How to select polynomial models with an accurate derivative

Abstract

Derivatives of estimated static relations are often used for linearization in control and in extended Kalman filtering. However, the structure of selected models may only be an approximation to the true relationship, which can cause problems in taking derivatives. Polynomial models, estimated from noisy observations, may give accurate descriptions of the data while at the same time their derivatives may be poor approximations of the true derivative. The explanation of the strong degradation of the derivative of selected models is straightforward: estimating polynomial models of increasing order from a set of data gives not only a description of the true underlying process, but also of the accidental realization of the additive noise. The higher order polynomial models will crinkle around the true process; therefore, they will mostly have an irregular derivative. Models with a better derivative can be selected by using a higher penalty factor in the selection criterion.