stochastic system is important for many applications. For example, a full-blown nonlinear filter is needed in general if the system is highly nonlinear, but a quasi-linear filter (e.g., an extended Kalman filter) is sufficient if the system is only slightly nonlinear. We first briefly survey various measures of nonlinearity for different representations of problems. Unfortunately, the conclusion of our survey is that a good quantitative measure of nonlinearity for stochastic systems is still lacking and existing measures designed for other applications are not suitable here. In view of this, we propose a general measure of nonlinearity for stochastic systems based on the idea of quantifying its deviation from linearity. It can be interpreted as a measure of the mean-square distance between a point (i.e., the given nonlinear system) and a subspace (i.e., the set of all linear systems) in a functional space. Properties and computation of this measure are explored. A numerical example is given in which the measure is applied to a target tracking problem.