Visual Display of Data

1. Visual Display of Data R. C. T. Lee National Chi Nan University Rctlee@ncnu.edu.tw

2. Point 1 = (0, 20, 0) Point 2 = (10, 10, 14.14) Point 3 = (20, 0, 0) Point 4 = (10, –10, 14.14) Point 5 = (0, -20, 0) Point 6 = (-10, -10, 14.14) Point 7 = (-20, 0, 0) Point 8 = (-10, 10, 14.14) Can you figure out what they are?

3. Sammon Method (back p10)

4. HT A4 pAD14 HMA 8884H 7768H 70H W4B Fou B9203 DV Kou Bas L24071 pJDW233 HBVKK2 pMND122 LS PI TA A938 DL DA hb321 GM AI ATGGAGAACACAACATCGGATTCCTAGGACCCCTGCTCGTGTTAA ATGGAGAGCACAACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGAAACGATCACATCAGGATTCCTAGGACGCCCTGCTCGTGTTA ATGGAGAACACAACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGACAACATCACATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGACAACATCCATCAGGACTCACTAGGACCCCTGCTCGTGTTA ATGGACAACATCACATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGACAACATTAATCAGGACTCCTAGGACCCCTCGCTCGTGTTA ATGGACAACATCACATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGACTCCTAGGACCCCTGCGCGTGTTA ATGGAAGGCATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAAAGCATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAAAGCATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAAAGCATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCGCATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCGCATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCGCATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCGCATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAAAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTTCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA

5. HT A4 pAD14 HMA 8884H 7768H 70H W4B Fou B9203 DV Kou Bas L24071 pJDW233 HBVKK2 pMND122 LS PI TA A938 DL DA hb321 GM AI ATGGAGAACACAACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAGCACAACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACACAACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGACAACATCACATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGACAACATCACATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGACAACATCACATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGACAACATTACATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGACAACATCACATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGACTCCTAGGACCCCTGCGCGTGTTA ATGGAAGGCATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAAAGCTCACATCAGGATTACCTAGGACCCCTGCTCGTGTTA ATGGAAAGCATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAAAGCATCACATAGGATTCCTCAGGACCCCTGCTCGTGTTA ATGGAGAACATCGCATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCGCATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCGCATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCGCATCAGGACTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAAAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTTCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA ATGGAGAACATCACATCAGGATTCCTAGGACCCCTGCTCGTGTTA

6. 0 11 15 16 53 56 56 59 59 56 38 38 39 38 42 42 44 44 46 44 44 43 43 45 44 45 11 0 8 11 48 51 51 54 56 53 34 34 35 34 39 39 41 41 43 41 41 42 38 40 41 40 15 8 0 15 44 47 47 49 50 47 32 32 33 32 35 35 37 37 39 37 37 38 36 36 37 36 16 11 15 0 44 45 45 48 50 49 30 30 31 30 32 32 34 34 38 38 38 39 35 37 38 37 53 48 44 44 0 3 3 12 14 16 45 45 44 45 47 47 49 48 48 49 48 48 44 45 45 44 56 51 47 45 3 0 2 11 13 15 46 46 45 46 46 46 48 47 49 50 49 49 41 42 42 41 56 51 47 45 3 2 0 11 13 15 46 46 45 46 48 48 50 49 51 52 51 51 43 42 44 41 59 54 49 48 12 11 11 0 10 12 46 46 45 46 53 53 55 54 54 56 55 55 48 49 49 48 59 56 50 50 14 13 13 10 0 8 46 46 45 46 46 46 48 47 52 53 52 50 46 45 43 44 56 53 47 49 16 15 15 12 8 0 50 50 49 50 49 49 51 50 55 56 55 54 48 49 47 48 38 34 32 30 45 46 46 46 46 50 0 4 3 4 36 36 36 34 37 39 38 36 31 32 32 32 38 34 32 30 45 46 46 46 46 50 4 0 5 6 36 36 36 34 39 41 40 38 29 32 32 32 39 35 33 31 44 45 45 45 45 49 3 5 0 5 33 33 33 31 36 38 37 35 30 31 31 31 38 34 32 30 45 46 46 46 46 50 4 6 5 0 36 36 36 34 36 38 37 35 32 34 33 33 42 39 35 32 47 46 48 53 46 49 36 36 33 36 0 2 8 8 32 33 32 31 39 39 37 36 42 39 35 32 47 46 48 53 46 49 36 36 33 36 2 0 8 8 32 33 32 31 39 39 37 36 44 41 37 34 49 48 50 55 48 51 36 36 33 36 8 8 0 4 34 35 34 33 39 39 37 38 44 41 37 34 48 47 49 54 47 50 34 34 31 34 8 8 4 0 32 33 32 31 37 37 35 36 46 43 39 38 48 49 51 54 52 55 37 39 36 36 32 32 34 32 0 5 4 5 35 35 35 34 44 41 37 38 49 50 52 56 53 56 39 41 38 38 33 33 35 33 5 0 5 6 36 36 34 33 44 41 37 38 48 49 51 55 52 55 38 40 37 37 32 32 34 32 4 5 0 5 33 33 33 32 43 42 38 39 48 49 51 55 50 54 36 38 35 35 31 31 33 31 5 6 5 0 34 34 34 33 43 38 36 35 44 41 43 48 46 48 31 29 30 32 39 39 39 37 35 36 33 34 0 10 11 10 45 40 36 37 45 42 42 49 45 49 32 32 31 34 39 39 39 37 35 36 33 34 10 0 5 6 44 41 37 38 45 42 44 49 43 47 32 32 31 33 37 37 37 35 35 34 33 34 11 5 0 7 45 40 36 37 44 41 41 48 44 48 32 32 31 33 36 36 38 36 34 33 32 33 10 6 7 0 Can you figure out anything based upon the matrix?

