Simulating signal noise

The multiple linear regression problem to be solved in fitting FP models can fail for several reasons. Firstly, in the notation of section 2.2.3, if the predictors contain collinearities, then $\mathbf{X}^\prime\mathbf{X}$ is of less than full rank and the matrix inverse in equation 2.67 will not exist, meaning that the regression coefficients will be undetermined. Even if there is a near collinearity, some of the eigenvalues of the matrix inverse will be extremely large and this renders the estimates for the regression coefficients unstable [61]. This danger is avoided by performing a PCA and retaining only those eigenvectors whose eigenvalues exceed an appropriate critical value.

However, the simulated measurements in the database are idealized and free of the errors which are inevitable in experiment. Even a well-conditioned regression problem can produce poor parameter estimates from using the idealized models with noisy real data. Therefore, our estimates of the regression coefficients must take into account the expected experimental noise in the raw measurements which is done by perturbing them prior to carrying out the regression analysis. More precisely, the perturbation of the magnetic data is carried out by firstly performing the PCA to obtain eigenvectors for the noise free case, then adding simulated noise to the raw measurements and recomputing the PC scores with the unperturbed eigenvectors. This stabilizes the recovery by helping to remove small random correlations in the database that might otherwise lead to false inflation of the regression coefficients (see [61]). This procedure has the effect of damping the coefficients of higher order PC combinations as described in [30], and is similar to the use of ridge regressions described in [29].

Examining our situation, the external coil currents are measured accurately (25A resolution for $I_{\mathrm{mod}}, I_{\mathrm{s}}, I_{\mathrm{tor}}$) and are thus effectively exact. The error in the limiter position is also assumed to be negligible. The main source of error will therefore originate in the magnetic data, so these are perturbed by a certain fraction of their standard deviation. Here, we have assumed equal relative errors with a Gaussian distribution throughout. This is a reasonable approximation since the sampling electronics are normally calibrated such that the largest observable signal corresponds to the saturation voltage of the integrator. The experimental noise will thus be the uncertainty in the last few information bits of the digitized value of each signal.