# Characterization and Separation of Traditional and Bio-Plastics by Hyperspectral Devices
## 3 Results and Discussion
### Tools for Determining Spectral Indices Useful for Polymers' Separation
#### 3.1.2 Correlation Matrix  
The correlation between two random variables is a measure of their dependence. The correlation value is -1 when the variables are adversely correlated, whereas it is 1 when the maximum positive correlation is achieved and 0 when the variables are uncorrelated. In the contest of hyperspectral investigations, the computation and analysis of correlation matrices of spectral signatures make it possible to reduce the dataset dimensionality by providing a set of self-consisting bands and to ignore redundant and useless information for separating couples of polymers. This in turn allows the detection of feature wavelengths for the construction of robust hyperspectral indices. Each couple of polymers is considered for the computation of correlation matrices, i.e. :  
\(\bullet\) PET_V - PLA_V  
\(\bullet\) PET_V - PS_V  
\(\bullet\) PLA_V - PS_V  
From the hyperspectral cube of each sample, 3000 hyperspectral signatures are randomly extracted from the ROI defined for each sample. The correlation matrix, R(\(\lambda_{i}\), \(\lambda_{j}\)), is computed as follows:  
\[R\Big{(}\lambda_{i},\lambda_{j}\Big{)}=\sum_{n}\frac{\Big{(}\rho(\lambda_{i})- \overline{\rho(\lambda_{i})}\Big{)}\Bigg{(}\rho\Big{(}\lambda_{j}\Big{)}- \overline{\rho\Big{(}\lambda_{j}\Big{)}}\Bigg{)}}{\sigma_{\rho(\lambda_{i})} \sigma_{\rho(\lambda_{j})}}. \tag{2}\]  
where \(\rho(\lambda_{k})\) is the reflectance at the generic wavelength \(\lambda_{k}\), \(\overline{\rho(\lambda_{k})}\) the mean reflectance at \(\lambda_{k}\) and \(n\) the number of hyperspectral signatures employed to compute the correlation matrix. Figure 8 shows the 2-D correlation matrices for the polymer pairs PET-PLA, PET-PS and PLA-PS. The elements of the matrix represent the R\({}^{2}\) value for each pair of wavelengths. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**n** & **Sample Name** & \(\lambda\) **(nm)** & \(\overline{CRR_{A}}\) & \(\sigma_{A}\) & **BD\({}_{\lambda}\)** \\ \hline \multirow{3}{*}{3648} & \multirow{3}{*}{PET\_V} & 1130 & 0.685 & 0.038 & 0.315 \\  & & 1170 & 0.760 & 0.032 & 0.240 \\  & & 1420 & 0.666 & 0.053 & 0.334 \\  & & **1660** & 0.492 & 0.075 & **0.508** \\ \hline \multirow{3}{*}{6364} & \multirow{3}{*}{PLA\_V} & **1170** & 0.491 & 0.051 & **0.509** \\  & & 1380 & 0.548 & 0.056 & 0.452 \\ \cline{1-1}  & & 1410 & 0.575 & 0.061 & 0.425 \\ \cline{1-1}  & & 1670 & 0.512 & 0.092 & 0.488 \\ \hline \multirow{3}{*}{3456} & \multirow{3}{*}{PS\_V} & **1150** & 0.478 & 0.055 & **0.522** \\  & & 1210 & 0.735 & 0.070 & 0.265 \\ \cline{1-1}  & & 1420 & 0.775 & 0.075 & 0.225 \\ \cline{1-1}  & & 1670 & 0.519 & 0.083 & 0.481 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Band Depth (BD\({}_{\lambda}\)) of virgin material samples, computed from spectral signatures processed with the CR method (n is the number of spectral signatures per sample, \(\lambda\) the absorption band wavelength, CRR the average Continuum Removal Reflectance, \(\sigma\) the standard deviation, BD the Band Depth). The most evident absorption peaks are bolded. The smaller values within each matrix identify the pairs of wavelengths characterized by a low correlation between the reflectance values. The elements of the correlation matrix with values close to zero are represented in blue whereas values close to one are represented in red. Table 3 shows the wavelength pairs identified via the correlation matrices and the related values of R\({}^{2}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Polymer Pair** & **Wavelength Pairs (nm)** & **R\({}^{2}\)** \\ \hline \multirow{3}{*}{PET-PLA} & 1120–1170 & \(-\)0.055 \\  & 1120–1370 & \(-\)0.030 \\  & 1370–1650 & \(-\)0.483 \\  & 1170–1650 & \(-\)0.566 \\ \hline \multirow{3}{*}{PET-PS} & 1120–1150 & \(-\)0.035 \\  & 1150–1660 & \(-\)0.198 \\  & 1210–1660 & 0.006 \\ \hline \multirow{3}{*}{PLA-PS} & 1430–1140 & 0.058 \\  & 1160–1140 & 0.164 \\ \cline{1-1}  & 1340–1640 & 0.017 \\ \cline{1-1}  & 1470–1640 & 0.011 \\ \hline \hline \end{tabular}
\end{table}