# Gaussian Distribution Model for Detecting Dangerous Operating Conditions in Industrial Fish Farming
## 4 Discussion  
Our Gaussian distribution model provides three possible working environments--normal, warning, and dangerous, and to differentiate between them, we adopted the fish mortality rate as a demarcation criterion. The choice of this parameter as a separation boundary is based on two fundamental justifications. First, mortality is one of the main concerns of fish farmers in several countries, as we have already covered in the introduction section. Second, various physical parameters such as temperature, pH, nitrite, ammonia, salinity, alkalinity, bromine, among others, are measured daily, and a dangerous health condition for fish does not occur with a change in a single of these parameters, as changing one alters others too. Therefore, an inappropriate situation occurs when all of them assume a specific combination of values that originate an unsuitable environment for the health and welfare of the fish, making them sick and consequently die. In other words, there is a  
Figure 7: (**a**) Standard deviation of Gaussian distributions versus mortality rate with \(R^{2}=0.997\). (**b**) 2D image visualization method when fish production is under normal, warning, or dangerous operating conditions from left to right, respectively.  
direct relationship between the general change in all parameters that assess water quality in fish hatchery with mortality. A deeper discussion about the correlation between the gathered data and the fish mortality can be found in our previous work [19]. However, it is necessary to stress here that the proposal specified here takes into account the parameters measured by the intensive aquaculture facilities from Safiestela SA. (from SEA8 group), in Portugal. In this way, the results may differ slightly from fish farming facilities that measures other parameters than those described in this research.  
A comparison between the results obtained with the data analysis from the prefattening and weaning phases of Sengealese sole reveals some points worth mentioning. Initially, Figure 5b confirms that after applying the PCA, the resulting principal components have a very well-defined Gaussian distribution. Second, when comparing Figures 4b and 7b, we perceive that the normal and dangerous operating conditions for these two phases of Sengealese sole development are very similar to each other, so that only the warning condition presents a slight discrepancy (small difference in the Gaussian distribution) produced by the fact that in each of these analyzes the monitoring parameters are not exactly the same. Furthermore, it is not our intention to determine the fish mortality rate. The purpose of the analysis presented in Figures 4a and 7a is only to demonstrate that we can use the mortality rate as a criterion indicating a dangerous situation for fish. Besides, the practical use of our method does not require any additional instrumentation costs, as this is based on the statistical analysis of physical parameters that are already routinely measured. The application of the PCA to the monitoring data before looking for a standard probability distribution that describes the data is also one of the strengths of our approach. PCA is a multivariate statistical data technique that has already been successfully applied in several scientific fields to assist in the detection and interpretation of information hidden in the data. Specifically in our case, with the PCA we were able to determine a Gaussian distribution pattern that was previously not noticeable in the raw data.