## 5 Experiments
### Mathematical Functions  
There are several types of benchmark mathematical function; in this case, the functions chosen for the experiment are summarized in Table 3.  
The parameters used for the experimentation are the following: 100 individuals or harmonies, 30 executions, 2000 iterations or generations, and \(10,50\) dimensions for most of the functions, except for the _Power sum_ function that only uses 4 dimensions. The statistical test that is used is the Z-test that is based on Equation (24), and the parameters used for this test are an alpha of 0.05, a level of confidence of 95%, and a sample size of 30. The main goal is to verify that by using the methods for dynamic parameter adjustment with Shadowed Type-2, we can obtain a better result with respect to the original methods for all values lower than \(-1.645\):  
\[Z=\ \frac{\left(\overline{X}_{1}-\overline{X}_{2}\right)-\left(\mu_{1}-\mu_{2} \right)}{\left(\sigma_{1}-\sigma_{2}\right)}. \tag{24}\]  
Thirty experiments were performed using the HS, DE, ST2 FHS, and ST2 FDE methods. Tables 4-12 summarize the results obtained for the HS and ST2 FHS methods, and Tables 13-20 show the results obtained for the DE and ST2 FDE methods. In these tables, the best, worst, average, standard deviation, and Z-values obtained for each mathematical function are presented. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dimension** & & **10** & **Z-Value** & & **50** & **Z-Value** \\ \hline Method & HS & ST2 FHS & & HS & ST2 FHS \\ Best & \(3.80\times 10^{-1}\) & \(9.76\times 10^{-4}\) & & \(5.89\times 10^{-1}\) & \(1.90\times 10^{-3}\) & \\ Worst & \(1.26\times 10\) & \(3.50\times 10^{-3}\) & \(-16.78\) & \(1.02\times 10\) & \(3.00\times 10^{-3}\) & \(-52.09\) \\ Average & \(7.07\times 10^{-1}\) & \(2.02\times 10^{-3}\) & & \(8.10\times 10^{-1}\) & \(2.28\times 10^{-3}\) & \\ SD & \(2.30\times 10^{-1}\) & \(5.19\times 10^{-4}\) & & \(8.48\times 10^{-2}\) & \(2.67\times 10^{-4}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results for the Sum Squares function. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Search** & **f min** & **Equation** \\ \hline Sum Squares & [\(-10\), \(10\)]n & 0 & \(f(x)=\frac{n}{i-1}\,ix_{i}^{2}\) \\ Trid & [\(-100\), \(100\)]n & \(-200\) & \(f(x)=\frac{n}{i-1}\,(x_{i}-1)^{2}-\sum\limits_{i=2}^{n}x_{i}x_{i-1}\) \\ [PERSON] & [\(-5\), \(10\)]n & 0 & \(f(x)=\sum\limits_{i=1}^{n}x_{i}^{2}+\left(\frac{n}{i+1}\,0.5 ix_{i}\right)+ \left(\frac{n}{i-1}\,0.5 ix_{i}\right)^{4}\) \\ [PERSON] & [\(-15\), \(30\)]n & 0 & \(f(x)=a\cdot exp\left(-b\times\sqrt{\frac{1}{\mu}\,\frac{n}{i-1}\,\cos(cx_{i})} \right)+a+exp(1)\) \\ [PERSON] \& [PERSON] & [\(-10\), \(10\)]n & 0 & \(f(x)=(x_{1}-1)^{2}+\sum\limits_{i=2}^{n}(2x_{i}^{2}-x_{i-1})^{2}\) \\ [PERSON] & [\(-10\), \(10\)]n & 0 & \(f(x)=\sin^{2}(\pi w_{1})+\sum\limits_{i=1}^{n-1}\left((w_{i}-1)^{2}\left[1+10 \sin^{2}(\pi w_{i}+1)\right]+\right.\) \\  & & & \((w_{n}-1)^{2}\left[1+\sin^{2}\left(2\pi w_{n}\right)\right]\), _where_ \(w_{i}\) \\  & & & \(=1+\frac{w_{i}-1}{n}\), _for all_ \(i=1,\ldots,n\) \\ [PERSON] & [\(-600\), \(600\)]n & 0 & \(f(x)=\frac{1}{400}\,\sum\limits_{i=1}^{n}\chi_{i}^{2}-\sum\limits_{i=1}^{n} \cos\left(\frac{x_{i}}{\sqrt{i}}\right)+1\) \\ [PERSON] & [\(-4\), \(5\)]n & 0 & \(f(x)=\sum\limits_{i=1}^{n/4}[(x_{i-3}+10x_{i+2})^{2}+5(x_{i-1}-x_{i})^{2}+ \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dimension** & **10** & **Z-Value** & \multicolumn{2}{c}{**50**} & **Z-Value** \\ \hline Method & HS & ST2 FHS & HS & ST2 FHS & \\ Best & \(7.34\times 10^{-11}\) & \(4.00\times 10^{-13}\) & \(2.04\times 10^{-4}\) & \(3.12\times 10^{-4}\) & \\ Worst & \(1.35\times 10^{-8}\) & \(2.0869\times 10^{-9}\) & \(-4.72\) & \(6.97\times 10^{-3}\) & \(9.47\times 10^{-4}\) & \(-2.41\) \\ Average & \(3.27\times 10^{-9}\) & \(3.63\times 10^{-10}\) & & \(1.42\times 10^{-3}\) & \(6.37\times 10^{-4}\) & \\ SD & \(3.33\times 10^{-9}\) & \(4.73\times 10^{-10}\) & & \(1.76\times 10^{-3}\) & \(1.49\times 10^{-4}\) & \\ \hline \hline \end{tabular}
\end{table}