Fig. 2

Learning E-GPS- and E-GPS-based analysis a Learning E-GPS with \(J(\mathcal{S})\) given by Eq. (4) simplified into \(J(a_1, a_2)\), where \(\eta _S\) is given by Eq. (6) and \(\lambda \ge 0\) is a weight for the role of \(\eta _S\). b Two lines in special cases with only two free parameters. c We may get three convex subsets by considering two parallel lines featured by three free parameters. One way for learning the two lines is using support vector machine (SVM) (Suykens and Vandewalle 1999; Suykens et al. 2002) for several different choices of the margin a among which we pick the best one according to J. d Alternatively, two parallel lines may also be obtained in two steps. First, we find the normal direction of the two lines by SVM and then project all the samples orthogonally onto the direction. Second, we treat the projected samples in the same way as in (b). Instead of SVM, the normal direction may also be determined by either Fisher discriminative analysis (FDA) or principal component analysis (PCA). e Recursively, we perform the division as illustrated in (c) or (d) on each c-state, ..., so on so forth, until all the remaining c-states are noncuttable, see Table 1(1)(c). Finally, we get a tree with each d-state (e.g., \(s_{11}\)) as a leaf. f Defragment is made from time to time by merging adjacent d-states \(s_2, s_3, s_4\) and merging c-states \(s_5, s_6\), see Table 1(3)