## 3 Fuzzy Kernel Nearest Feature Line Embedding (FKNFLE)
### Classification Results
Figure 4: The classification accuracy _versus_ the reduced dimension on three benchmark datasets using the various algorithms: (**a**) IPS-10; (**b**) Pavia University; (**c**) Pavia City Center. Figure 5: The accuracy rates _versus_ the number of training samples for datasets (**a**) IPS-10; (**b**) Pavia University; and (**c**) Pavia City Center. Figure 6: _Cont._  
Figure 6: The classification maps of dataset IPS using various algorithms: (**a**) The ground truth; (**b**) FKNFLE; (**c**) KNFLE; (**d**) FNFLE; (**e**) NFLE; (**f**) NRS (nearest regularized subspace); (**g**) LFDA-NRS (local Fisher discriminant analysis-NRS); (**h**) LFDA; (**i**) supervised LPP; (**j**) LDA; and (**k**) PCA. Figure 7: Cont. Figure 7: The classification maps of dataset Pavia University using various algorithms: (**a**) The ground truth; (**b**) FKNFLE; (**c**) KNFLE; (**d**) FNFLE; (**e**) NFLE; (**f**) NRS; (**g**) LFDA-NRS; (**h**) LFDA; (**i**) supervised LPP; (**j**) LDA; and (**k**) PCA. Figure 8: _Cont._  
Figure 4:0 The classification maps of dataset Pavia City Center using various algorithms: (**a**) The ground truth, (**b**) FKNFLE; (**c**) KNFLE; (**d**) FNFLE; (**e**) NFLE; (**f**) NRS; (**g**) LFDA-NRS; (**h**) LFDA; (**i**) supervised LPP; (**j**) LDA; and (**k**) PCA. The proposed method was compared with various classification methods on computational time. All methods were implemented by MATLAB codes on a personal computer with an i7 2.93-GHz CPU and 12.0 gigabyte RAM. The comparisons of various algorithms on computational time were tabulated in Table 2 for the IPS-10, Pavia University, and Pavia City Center datasets. Considering the training time, the proposed FKNFLE algorithm was generally faster than NRS and NRS-LFDA by two times and 15 times, respectively. Due to the fuzzification process, algorithms FKNFLE and FNFLE were slower than KNFLE and NFLE, by 13 times and 15 times, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Datasets** & \multicolumn{2}{c}{**IPS-10**} & \multicolumn{2}{c}{**Pavia University**} & \multicolumn{2}{c}{**Pavia City Center**} \\ \hline
**Algorithms** & **Training** & **Testing** & **Training** & **Testing** & **Training** & **Testing** \\ \hline  & 900 & 8720 & 810 & 8046 & 810 & 9529 \\ NFLE-NN & 10 & 18 & 9 & 16 & 9 & 20 \\ KNFLE-NN & 12 & 18 & 11 & 16 & 11 & 20 \\ FNFLE-NN & 155 & 18 & 140 & 16 & 140 & 20 \\ FKNFLE-NN & 156 & 18 & 141 & 16 & 141 & 20 \\ NRS & 326 & 326 & 294 & 300 & 294 & 351 \\ LFDA-NRS & 2331 & 327 & 2098 & 301 & 2098 & 352 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The training and testing times of various algorithms for the benchmark datasets (s). Figure 9: The thematic maps of (**a**) Pavia University, and (**b**) Pavia City Center using the proposed FKNFLE algorithm. From Tables 3-5, the producer's accuracy, overall accuracy, kappa coefficients, and user's accuracy defined by the error matrices (or confusion matrices) [27] were calculated for performance evaluation. They are briefly defined in the following. The user's accuracy and the producer's accuracy are two widely used measures for class accuracy. The user's accuracy is defined as the ratio of the number of correctly classified pixels in each class by the total pixel number classified in the same class. The user's accuracy is a measure of commission error, whereas the producer's accuracy measures the errors of omission and indicates the probability that certain samples of a given class on the ground are actually classified as such. The kappa coefficient, also called the kappa statistic, is defined to be a measure of the difference between the actual agreement and the changed agreement. The overall accuracies of the proposed method were 83.34% in IPS-10, 91.31% in Pavia University, and 97.59% in Pavia City Center with the kappa coefficients of 0.821, 0.910, and 0.971, respectively. Subset IPS-10 of 10 classes is used for fair comparisons with other algorithms. Another alternative classification on the whole IPS dataset of 16 classes was performed.