Measure of Nonlinearity for Estimation

Nonlinearity, among other factors, is often the root cause of difficulties in nonlinear problems. It is important to quantify a problem's degree of nonlinearity to decide a proper solution. For example, a full-blown nonlinear filter is needed in general if the estimation problem is highly nonlinear, but a quasi-linear filter (e.g., an extended Kalman filter) is sufficient for a weakly nonlinear case. This paper first surveys various measures of nonlinearity (MoNs) for different applications. For nonlinear estimation, we conclude that these MoNs are not suitable and a better measure is needed. In view of this, we propose a general MoN for estimation. It measures the mean-square closeness between a point and a subspace in a functional space. Properties and computation of this measure are studied. Numerical examples of static models for parameter estimation and dynamic models for process estimation are given to illustrate our measure.