7. Points 1~ 4: Japan Points 5~10: South America Points 11~14: Africa Points 15~18: the Pacific Ocean Points 19~22: Europe Points 23~26: Europe (back p10) (back p34) (back p72)

8. PID:1P1PPosition:1~26 Original 3D graph

9. PID:1P1PPosition:1~26 Experiment result

10. The visual display problem is defined as following: • Given a set S of n high dimensional points, map them onto the 2-space such that the distances among them are preserved as much as possible. (back p2) • 2. Given the distance matrix of a set S of n data points, map the points, totally based upon the distance matrix, to the 2-space, such that the distances among them are preserved as much as possible. (back p6) • Throughout this presentation, we assume that our distances satisfy triangular inequality.

11. There are many different approaches for visual display. The first approach we shall introduce is the MST preservation approach.

12. 1 6 5 7 4 2 8 3 9 10 In some cases, the only distances that we care are local distances.

13. 1 6 5 7 4 2 8 3 9 10 A Minimal Spanning Tree of the Data

14. The MST preservation approach makes sure that all distances on the MST are exactly preserved. Given a set of n points, there are distances. There are only distances on an MST. Obviously, we can preserve some other distances exactly.

15. 1 4 2 5 3 6 8 7 10 9 Given an MST, we can always consider it as a rooted tree.

16. 1 4 2 5 3 6 8 7 10 9 Once the root is given, we can conduct a breadth-first search and determine a sequence of points. Sequence of points to be mapped: 1,2,4,3,5,8,6,10,9,7.

17. 2 1 4

18. The general rule: • Let be a point to be mapped. Let be the point which has already been mapped and is connected to • in the MST. Then the distance will be exactly preserved. • (2) Let be a point, among all points having been mapped, which is nearest to . Then is to be exactly preserved.

19. 1 6 5 7 4 2 8 3 9 10 4 1 2 3 4 1 2 3’ Preserve d(3,4) and d(3,2)

20. This method is called the triangulation method.

21. Note that there are two locations for . We choose the one which causes smaller total error and some other consideration.

22. Detection: Alpha helix 46~54 PDB: 45~55 PID:1KUY Position:45~55 Experiment result Original 3D graph

23. Detection: Beta sheet14~18 22~26 PDB: 14 ~18 21~30 PID:1JI2Position:14~30 Experiment result Original 3D graph

24. Let us consider the following problem: We are given the distance matrix of a set of n points in k-dimensional points and we want to recover these points. It has been proved that this can be done.

25. We are given

26. Step 1: FindB = HAH where H

27. Step 2: Find eigenvectors of B. Let the eigenvalue of be . Step3: Let

28. Let V= . Then X= .

29. Example:

31. = (-0.7071, -0.23572) = ( 0, 0.47138) = ( 0.7071, -0.23572)

32. For visual display, we only select the two eigenvectors with the highest two eigenvalues.

33. Points mapped from a sphere defined by .

34. Multidimensional Scaling over 26 Hepatitis B Viruses. (back p5)

35. Iterative Methods • We initially assign points to the plane arbitrarily and randomly. • (2) We adjust the points, based • upon some values to minimize • the error.

