# A Learning-Based Methodology to Optimally Fit Short-Term Wind-Energy Bands
## 4 Combining Forecasts  
At first sight, we might think that a convex combination of forecasts and their bands would inherit the width of each one, and that we cannot improve bands quality by means of combining them. The only mechanism we have seen that can get narrower bands goes by reducing \(\lambda\). As a drawback, this also increases the percentage of anomalous days. However, we might regain confidence if anomalous days -for the different forecasts- were somehow independent, since a combination of anomalous situations in all bands would be rarer than in any of them by separate. That's the idea behind this section. To check the consistency of this idea we analyze how independent anomalous days are, by its correlation matrices. Table 3 recapitulates figures of anomalous days for different values of \(\lambda\) with \(\theta=0.035\). These numbers were computed over the test-set for [PERSON]'s random variables, \(Mt(d)\), \(Gt(d)\) and \(Pt(d)\), indicator of the event of finding an anomalous day: they evaluate to 1 (respectively 0) if and only if the forecast for day \(d\) of the respective corresponding -Metologica, Garrad Hassan and PSF44- classifies as anomalous (resp. regular). From these correlation values, we infer that anomalous days of Metologica and Garrad Hassan are positively but weakly correlated. By running simple simulations with two sets of dependent [PERSON]'s random variables with the same expected value, we observe that the correlations values of the table appeared when 1 out of between 3 to 4 samples of one set copy the value \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **MT** & **GH** & **PS** & **MT** & **GH** & **PS** & **MT** & **GH** & **PS** \\ \hline
**MT** & 1.0000 & 0.2621 & \(-\)0.0178 & **MT** & 1.0000 & 0.2854 & 0.0147 & **MT** & 1.0000 & 0.2797 & \(-\)0.0093 \\ \hline
**GH** & 0.2621 & 1.0000 & \(-\)0.0648 & **GH** & 0.2854 & 1.0000 & \(-\)0.0720 & **GH** & 0.2797 & 1.0000 & \(-\)0.0308 \\ \hline
**PS** & \(-\)0.0178 & \(-\)0.0648 & 1.0000 & **PS** & 0.0147 & \(-\)0.0720 & 1.0000 & **PS** & \(-\)0.0093 & \(-\)0.0308 & 1.0000 \\ \hline  & \multicolumn{4}{c}{\(\lambda=0.60\)} & \multicolumn{4}{c}{\(\lambda=0.65\)} & \multicolumn{4}{c}{\(\lambda=0.70\)} \\ \hline  & **MT** & **GH** & **PS** & **MT** & **GH** & **PS** & **MT** & **GH** & **PS** \\ \hline
**MT** & 1.0000 & 0.2450 & 0.1127 & **MT** & 1.0000 & 0.2488 & 0.0581 & **MT** & 1.0000 & 0.2507 & 0.0109 \\ \hline
**GH** & 0.2450 & 1.0000 & 0.0817 & **GH** & 0.2488 & 1.0000 & 0.0810 & **GH** & 0.2507 & 1.0000 & 0.0929 \\ \hline
**PS** & 0.1127 & 0.0817 & 1.0000 & **PS** & 0.0581 & 0.0810 & 1.0000 & **PS** & 0.0109 & 0.0929 & 1.0000 \\ \hline  & \multicolumn{4}{c}{\(\lambda=0.75\)} & \multicolumn{4}{c}{\(\lambda=0.80\)} & \multicolumn{4}{c}{\(\lambda=0.85\)} & \multicolumn{4}{c}{\(\lambda=0.85\)} \\ \hline  & **MT** & **GH** & **PS** & **MT** & **GH** & **PS** & **MT** & **GH** & **PS** \\ \hline
**MT** & 1.0000 & 0.3246 & 0.0262 & **MT** & 1.0000 & 0.2830 & \(-\)0.0926 & **MT** & 1.0000 & 0.2921 & \(-\)0.0707 \\ \hline
**GH** & 0.3246 & 1.0000 & 0.1119 & **GH** & 0.2830 & 1.0000 & 0.0281 & **GH** & 0.2921 & 1.0000 & 0.0243 \\ \hline
**PS** & 0.0262 & 0.1119 & 1.0000 & **PS** & \(-\)0.0926 & 0.0281 & 1.0000 & **PS** & \(-\)0.0707 & 0.0243 & 1.0000 \\ \hline  & \multicolumn{4}{c}{\(\lambda=0.9\)} & \multicolumn{4}{c}{\(\lambda=0.95\)} & \multicolumn{4}{c}{\(\lambda=1.00\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Correlation matrices for anomalous days for different \(\lambda\)’s and \(\theta=0.035\). of the other. PSF44 is basically independent of the others providers, and in fact can either be positively or negatively correlated with them.