Fig. 1. Triangular and Gaussian kernels.

Fig. 2. KNN and KNN-kernel estimation on the sample data set containing 500 objects generated from one Gaussian distribution (mean=0 and s=1), with k=100. The dotted line is the theoretical pdf function for the data set.

Fig. 3. Density estimation functions for the data set of two classes of different densities.

Fig. 4. A simple example shows how KNNCLUST works with k=2. Each row plots objects in one particular step of the process with their indices and symbols, illustrated class memberships. The symbols: *, o, x, □, , and Δ stand for cluster membership, in which objects belonging to the same cluster have the same symbol.

Fig. 5. The simulated data set. Class one is a mixture of two Gaussians and the other three are generated from three single Gaussian distributions with very different in cluster densities.

Fig. 6. Clustering result by KNNCLUST with k=180; the total accuracy is 95.9%.

Fig. 7. DBSCAN (a) min-points=10, ε=20; (b) min-points=20, ε=950.

Fig. 8. The best of 100 runs of EM to (a) 4 clusters; (b) 5 clusters.

Fig. 9. The gray-scale images of the first two principal components (PC1 on the left, PC2 on the right), and the six main object classes that have been identified in the area.

Fig. 10. Score plot of PC1 and PC2.

Fig. 11. Score-plots of two first PCs by DBSCAN with parameter min-points is 25 (a) 8 clusters found by ε=300, (b) 6 clusters found by ε=400, (c) 4 clusters found by ε=500 and (d) 2 clusters found by ε=900.

Fig. 12. Score-plots of two first PCs and result images of six clusters obtained by (a) K-means (the best of 100 runs); (b) EM (the best of 100 runs) and (c) KNNCLUST with k=550.

Fig. 13. Compactness index of K-means in 100 runs compared to the index of the KNNCLUST result.

Commonly used univariate kernels; where z=(x-x_i)./H