How to select polynomial models with an accurate derivative

Polynomial models, estimated from noisy observations, can give an accurate description, of the data while at the same time the derivative of the estimated polynomial can be a poor model for the true derivative of the process that generated the data. This derivative is used for example for linearisation with a Taylor expansion in control theory and in extended Kalman filtering. The explanation for the strong degradation of the derivative of the best selected data model is straightforward. Estimating polynomial models of increasing order from the data gives not only a description of the true underlying process, but also of the accidental realisation of the additive noise. Therefore, high order models that crinkle around the true process will mostly have an irregular derivative. Models with an accurate derivative can be selected by using a higher penalty factor in the selection criterion than the usual factor two for every additional regressor. A distinction has to be made between hierarchical models with a prefixed sequence in which variables are entered into the model and subset models where the sequence is free and each regressor may be the next candidate