## 4 Results
### Fitting Distribution to the Dataset
For validation purposes, we elaborated a testing environment for verifying the distribution results. There are various tests available to check if a sample conforms to a given distribution. In our study, we used six different techniques to test our null hypothesis. The null hypothesis is that X comes from Gaussian distribution. The results of the tests are presented in Table 3. We see that 5 out of 6 tests rejected the null hypothesis, so now we can say with confidence that the distribution is not Gaussian. Based on the results, we needed to find another distribution that describes more accurately the execution times. We used the scipy package in Python that has more than 100 different distributions implemented. After testing every distribution by fitting a curve to our histogram with the lowest SSE, we kept the five best results and found that the Johnson's SU distribution [40] produced a relatively low SSE (111.51). Its PDF can be seen in Equation (10). \[\frac{\delta}{\lambda\sqrt{2\pi}}\frac{1}{\sqrt{1+\left(\frac{\chi-\xi}{ \lambda}\right)^{2}}}e^{-\frac{1}{2}\left(\gamma+\delta\sinh^{-1}\left(\frac{ \chi-\xi}{\lambda}\right)\right)^{2}},\text{ {where }}\gamma,\xi\in\mathbb{R}\text{ and }\delta,\text{ }\lambda\in \mathbb{R}^{+} \tag{10}\]  
To validate whether it describes our execution time histogram more accurately than the Gaussian PDF, we executed the GA, PSO and SA algorithms again to find the best parameter values producing the lowest possible SSE. The [PERSON]'s SU PDF has four parameters as follows:  
* \(\gamma\) controls the flattening of the tail and it must be a real number: if \(\gamma>0\) then it flattens the left tail, if \(\gamma<0\), it flattens the right tail
* \(\delta\) adjusts the width and it must be a positive real number: the smaller \(\delta\), the wider the function
* \(\xi\) sets the position of the function on axis-X and it must be a real number: if \(\xi>0\), then the function is moved to a more positive domain, if \(\xi<0\), then the function is moved to a more negative domain
* \(\lambda\) controls the apex and and it must be a positive real number: if \(\lambda\) is higher, then the apex is moved to the positive domain, if \(\lambda\) is lower, then the apex is moved to the negative direction  
In Table 4, we collected the results of the algorithms. GA found four parameters with 15.4543 SSE, PSO produced moderately different parameters with lower 15.4407 SSE and unfortunately SA worked with an outlier 53.9464 SSE that is definitely the worst result. The results were drawn back to the original histogram and it is seen that the [PERSON]'s SU PDF approximates slightly better the long-tail feature. Figure 9 depicts the histogram and the calculated PDFs. \begin{table}
\begin{tabular}{c c} \hline
**Test** & **Result** \\ \hline D’Agostino K2 test & rejected \\ Shapiro-Wilk test & rejected \\ Anderson-Darling test & rejected \\ CHI2 test & not rejected \\ Jarque-Bera test & rejected \\ Kolmogorov-Smirnov test & rejected \\ \hline \end{tabular}
\end{table} 
Table 3: Results of null hypothesis tests for Gaussian distribution. Using the distribution functions, we generated random samples for the execution time of the first algorithm. In the case of the second algorithm, the idea is identical with that presented here. The histogram can be seen in Figure 10.