## 3 Fuzzy Kernel Nearest Feature Line Embedding (FKNFLE)
### Classification Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{11}{c}{**Reference Data**} & \multicolumn{11}{c}{**UA**} \\ \cline{2-13}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** & **16** & \\ \hline
1 & 78.22 & 0 & 0 & 0 & 4.35 & 0 & 0 & 17.43 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 78.22 \\
2 & 0 & 77.15 & 2.22 & 0.69 & 0 & 0.18 & 0 & 0 & 0.07 & 5.17 & 13.26 & 1.19 & 0 & 0 & 0.07 & 0 & 77.15 \\
3 & 0 & 3.32 & 73.03 & 3.04 & 0 & 0 & 0 & 0 & 0.71 & 15.15 & 4.75 & 0 & 0 & 0 & 0 & 73.03 \\
4 & 0 & 13.91 & 8.84 & 65.83 & 0.42 & 0 & 0 & 0.82 & 0 & 1.29 & 7.59 & 1.29 & 0 & 0 & 0 & 0 & 65.83 \\
5 & 0 & 0.21 & 0.23 & 0.24 & 94.61 & 0.22 & 0 & 0 & 0.80 & 0.81 & 1.04 & 0 & 1.85 & 0 & 0 & 94.61 \\
6 & 0 & 0.12 & 0.14 & 0 & 0.19 & 97.11 & 0 & 0 & 0 & 0 & 0.68 & 0 & 0 & 0.58 & 1.18 & 0 & 97.11 \\
7 & 0 & 0 & 0 & 0 & 3.61 & 0 & 92.81 & 3.58 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 92.81 \\
8 & 1.81 & 0 & 0 & 0 & 0 & 0 & 0 & 98.19 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 98.19 \\
9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 94.99 & 0 & 0 & 0 & 5.01 & 0 & 0 & 0 & 94.99 \\
10 & 0 & 3.83 & 0.31 & 0 & 0.32 & 0.31 & 0 & 0 & 0.11 & 81.74 & 12.95 & 0.43 & 0 & 0 & 0 & 0 & 81.74 \\
11 & 0 & 4.62 & 3.52 & 0.22 & 0.32 & 0.31 & 0 & 0 & 0.08 & 5.45 & 83.95 & 1.37 & 0 & 0 & 0.16 & 0 & 83.95 \\
12 & 0 & 4.93 & 7.93 & 0.61 & 0.12 & 0.14 & 0 & 0 & 0 & 2.05 & 9.79 & 74.25 & 0 & 0 & 0.17 & 0 & 74.25 \\
13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.49 & 99.51 & 0 & 0 & 0 & 99.51 \\
14 & 0 & 0 & 0 & 0 & 0.47 & 0.08 & 0 & 0 & 0 & 0 & 0 & 0 & 0.08 & 96.03 & 3.34 & 0 & 96.03 \\
15 & 0 & 0 & 0.54 & 0.54 & 7.25 & 15.02 & 0 & 0 & 0.20 & 1.85 & 2.59 & 0.25 & 0.25 & 16.85 & 54.66 & 0 & 54.66 \\
16 & 0 & 1.03 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.06 & 3.28 & 0 & 0 & 0 & 0 & 94.63 & 94.63 \\ PA & 97.73 & 70.70 & 75.47 & 92.49 & 84.73 & 85.65 & 1 & 81.81 & 99.51 & 81.64 & 56.21 & 87.29 & 94.90 & 83.27 & 91.74 & 1 \\ \multicolumn{11}{c}{Kappa Coefficient: **0.826**} & \multicolumn{11}{c}{Overall Accuracy: **83.85\%**} \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The classification error matrix for data set IPS of 16 classes (in percentage). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Classes**} & \multicolumn{11}{c}{**Reference Data**} & \multicolumn{11}{c}{**User’s**} \\ \cline{2-13}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **Accuracy** \\ \hline
7 & 0 & 1.12 & 0.51 & 0.61 & 2.24 & 0 & 93.25 & 1.22 & 1.02 & 93.25 \\
8 & 0.47 & 1.42 & 0.95 & 1.42 & 2.38 & 1.90 & 0 & 90.76 & 0.66 & 90.76 \\In this study, since we focused on the performance of kernelization and fuzzification, the _k_-NN classifier was adopted rather than the complex support vector machine (SVM) classifier. An analysis of various \(k\) values is given to demonstrate the performance of the _k_-NN classifier as shown in Table 7. Here, value \(k\) was set as values 1, 3, and 4, and the voting strategy was used in this analysis. Obviously, an adaptive higher value of the _k_-NN classifier can achieve more competitive performances. Next, the empirical parameters \(K_{1},K_{2},K_{3}\), and \(K_{4}\) were properly determined by a cross-validation technique. Training samples were separated into two groups: the training and validation subsets, where, for example, 50% of the samples for training and the other for validation. The validation results were generated under various parameters, and the proper setting was determined by selecting the best results. From the cross-validation experiment, the proper parameters \(K_{1}=8\), \(K_{2}=28\), \(K_{3}=14\), and \(K_{4}=28\) were chosen. After that, the transformation was obtained from the whole training set. A sensitivity analysis on four parameters \(K_{1},K_{2},K_{3}\), and \(K_{4}\) was done as shown in Figure 10. In Figure 10a, the variances of classification rates were relatively low for parameters \(K_{1}\)_versus_\(K_{2}\).