## 3 Results and Discussion
### Data Analysis  
#### 3.2.1 Pearson and Spearman Correlation Coefficients  
Pearson moment correlation coefficient and Spearman rank correlation coefficient were calculated to evaluate a possible correlation between the monitored parameters and calculated performances of the thermophilic biological process. Spearman's correlation index S is a non-parametric statistical measure of correlation, it measures the degree of relationship between two variables. The Pearson correlation index P between two statistical variables is an index that expresses a possible linearity relationship between them.  
From the analysis of Pearson's correlation index (Figure 2), a positive correlation between the COD fed at the system and nitrates were pointed out (0.44), This consideration explains the value of 0.36 between the yield of removal nitrates and the yield of removal COD. The COD entering the plant is positively correlated to the amount of oxygen consumed (0.42) and the total nitrogen treated (0.36). CODin and COD are related to oxygen consumption; as the organic substance increases, the amount of oxygen needed to degrade it also increases. It is also possible to note some negative correlations, such as the ammonia entering the plant seeming to negatively influence the total nitrogen and COD abatement yields, suggesting a treatment upstream of the biological process. High concentrations of ammonia entering the TBR can lead to a reduction in total nitrogen removal yields and COD [29].  
[PERSON]'s correlation matrix shows very similar results (Figure 3). Further, in this case, there is a negative correlation between ammonia entering the plant and the reduction yields of removal COD and total nitrogen. We can see that the correlation coefficient passes from the value of \(-\)0.16 in the matrix P to a value \(-\)0.29 in the matrix S. These results confirm the negative correlation between these two parameters, ammonia, and fed COD. It is possible to see a solid positive correlation between the nitrates fed to the system and the total nitrogen fed. This correlation can be explained considering that nitrates are the  
Figure 2: [PERSON]’s correlation matrix.  
most considerable portion of the nitrogen forms treated by the thermophilic biological system. Also, a negative correlation index of total phosphorus removal vs. TS (\(-0.03\)) was pointed out.  
From the analysis of the two correlation matrices, a positive correlation emerges between all the polluting parameters monitored at the entrance to the TBR. This management choice is positively reflected in the performance analysis, shown in Table 1, which shows very high-performance values for all the polluting parameters analyzed except for ammonia nitrogen.  
If Pearson (P) < Spearman (S), this means a monotone correlation, but not a linear one [30]. Results could help in the future in the correct modeling of the observed parameters, giving information on the type of correlation between the data, whether linear or monotone. The negative correlation index between the total phosphorus abatement yields and the total solids concentration increase in the Spearman's matrix means the correlation between the two data is monotone but not linear.  
The results obtained can be compared with the results presented by [PERSON] et al. [19], in which the bacterial species are correlated with the COD and BOD\({}_{5}\) removal yields in our work instead the Spearman and Pearson matrices are made between the diverse environmental parameters of the plant.