# A New Fuzzy Harmony Search Algorithm Using Fuzzy Logic for Dynamic Parameter Adaptation
## 6 Statistical Comparison  
A statistical comparison is also made between the fuzzy bat algorithm (FBA) and the fuzzy harmony search algorithm (FHS), to evaluate the efficiency of each method on each of the six mathematical functions applied.  
The statistical test for comparison is the Wilcoxon matched pairs test, which can be used to analyze the data, and whose parameters are given in Table 19.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{FBA} & \multicolumn{3}{c}{FHS} \\ \hline
**Num.** & \multirow{2}{*}{**Best**} & \multirow{2}{*}{**Worst**} & \multirow{2}{*}{**Average**} & \multicolumn{2}{c}{**Num.**} & \multirow{2}{*}{**Best**} & \multirow{2}{*}{**Worst**} & \multirow{2}{*}{**Average**} \\
**Bats** & & & & & & & \\
**2** & & & & & & \\
**3** & & & & & & \\
**40** & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 16: Experiment results for the Zakharov function with the FBA and FHS algorithms.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{FBA} & \multicolumn{3}{c}{FHS} \\ \hline
**Num.** & \multirow{2}{*}{**Best**} & \multirow{2}{*}{**Worst**} & \multirow{2}{*}{**Average**} & \multicolumn{2}{c}{**Num.**} & \multirow{2}{*}{**Best**} & \multirow{2}{*}{**Worst**} & \multirow{2}{*}{**Average**} \\
**Bats** & & & & & & & \\
**2** & & & & & & \\
**5** & & & & & & \\
**10** & & & & & & \\
**20** & & & & & & \\
**30** & & & & & & \\
**40** & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 17: Experiment results for the Rastrigin function with the FBA and FHS algorithms.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{FBA} & \multicolumn{3}{c}{FHS} \\ \hline
**Num.** & \multirow{2}{*}{**Best**} & \multirow{2}{*}{**Worst**} & \multirow{2}{*}{**Average**} & \multicolumn{2}{c}{**Num.**} & \multirow{2}{*}{**Best**} & \multirow{2}{*}{**Worst**} & \multirow{2}{*}{**Average**} \\
**Bats** & & & & & & & \\
**2** & & & & & & & \\
**5** & & & & & & \\
**10** & & & & & & \\
**20** & & & & & & \\
**30** & & & & & & \\
**40** & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 18: Experiment results for the function sum squared with FBA and FHS algorithms.  
The alternative hypothesis states that the average of the results of the fuzzy harmony search algorithm is different than the average performance of the fuzzy bat algorithm, and therefore the null hypothesis tells us that the average of the results of the fuzzy harmony search algorithm is equal to the average of the fuzzy bat algorithm.  
To test the hypothesis, first the absolute values \(|Z_{i}|\ldots|Z_{n}|\) are sorted and assigned their range rank; the sign column indicates that all values obtained are positive; the signed rank column indicates the order of these values from lowest to highest.  
The formula of the statistical test is applied:  
\[W^{+}=\sum_{n\
eq i>0}R_{i} \tag{7}\]  
That is, the sum of the ranges \(R_{i}\) corresponding to positive values \(Z_{i}\).  
The value of \(W^{+}\) is the sum of the positive ranks, the value \(W^{-}\) is the sum of the negative ranks, W is the differences between two data samples, and W0 indicates the value of the table for a two-tailed test using 30 samples.  
The test to evaluate is as follows:  
If \(W\leq\) W0, then reject Ho.  
Table 20 shows a statistical test applied to the two fuzzy methods. In applying the Wilcoxon statistic, with a confidence level of 95% and a value of W = 0 and W0 = 1, the W \(\leq\) W0 condition is satisfied, so the statistical test results are as follows: for the fuzzy harmony search, there is significant evidence to reject the null hypothesis and the alternative hypothesis is accepted stating that the average fuzzy harmony search algorithm is different than the average performance of the fuzzy bat algorithm. In summary, one can state that the fuzzy harmony search outperforms the fuzzy bat algorithm for this set of benchmark functions.