## 2 Materials and Methods
### Statistical Analysis of Input Attributes  
A simple statistical analysis was carried out in order to obtain an insight about the main characteristics of the dataset and potential correlations or dimensional reduction. We calculated the mean values (\"avg\") and the standard deviations (\"std\") for each MME, for a given material label. We introduced the coefficients of variation, a ratio matrix, \(CV\), between the standard deviation and the corresponding mean value. These parameters are used to define whether an attribute brings relevant information or not [34]. The \(CV\) entry \((i,j)\) of a MME \(M_{ij}\), for a given material \(m\) and an Aol \(\alpha\) is given by  
\[CV_{ij}^{m,\alpha}=log_{10}\frac{std(M_{ij}^{m,\alpha})}{|avg(M_{ij}^{m, \alpha})|}\,. \tag{3}\]  
Positive \(CV\) values indicate uncertainty, which means that the information entry may not be conclusive. On the contrary, a high signal-to-noise ratio corresponds to a negative \(CV\) value, which can be used as a criterion to select the most relevant Muller matrix elements for classification. Figure 2 shows the \(CV\) parameters for four MMEs, for each of the 7 classes studied.  
From Figure 2 we can see that the above criterion is always fulfilled for \(M_{11}\). It also applies to the other diagonal MMEs, which means that they are the most robust attributes for classification algorithms. As expected, the absolute values of \(CV<0\) are maximal for the most regular objects (samples of car paints and traffic signs) and they peak at \(\alpha=0\), which corresponds to specular reflection. On the other hand, the criterion \(CV<0\) is almost  
Figure 2: Coefficients of variation for Mueller matrix elements \(M_{11}\) (**top left**), \(M_{02}\) (**top right**), \(M_{01}\) (**bottom left**), and \(M_{10}\) (**bottom right**) versus the angle of incidence for seven classes of materials/objects.  
never fulfilled in the case of \(M_{02}\), which means that considering \(M_{02}\) as an attribute for a ML algorithm will probably not provide much reliable additional information. Other non-diagonal components have also been analyzed using this criterion and, generally, appear less relevant than the diagonal ones. This can be due to the fact that their mean values (for some classes of materials) are close to zero because these materials almost do not polarize, diattenuate, or introduce retardance. However, the case of \(M_{01}\) and \(M_{10}\) shown in the bottom panels of Figure 2 is intermediate, since most of the materials partially convert linearly polarized light into orthogonal polarization, as exemplified by the well-known cases of metal and dielectric spheres [35].  
By analyzing the full set of 15 normalized MMEs, we verified that the attributes that present the best \(CV\) ratios are \(M_{11}\), \(M_{22}\), \(M_{33}\), \(M_{01}\), and \(M_{10}\), and these can be regarded as the most relevant attributes. Even though one should theoretically expect \(M_{01}=M_{10}\) under backscattering conditions by virtue of the reciprocity theorem [6], in practice they are not equal (as are different their statistical \(CV\)-s in Figure 2), in part because of deviations from the strict backscattering conditions even in laboratory measurements, beyond the inevitable measurement errors. Moreover, as mentioned above, retaining more information, despite its partial redundancy and low signal-to-noise ratio characteristic of some measured MMEs, may be helpful in ML algorithms. We will show evidence of this in the next section, although it should be seen from the cost/benefit point of view.