# Gaussian Distribution Model for Detecting Dangerous Operating Conditions in Industrial Fish Farming
## 3 Results
### Data Investigation of the Weaning Phase of the Senegalese Sole Development  
The results presented in Figures 2-4 refer to the pre-fattening phase of the Senegalese sole collected throughout 2019. In order to confirm the viability of our proposal, an identical investigation was carried out on the data referring to the phase of weaning of this fish species collected throughout 2018, given that the mortality is a problematic factor in this phase of Senegalese sole development [35]. This additional analysis is shown in Figures 5-7, in the same structure and sequence as the previous one. The results of this subsection confirm the effectiveness and the promising potential that our Gaussian distribution model offers not only for the fish farming industry, but also for other biological, physical, and chemical fields.  
It is important to emphasize that each fish development phase has different values for the physiochemical monitoring parameters that assess the quality of the water in the fish tanks. Moreover, even with this difference, our mathematical model can successfully reveal regardless of the fish development phase that the system (here represented by the physiochemical parameters) has an internal structure with a very well-defined probabilistic nature as displayed in Figure 5b, which evolves while preserving this characteristic, as shown in Figure 6, and consequently supports the building of a highly precise Gaussian distribution model as exhibited in Figure 7. Furthermore, these results suggest that non-standard variations of a wide range of parameters can be monitored with our method. For example, we can identify when the fish's stress level is high (through cortisol measurements [36; 37]), if the amount of food provided is not adequate for the fish growth, or even the rate of water renewal in the tanks it is not ideal for fish welfare, among other possibilities.  
Figure 2: Distribution of aquaculture data containing an analysis of temperature, salinity, pH, ammoniacal nitrogen (NH\({}_{3}\)-N), nitrogen dioxide (NO\({}_{2}\)), redox, bromine, amount of food provided, and rate of water renewal in the fish hatchery. (**a**) No defined distribution of raw data (all parameters). (**b**) Well-defined Gaussian distribution of the principal components of the PCA. Gaussian fitting with mean \(\mu\) = 0 and standard deviation \(\sigma\) = 0.124.  
Figure 3: Data regarding operating conditions for fish farming production. (**a**) Low mortality rate. Gaussian fitting with \(\mu=0\) and \(\sigma=0.114\). (**b**) Moderate mortality rate. Gaussian fitting with \(\mu=0\) and \(\sigma=0.195\). (**c**) High mortality rate. Gaussian fitting \(\mu=0\) and \(\sigma=0.290\).  
Figure 4: (**a**) Standard deviation of Gaussian distributions versus mortality rate with \(R^{2}=0.999\). (**b**) 2D image visualization when fish production is under normal, warning, or dangerous operating conditions from left to right, respectively.  
Figure 5: Distribution of aquaculture data containing an analysis of temperature, salinity, pH, ammoniacal nitrogen (NH\({}_{3}\)-N), nitrogen dioxide (NO\({}_{2}\)), and transmittance at 400 \(\upmu\)m and 500 \(\upmu\)m. (**a**) No defined distribution. (**b**) Well-defined Gaussian distribution. Gaussian fitting with mean \(\upmu=0\) and standard deviation \(\sigma=0.11\).  
Figure 6: Data regarding operating conditions for fish farming production. (**a**) Low mortality rate. Gaussian fitting with \(\mu=0\) and \(\sigma=0.105\). (**b**) Moderate mortality rate. Gaussian fitting with \(\mu=0\) and \(\sigma=0.127\). (**c**) High mortality rate. Gaussian fitting \(\mu=0\) and \(\sigma=0.252\).