Formulating the geometric information with the Fourier coefficients

Figure 5.4: First 50 PCA eigenvalues of the 23 skeleton Fourier coefficients [mm] at 10 $s$-values

We first attempted to formulate the geometric information in terms of principle components of the skeleton set of Fourier coefficients taken at certain points on $s$ from boundary to axis. Given the 23 Fourier coefficients on each of (say) 10 flux surfaces, we are including almost as many variables as there are observations in the database. This is questionable from a statistical point of view, as the pool of variables from which we choose our predictors (i.e. the PCs) should be comfortably smaller than the size of the database. For the purposes of illustration, however, we proceed. The eigenvalues from a covariance-based^5.1 PCA of the 210 (the $m>1$ coefficients vanish for $s=0$, yielding a total of $23 \times 10 - 20 = 210$) variables is shown in figure 5.4, revealing that somewhere in the region of 30 PCs are significant. Note that here, we are loath to discard higher order PCs since the pressure may very well depend on very small variations in the geometry. For the same reason, we do not include noise in the predictors. We cannot retain all 30 PCs and employ a quadratic model for recovering the pressure, since this requires 496 cross-combinations (or $(n+1)(n+2)/2$ where $n$=30) which is greater than the number of cases in the database.

Figure 5.5: Pressure recovery RMSE about the database mean profile with one standard deviation lines (left axis) and percentage error $\varepsilon$ indicated as crosses (right axis) for a linear model with 20 PCs of the skeleton set of Fourier coefficients
Figure 5.6: Pressure recovery RMSE about the database mean profile with one standard deviation lines (left axis) and percentage error $\varepsilon$ indicated as crosses (right axis) for a linear model with 30 PCs of the skeleton set of Fourier coefficients

Attempting the $p(s)$ recovery with a linear model in the PCs works reasonably well for the centre but deteriorates rapidly towards the boundary, as is shown in figures 5.5 and 5.6 for linear models with 20 and 30 PCs, respectively. Our remarks in section 5.1 regarding the profile regularization apply here also. However, we also observe that, on average, the greatest pressure-induced changes to the geometry are in the plasma core. Thus, when given this geometric information, it is more straightforward to diagnose the pressure here than towards the edge, where the geometrical deformations are much weaker. Indeed the RMSE remains constant to within a factor of 1.5 across most of the profile. It may be that including the Fourier coefficients at more surfaces towards the edge would improve the situation, however we have no chance to test this due to the constraints on the maximum allowable numbers of variables in the PCA imposed by the database size. We also tried a fully quadratic model with fewer PCs (we can use up to 18 whilst maintaining a 1:2 ratio of degrees of freedom in the model to observation in the database). This gave results which were far inferior to the linear model, indicating that the database is not large enough to allow the construction of an adequate model using the skeleton Fourier coefficients. In conclusion, although $p(s)$ is well predicted towards the core, the results over the outer portion of the plasma are less than convincing.