Scalar parameter recovery

We begin with the simplest case of scalar parameters, including global quantities such as $W_{\mathrm{p}}$ and even profile parameters taken at a fixed radial location. In this case, a linear model in the FP predictors is sufficient for a reasonable recovery in some cases, but we note a marked improvement for a second order model. The small improvement in fit obtained by using a cubic model does not justify the much larger model size, thus the optimum balance between accuracy and compactness seems to be a quadratic dependence on the independent variables $X_i$. The improved conditioning of the problem through PCA and inclusion of simulated noise in regressions ensures stability even for parameters which display near-linear behaviour. The model for an arbitrary global parameter $(y)$ on the $N$ independent fitted parameters $X_i$ thus takes the form:

$$y = \mathrm{constant}\ + \ \underbrace{ \sum_{i=1}^N {a_i X_i} }_{\rm {linear}}\ + \ \underbrace{ \sum_{i=1}^N {b_i X_i^2} }_{\rm {quadratic}}$$

$$\mbox{}+ \ \underbrace{ \sum_{i=1,\ j=i+1}^{N} { c_{ij} X_i X_j } }_{\rm {mixed\ second\ degree}}$$ (3.2)

This can be written more succinctly if we define $X_0$ to be unity, as:

$$y = \sum_{i=0 \atop j=i}^{N} {a_{ij} X_i X_j}$$ (3.3)

and there are $(N+1)(N+2)/2$ fitted parameters in the model.