FP results

We divide results into appropriate sections according to the predictors used. Each recovered quantity is shown with its database mean value, spread, root-mean-square error (RMSE) and the $\mathrm{R}^2$ or $\varepsilon$ statistic adjusted for the degrees of freedom in the model, which we now define. For a given variable with spread $\sigma$, we define the sum-of-square error (SSE) for a model fitted over $n$ observations with $p$ fitted variables to be the sum of squared differences between actual and recovered values. Then the RMSE = $\sqrt{\mbox{SSE}/(n-p)}$ and $\mathrm{R}^2= 1 - \mbox{RMSE}^2/\sigma^2$. The percentage error $\varepsilon$ is defined as 100$\times$(RMSE/$\sigma)^2$ or equivalently 100 $\times\sqrt{1 - \mathrm{R}^2}$.

We remark that all of the above statistical quantities are adjusted for the number of parameters in the model. Unlike their uncorrected analogues, which generally improve monotonically with each new parameter added to a model since they depend only on the absolute explained variance, these are not artificially enhanced by an overfitted model (i.e. a model containing many predictors which have no explanatory worth). Indeed, adding parameters of no predictive value to a model can often cause a deterioration in the corrected quantities. Note that the physically relevant quantity is actually the RMSE, however statistics such as $\mathrm{R}^2$ and $\varepsilon$ render the comparison of results for parameters of different dimensions and widely differing spreads to be made on an equal footing. Also, $\varepsilon$ is sometimes a more convenient measure than $\mathrm{R}^2$, as the latter can often be very close to unity for a good fit and the former allows greater information resolution.