# Ensemble Modelling for Predicting Fish Mortality +
## 5 Results and Discussion
\[Deaths\_Past_{+3}=\frac{Deaths\_Past_{0}}{\sqrt{\frac{Deaths\_Past_{-6}}{Temp _{-3}}+\frac{Deaths\_Past_{-6}}{VACCINE\_I_{-6}\_VACCINE\_II_{-3}Temp_{-3}}}}\]  
\[Deaths\_Vib_{+3}=MED\_FOOD\_I_{-7}\]  
\[Deaths\_Myx_{+3}=MED\_FOOD\_II_{-7}\]  
\[Deaths\_VNN_{+3}=VACCINE\_I_{0}+MED\_FOOD\_III_{-4}\]  
Some comments on the above mathematical equations are presented in order. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Infectious** & **Final Stage** & **Intermediate** & **ANN Model** & **SR Model MAE** \\
**Disease** & **Enseble** & **Model MAE** & **Model MAE** & **MAE of [11]** & **of [11]** \\ \hline Pasteurella & 52.18 & 58.55 & 60 & 65.1 \\ \hline Vibrio Harveyi & 13.13 & 13.24 & 18 & 20 \\ \hline Myxobacteria & 2.95 & 2.98 & 3 & 9.13 \\ \hline Viral Nervous Necrosis (VNN) & 1.27 & 1.54 & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean Absolute Error (MAE) in the validation dataset of the (two-staged) ensemble models, as well as of the standalone ANN and SR models developed in [11], for each infectious disease. Note that VNN was not studied in [11]. Figure 4: Estimation (in the validation dataset) of the intermediate-stage (**a**) and final-stage (**b**) ensemble models for the number of fish deaths on the 3 rd forthcoming day, due to the considered infectious diseases. * The first equation asserts that the estimated number of fish deaths due to Pasteurella on the 3 rd forthcoming day (\(Deaths\_Past_{+3}\)) is a function of the number of fish deaths due to Pasteurella recorded today (\(Deaths\_Past_{0}\)) and six days ago (\(Deaths\_Past_{-6}\)), of the water temperature recorded three days ago (\(Temp_{-3}\)), as well as of the amount of the vaccines \"VACCINE_I\" and \"VACCINE_II\" administered six and three days ago (\(VACCINE\_I_{-6}\) & \(VACCINE\_I_{-3}\)). * The second expression is a very simple equation, asserting that the estimated number of fish deaths due to Vibrio Harveyi on the 3 rd forthcoming day (\(Deaths\_Vib_{+3}\)) is a function of the amount of the medicated food \"MED_FOOD_I\" administered seven days ago (\(MED\_FOOD\_I\_\)). * The third expression is again a very simple equation, asserting that the estimated number of fish deaths due to Myxobacteria on the 3 rd forthcoming day (\(Deaths\_Myx_{+3}\)) is a function of the amount of the medicated food \"MED_FOOD_II\" administered seven days ago (\(MED\_FOOD\_II\_\)). * Lastly, the fourth equation states that the estimated number of fish deaths due to VNN on the 3 rd forthcoming day (\(Deaths\_VNN_{+3}\)) is a function of the amount of the vaccine \"VACCINE_I\" administered today (\(VACCINE\_I_{0}\)), as well as of the medicated food \"MED_FOOD_III\" administered four days ago (\(MED\_FOOD\_III_{-4}\)). It should be evident that the mathematical equations of the SR models encompassed into the ensemble models, essentially, provide fish farmers with a precise and quantifiable method for describing the development of fish mortalities resulting from infectious diseases. Thereafter, the Decision Trees of the DT models encompassed into the trained final stage ensemble models are concisely presented in Figure 5. Note that, for an efficient presentation, only the root node and the next two levels of each Decision Tree are shown. The depicted Decision Trees exemplify the capability of the ensemble models to tailor their predictive mechanisms to the peculiarities of each pathogen, namely, Pasteurella, Vibrio Harveyi, Myxobacteria, and VNN. The Decision Trees shownin Figure 5 not only furnish a transparent decision-making process but also reveal critical biological and environmental thresholds that significantly influence mortality outcomes. For instance, factors like water temperature and administered amounts of food and medicated food, which are prominently featured in the Decision Tree concerning Pasteurella, are identified as pivotal in the disease's propagation. Such insights are indispensable for aquaculture operators, as they highlight specific conditions under which preventive measures could be most effectively deployed.