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Figure 21.15(a) to (c) show a 2D nonseparable case in a 4D space using the kernel given by Equation (21.55). We can appreciate that our method works very well. Figures 21.15(d) (e) shows two nonlinear separable cases in an 8D space using in both of them, the kernel given by Equation (21.56). Figure 21.15(f) shows a 3D nonseparable case again using the kernel given by Equation (21.56), which works in an 8D space. We have shown that the kernels involving the Clifford product find optimal hyperplanes.
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Figure 21.15 (a),(b),(c) Nonlinear 2D separable classification using the kernel of Equation (21.55), which works in 4D; (d),(e),(f) nonlinear separable classification cases using the kernel of Equation (21.56), which works in 8D. The first two experiments are 2D cases and the third a 3D case.
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Figure 21.15 (continued)
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9.5 An SMVM Using Clustering Hyperspheres
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The last experiment clustered 3D data using the method explained in Section 7.5. Figure 21.16 shows the spheres that were found via a self-organizing neural network. They were coded as spheres in the conformal geometric algebra G4 1 and only these were given to the SMVM for finding the optimal nonlinear hyperplane. Note that after training, all the vectors inside the spheres are classified correctly. Even points near to both sides of the nonlinear optimal hyperplane are classified correctly. The coding of the spheres, points and points near to the optimal hyperplane was as follows: we used the conformal geometric algebra G4 1 and vectors were represented using the basis 1 2 3 e+ e_ . Table 21.2 and Figure 21.16 show the coding and classification results. Note the successful classification of the points near to the decision border.
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Figure 21.16 Nonlinear separable classification cases using the conformal kernel. (a) Spheres; (b) the patterns inside the spheres.
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Table 21.2 Classification of spheres, points in and out of the spheres. Object s1 = s2 = s3 = s4 = s5 = 1 2 0 15 25 0 0 0 15 15 1 5 1 5 1 5 2 155 3 155 3 1 1 4 755 5 755 3 3 2 5 11 22 12 22 Expected 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Obtained 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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sphere inside sphere s1 s6 = 1 2 5 3 09375 4 09375 points inside S1 ps11 = 1 2 1 1 5 3 5 ps12 = 1 2 5 0 3 125 4 125 ps13 = 1 1 5 0 1 125 2 125 ps14 = 1 5 2 0 2 625 3 625 ps15 = 0 5 2 2 0 2 045 3 045 points inside S2 ps21 = 0 0 1 5 0 625 1 625 ps22 = 0 0 5 1 5 0 75 1 75 ps23 = 0 0 5 1 5 0 75 1 75 ps24 = 0 5 0 1 5 0 75 1 75 points inside S3 ps31 = 1 5 1 5 1 5 2 875 3 875 ps32 = 2 2 1 2 2 6 125 7 125 ps33 = 1 1 5 1 2 1 845 2 845 ps34 = 1 5 1 5 0 5 1 875 2 875 points inside S4 ps41 = 3 1 1 5 6 ps42 = 2 4 1 2 1 3 3 945 4 945 ps43 = 3 0 5 0 5 4 25 5 25 ps44 = 3 0 4 1 4 58 5 58 points inside S5 ps51 = 3 3 2 5 11 625 12 625 ps52 = 3 7 3 2 5 13 97 14 97 ps53 = 3 3 5 2 5 13 25 14 25 ps54 = 3 3 3 2 13 62 14 62 outside the spheres and near to the optimal hyperplane po1 = 2 5 2 0 4 625 5 625 po2 = 1 0 5 0 0 125 1 125 po3 = 0 0 9 1 5 1 03 2 03 po4 = 0 3 0 3 1 5 0 715 1 715 po5 = 0 3 0 3 2 5 2 715 3 715 po6 = 2 1 0 3 1 2 25 3 25
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