## 3 Results and Discussion
### Data Analysis
#### 3.2.2 Multivariate Linear Regression  
Tables 2 and 3 show the matrices of coefficients of a set of multiple linear regressions, where each parameter is seen as a function of the other parameters monitored by the plant. It was decided to differentiate the parameters between two diverse groups: (i) fed loads of each polluting chemical parameter, and (ii) the yields of removal together with process parameters.  
Figure 3: [PERSON]’s correlation matrix.  
From Table 2 it is possible to highlight that Plot and N-NH4\({}^{+}\) removal yields were never used as a parameter useful for modeling, and Plot was not the result of the linear regression of any other parameter.  
Figure 4 shows the comparisons between the values measured in the plant and the values estimated by the multiple linear regressions. Thanks to these comparisons, it is possible to appreciate the excellent goodness-of-fit of the regression models. In particular, the yield value of removal COD and nitrogen forms, apart from ammonia, is well explained by the linear regression against the other monitored parameters. In regard to the COD entering the plant and the COD efficiency, the relationships between measured and estimated values have an angular coefficient of 1.059 and 1.003 with an R\({}^{2}\) of 0.98 and 0.99, respectively.  
Similar results were obtained by observing the graphs of the input and the reduction of nitrates. In this case the slope of the interpolating line was 0.923 and 0.982 associated with R\({}^{2}\) values of 0.928 and 0.996, respectively. Also, in this case, it is possible to state that the multiple linear regression model allows estimating the data with an excellent degree of precision. In the case of N-NO\({}_{\chi}\), however, the model tends to slightly underestimate the predicted value compared to the observed one.  
This type of analysis allows the AW utility manager to predict the number of certain pollutants or performances of the thermophilic biological system in presence of a limited amount of data.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **COD IN** & **N-NO\({}_{\chi}\) IN** & **N-NH4+IN** & **Not IN** & **Plot IN** & \(\alpha_{i}\) \\ \hline COD IN & - & 0.2927 & 0.1762 & - & 0.1119 & 0.4 \\ N-NO\({}_{\chi}\) IN & 0.4199 & - & - & 0.5212 & - & 0 \\ N-NH4+IN & 0.1532 & - & - & 0.7021 & - & 0 \\ Ntot IN & - & 0.2262 & 0.3015 & - & - & 0.02 \\
**Plot IN** & 0.3595 & - & - & -0.2545 & - & 0.4 \\ \hline \end{tabular}
\end{table}
Table 2: Coefficients of multivariate linear regressions of input loads. Rows are indicative of the modeled parameters, while the columns represent coefficients of the model. \(\alpha_{i}\), intercept value with ordinate axis.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(\mu\) COD & \(\mu\) N-NO\({}_{\chi}\) & \(\Delta\) N-NH4\({}^{+}\) & \(\mu\)**Not** & \(\mu\)**Plot** & \(\mathbf{Co_{2}}\) & **TS** & **VS** & \(\alpha_{i}\) \\ \hline \(\mu\)**COD** & - & 0.2722 & - & 0.1858 & - & 0.1563 & 0.1047 & - & 0.31 \\ \(\mu\)**N-NO\({}_{\chi}\)** & 0.3223 & - & - & - & - & - & -0.0994 & 0.73 \\ \(\Delta\)**N-NH4\({}^{+}\)** & - & - & - & 1.3883 & - & & & 0 \\ \(\mu\)**Not** & 0.735 & - & - & - & - & \(-0.1661\) & - & - & 0.16 \\ \(\mu\)**Plot** & - & - & - & - & - & - & - & - & 0.55 \\ \(\mathbf{Co_{2}}\) & 0.3959 & - & - & \(-0.1039\) & - & - & - & - & 0.09 \\
**TS** & 0.147 & - & - & - & - & - & - & 0.3268 & 0.43 \\
**VS** & - & 0.0493 & - & - & - & - & 0.2711 & - & 0.13 \\ \hline \end{tabular}
\end{table}
Table 3: Coefficients of multivariate linear regressions of yields and physical process parameters. Rows are indicative of the modeled parameters, while the columns represent coefficients of the model. \(\alpha_{i}\), intercept value with ordinate axis.  
Figure 4: _Cont._