## 3 Methodology
### The Well Sequence
For statistical comparison, widely known mean ranks obtained by Kruskal-Wallis and Friedman test, which are implemented to compare the implications between the DE-H algorithm and other algorithms in DE-KN, DE-WE, DE-S, DE-TO, and standard DE, are given in the Table 5.

### Results and Graphs on DE Approaches

The experimental simulation results on benchmark functions are shown in Table 6.

\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Functions** & **DIM \(\times\) Iter** & **DE** & **DE-H** & **DE-S** & **DE-TO** & **DE-WE** & **DE-KN** \\ \hline \multirow{4}{*}{F1} & \(10\times 1000\) & \(1.1464\times 10^{-14}\) & \(2.1338\times 10^{-44}\) & \(5.8561\times 10^{-44}\) & \(7.4117\times 10^{-5}\) & \(7.4827\times 10^{-80}\) & \(5.7658\times 10^{-39}\) \\  & \(20\times 2000\) & \(3.3550\times 10^{-46}\) & \(7.2338\times 10^{-46}\) & \(1.3545\times 10^{-46}\) & \(1.2426\times 10^{-5}\) & \(9.6318\times 10^{-6



### [PERSON] and Kruskal-Wallis Test on DE Approaches

In Table 7, the exhaustive statistical results are explained. ### Comparison of PSO and DE Regarding Data Classification

#### NN Classifications with PSO-Based Initialization Approaches

For further verification of performance of proposed algorithms TO-PSO, WE-PSO and KN-PSO, a comparative study for real-world benchmark data sets problem is tested for training of neural network. We have performed experiments using seven benchmark data sets (diabetes, heart, wine, seed, vertebral, blood tissue, and mammography) exerted from the world-famous machine-learning repository of University of California, Irvine (UCI). Training weights are initialized within interval \([-50,50]\). Accuracy of feed-forward neural network is tested in the form of root mean squared error (RMSE). Table 8 shows the characteristics of the data sets used. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Functions** & **DIM \(\times\) Iter** & **DE** & **DE-H** & **DE-S** &



Discussion

Multilayer feed-forward neural network is trained with a backpropagation algorithm, standard PSO, SO-PSO, and H-PSO, and proposed TO-PSO, KN-PSO, and WE-PSO. Comparison of these training approaches was tested on real classification problem data sets taken from UCI repository. The crossvalidation method was used to compare the performances of different classification techniques. In the paper, k-fold crossvalidation method used for comparison of classification performances for the training of neural network with back propagation, standard PSO, SO-PSO, and H-PSO and proposed TO-PSO, KN-PSO, and WE-PSO is used. The k-fold crossvalidation method was proposed and used in the experimental with value k = 10. Data set was divided into 10 chunks, such that each chunk of data contains the same proportion of each class of data set. One chunk is used for testing, while nine chunks are used for the training phase. The experimental results of algorithms such as with backpropagation, standard PSO, SOPSO, and H-PSO and proposed TO-PSO, KN-PSO, and WE-PSO are compared with each other on seven well-known real data sets taken from UCI, and their performances are evaluated. In Table 9, the simulation results show that the training of neural network with KN-PSO algorithm out performs in accuracy and is capable of providing a better classification accuracy than the other traditional approaches. KN-PSO algorithm may be used effectively for data classification and statistical problems in the future as well. Figure 39 represents the accuracy graph for seven data sets. The classification testing accuracy were imported from Microsoft Excel Spreadsheet to the software RStudio version 1.2.5001 (Boston, MA, USA) to get assurance of the winner approach statistically among all the other approaches. The testing accuracy of all seven variants of PSONN was analyzed by one-way ANOVA test and post hoc [PERSON]'s multi-comparison test with a 0.05 significance level [55]. Table 10 depicts the results of one-way ANOVA of the testing accuracy of classification data.