Profile (1D) parameter recovery

One approach for modelling profile parameters (those with a radial variation, such as flux surface effective minor radius ${r_{\mathrm{eff}}}(s)$) is to treat them as scalar variables and attempt to recover them at many fixed $s$ values along the profile. However, we prefer to exploit the fact that their radial dependence is relatively smooth and can be well described by a polynomial of degree $M$ in $s$. This leads us to apply a model of the form:

$$y = \sum_{k=0}^M { \sum_{i=0 \atop j=i}^{N} {a_{ijk} X_i X_j \rho^k} },$$ (3.4)

where $\rho = \sqrt{s}$ is a quantity which varies like the flux surface effective radius. This model is fitted as a single regression over all radial positions and thus yields a global function for each recovered profile variable that describes their behaviour for all configurations and at all radii.