# Variable Selection Using Adaptive Band Clustering and Physarum Network
## 4 Results and Discussion
### Diesel Data Set  
The convergence diagram of the AP algorithm is shown in Figure 5. With the increase of the number of iterations, the fitness (net similarity) of quantized intermediate solutions gradually increases. When the number of iterations increases to 386, and fitness no longer changes. The AP algorithm converges in the first round as well as in the rest of the rounds.  
The convergence diagram of the PN algorithm is shown in Figure 6. The \(D_{ij}\) represents the rate of change of the network continuity. With the increase of the number of the iterations, the \(D_{ij}\) gradually decreases. When the number of iterations increases to 18, all the \(D_{ij}\) are smaller than the threshold of stop. The PN algorithm is converged.  
The number of wavelengths selected by using AP-PN in the first round was 120, which is illustrated in Figure 7a. These 120 wavelengths were input into GA-PLS. The times of each wavelength selected by GA-PLS are illustrated in Figure 7b. By using the 25 most selected wavelengths, the root mean square errors of cross-validation (RMSECV) can reach the smallest value. These final 25 selected wavelengths determined by using AP-PN-GA-PLS are illustrated in Figure 7c. The scatter plot of the predicted value vs. the measured value is given in Figure 7d. The curve that fits the data points best is also given in Figure 7d.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Number of Input Wavelengths** & **Number of Selected Wavelengths** & **R** & **RMSEP(\%)** \\ \hline PLS & 700 & 700 & 0.9832 & 0.0902 \\ GA-PLS & 700 & 67 & 0.9989 & 0.0431 \\ PN-GA-PLS & 700 & 35 & 0.9960 & 0.0597 \\ AP-GA-PLS & 700 & 39 & 0.9941 & 0.0643 \\ AP-PN-GA-PLS & 700 & 25 & 0.9970 & 0.0397 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The average values of the number of selected wavelengths, correlation coefficient (R), and predicted root mean square error (RMSEP) using PLS, GA-PLS, PN-GA-PLS, AP-GA-PLS, and AP-PN-GA-PLS (corn data).  
All five algorithms, i.e., PLS, GA-PLS, PN-GA-PLS, AP-GA-PLS, and AP-PN-GA-PLS, were tested in the ten rounds, and the average number of selected wavelength, RMSEP, and R are summarized in Table 5.  
It is seen from Table 5 that AP-PN-GA-PLS can give the largest R (0.9744) and the smallest RMSE (0.1167%) among PN-GA-PLS (R = 0.9727, RMSEP = 0.1404%), AP-GA-PLS (R = 0.9722, RMSEP = 0.1356%), and PLS (R = 0.9716, RMSEP = 0.1370%). It is also observed that the AP-PN-GA-PLS achieved good prediction performance with the least number of wavelengths among all methods. The number of selected wavelengths by AP-PN-GA-PLS was 26. The number of selected wavelengths by AP-GA-PLS, PN-GA-PLS, GA-PLS, and PLS were 66, 40, 142, and 401, respectively. The AP-PN-GA-PLS model may, thus, achieve the least complexity without degrading the prediction accuracy. The AP-GA-PLS can further reduce the redundant information in variables, compared with GA-PLS, but its wavelength selection performance (66 wavelengths selected, RMSEP = 0.1356%) is not as good as that of PN-GA-PLS (40 wavelengths selected, RMSEP = 0.1404%).  
Figure 5: The convergence diagram of the AP algorithm (diesel data): Iterations vs. Fitness (net similarity) of quantized intermediate solution in the first round. When the iteration is 386, the network is converged.  
Figure 6: The convergence diagram of the PN algorithm (corn data): Iterations vs. \(D_{ij}\) (\(D_{\bar{y}}\) is the rate of change of the network continuity). When the iteration is 18, all the \(D_{\bar{y}}\) are smaller than the threshold of stop, and the network is converged.  
The AP-PN-GA-PLS selected 26 wavelengths. In general, the polycyclic aromatic hydrocarbon (PAHs) relates to the viscosity of diesel [32]. The final selected wavelengths of 942 nm and 1046 nm are similar to 934 nm and 1053 nm, respectively, which are the absorption wavelengths of methylene that relate to octane number. The selected wavelengths of 1422 nm and 1426 nm are the absorption wavelengths of the aromatic ring that relates to viscosity [32].