## 3 Results
### Double-Band Regression  
We present here the results for the case where two bands are predicted from the others. This case will, of course, produce higher estimation errors because for each predicted band there is one fewer predictor. Tables 15 and 16 present the results for the L1C and the L2A data. Each row in the tables presents the results for a pair of bands jointly predicted. The same quality metrics as for single-band regression are presented. Each table has 15 rows since we evaluate all possible combinations of pairs of bands. The rows in Table 15 are sorted in increasing order of the maximum relative error of the pair of bands. This allows one to quickly see that only one pair of L1C bands (B06 and B07) can be predicted with less than 3% error, and that another pair (B07 and B8A) is slightly above this threshold. For the L2A data presented in Table 16, the rows are sorted by increasing RMSE using the maximum of the pair in each row. In this case, only the pair (B5, B8A) fulfills the 0.01 error threshold, although the pair (B5, B6) is not much above this threshold. Figure 11 presents the scatterplots for the two best pairs of L1C bands (the two first rows in Table 15. Although the scatterplots are generated by subsampling the test dataset for readability, one can see that the estimations are unbiased and have a small dispersion around the regression lines. One can also see that part of the error comes from pixels with reflectances higher than one, for which there is underestimation. Since the regression algorithm is configured to yield reflectances in the \([-0.2,1.3]\) interval, we can expect that the error in this interval is smaller than what is reported in the tables. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Band** & \(\sigma\) **(68.27\%)** & **2\(\sigma\) (95.45\%)** & **3\(\sigma\) (99.73\%)** \\ \hline B05 & 68.80 & 92.26 & 98.29 \\ B06 & 70.23 & 93.14 & 98.53 \\ B07 & 70.82 & 93.27 & 98.43 \\ B8A & 70.36 & 91.99 & 97.57 \\ B11 & 56.86 & 84.11 & 94.94 \\ B12 & 61.57 & 87.50 & 96.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Probability of the absolute error being lower than \(n\times\sigma\) (L1C). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Band** & \(\sigma\) **(68.27\%)** & **2\(\sigma\) (95.45\%)** & **3\(\sigma\) (99.73\%)** \\ \hline B5 & 65.68 & 90.00 & 96.87 \\ B6 & 71.84 & 93.85 & 98.70 \\ B7 & 74.28 & 94.50 & 98.69 \\ B8A & 70.07 & 92.13 & 97.93 \\ B11 & 56.18 & 81.51 & 93.04 \\ B12 & 64.70 & 91.15 & 97.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Probability of the absolute error being lower than \(n\times\sigma\) (L2A). For L2A data, Figure 12 presents the scatterplots for the pairs of bands in the three first rows of Table 16. As with the L1C case, the scatterplots show unbiased estimations with small dispersions, except for the B12 band in the third pair. The random sample of the test set used for generating these scatterplots does not contain pixels showing the underestimation of reflectances higher than one, but they also exist. It is difficult to explain these results. First of all, the pairs of bands that are predicted the _best_ differ between L1C and L2A. This was already the case for the regression of a single band, but in that case we could clearly define two groups: red edge-NIR and SWIR. In the case of two bands, one could have expected that for a pair of bands to be correctly reconstructed, they would have to be nonadjacent so that the missing information could be reconstructed using the neighboring bands. However, we see that the best pair in L1C is (B06, B07) and that the second best pair in L2C is (B5, B6). With the same kind of reasoning, one could have expected that the pair (B11, B12) should be the one with the largest errors, since reconstructing the SWIR bands using only the VIS-NIR range should be nearly impossible. This is the case in terms of relative error, but not in terms of RMSE, which makes SWIR a better candidate for L2A reconstruction than more spectrally distant pairs.