Scoring Functions

A variety of scoring functions may be used to test the fit of a particular predictions of $\hat{Y}_i^g$ with the observed $Y_i$. The simplest scoring is to avoid computing any of predictions, and, recalling equation 5, simply declare that the $Y_i$ with the least magnitude value must reveal the active source. A threshold test could also be applied to make sure the minimum value were sufficiently close to zero. This test can fail, however, in some situations where more than one source is present. If the weighted linear combination $Y_i$ of two or more complex $S$ values in a particular bin happened to add up to a value near zero, the corresponding $Y_i$ function would claim an undeserved victory. A slightly more sophisticated cost function can solve this problem by comparing each $Y_i$ value to that predicted by guess $\hat{Y}_i^g$:

$$f(g) = \ensuremath{\frac{\sum_{i=1}^N \vert\hat{Y}_i^g - Y_i \vert}{\sum_{i=1}^N \vert Y_i\vert }}$$ (6)

which one may interpret as the overall fractional model error when we guess that only source $g$ is active. It is noted that the errors are summed before dividing by the sum of the magnitudes of the $Y_i$ values, because the near-zero values of the ``winning'' $Y_i$ can lead to arbitrarily large errors.