Parametric scaling of mathematical models

Parametric scaling, the process of extrapolation of a modelling result to new parametric conditions, is often required in model optimization, and can be important if the effects of parametric uncertainty on model predictions are to be quantified. Knowledge of the functional relationship between the model solution (y) and the system parameters (α) may also provide insight into the physical system underlying the model. This paper examines strategies for parametric scaling, assuming that only the nominal model solution y(α) and the associated parametric sensitivity coefficients (∂y/∂α, ∂²y/∂α², etc.) are known. The truncated Taylor series is shown to be a poor choice for parametric scaling, when y has known bounds. Alternate formulae are proposed which 'build-in' the constraints on y, thus expanding the parametric region in which the extrapolation may be valid. In the case where y has a temporal as well as a parametric dependence, the extrapolation may be further improved by removing from the Taylor series coefficients the 'secular' components, which refer to changes in the time scale of y(t), not to changes in y as a function of α.