36. Let the original data points be . Let be mapped to respectively on the 2-space.

37. Let The iterative methods adjust the points on the 2-space to reduce E.

38. The Relaxation Method

41. The relaxation method iteratively adjusts and for all and until some criterion about the error is reached. Original Points

42. (2) (1) Intermediate Initial Assignment

43. (3) (4) Intermediate Final

44. Sammon’s Method In this method, every is adjusted by considering iteself with all of the other points.(The Deepest Descent Method)

45. aqvintfdgvadylqtyhklpdnyitaseaqalgwvaskgnladvapgks vhftaeekavitglwsrvnveetggeavgrllvvypwtqrffdsfgnmss aqvintfdgvadyaqtyhklpdnyitkseaqalgwvaskgnladvapgks aqvintfdgvadylqtyhklpdnyitkseaqalgwvaskgnladvapgks msnftaedkaaitslwakvnvedaggetlgrllvvypwtqrffdsfgsls aqvintfdgvadylqtyhklpdnyitkseaqalgwvaskgnladvapgks aqvintfdgvadylqtyhklpdnyitkseaqalgwvaskgnladvapgks msnftaedkaaitslwakvnvedaggetlgrllvvypwtqrffdsfgsls msnftaedkaaitglwakvnvedaggetlgrllvvypwtqrffdsfgsls aqvintfdgvadylltyhklpdnyitkseaqalgwvaskgnladvapgks mtlrgaeklreeldflksvrrpeiiaaiaearehgdlkenaeyhaareqq mvhftaeekaaitslwgkmnveeaggealgrllvvypwtqrffdnfgnls msnftaedkaaitslwgkvnvedaggetlgrllvvypwtqrffdsfgsls msnftaedkaaitslwgkvnvedaggetlgrllvvypwtqrffdsfgsls

46. aqvintfdgvadylqtyhklpdnyitkseaqalgwvaskgnladvapgks msnftaedkaaitslwakvnvedaggetlgrllvvypwtqrffdsfgsls msnftaedkaaitslwgkvnvedaggetlgrllvvypwtqrffdsfgsls mskfpmtvqgaraleeevkhlkgvlrpqisqaiaearelgdlkenaeyha mrapmtlkgvrrlrdelehlksvkrpeiinaiaearahgdlkenaeyhaa msnftaedkaaitslwakvnvedaggetlgrllvvypwtqrffdsfgsls mnmnklvkssvatsmallllsntanaegkitpvsvkkvddkvtlykttat msnftaedkaaitslwgkvnvedaggetlgrllvvypwtqrffdsfgsls msnftaedkaaitslwakvnvedaggetlgrllvvypwtqrffdsfgsls mkmnklvkssvatsmallllsgtanaegkitpvsvkkvddkvtlykttat mkmkklvkssvassiallllsntvdaaqhitpvsekkvddkitlykttat mkmkklvkssvassiallllsntvdaaqhitpvsekkvddkitlykttat aqvintfdgvadylqtyhklpdnyitkseaqalgwvaskgnladvapgks

47. mkmkklvkssvassiallllsntvdaaqhitpvsekkvddkitlykttat mkmnklvkssvatsmallllsgtanaegkitpvsvkkvddkvtlykttat mkmnklvkssvatsmallllsgtanaegkitpvsvkkvddkvtlykttat mnyipmtlkgaeklreelkylkqtkrseiiksisearqygdlkenaeyqa mrapmtlkgvrrlrdelehlksvkrpeiinaiaearahgdlkenaeyhaa mrapmtskgaqrlreeldhlksvkrpavitaiaearahgdlkenaeyhaa mrppmtlkgvrrlrdelehlklvkrpeiinaiaearahgdlkenaeyhaa mtkfpmtvqgaraleeelkhlktelrpqitqaiaearelgdlkenaeyha mtlkgvrrlrdelehlksvkrpeiinaiaearahgdlkenaeyhaareqq mrapmtskgaqrlreelehlksvkrpevinaiaearahgdlkenaeyhaa vintldgvadylqtyhklpdnyitkseaqalgwvaskgnladvapgksig mqaipmtlrgaeklreeldflksvrrpeiiaaiaearehgdlkenaeyha mqaipmtlrgaeklreeldflksvrrpeiiaaiaearehgdlkenaeyha

48. mkieklgkssvassiallllsntvdaaqnitpkrekkvddkitlykttat msltkypmtvagelalreelerlkkvdrprivqaiaearehgdlkenaey mnmnklvkssvatsmallllsntanaegkitpvsvkkvddkvtlykttat mkmnklvkssvatsmallllsgtanaegkitpvsvkkvddkvtlykttat mvhftaeekstilslwgkvnveeaggealgrllvvypwtqrffdsfgnls mvhftaeekaavtslwskmnveeaggealgrllivypwtqrffdsfgnls mvhftaeekaavtslwskmnveeaggealgrllvvypwtqrffdsfgnls aqvintfdgvadylltyhklpdnyitkseaqalgwvaskgnladvapgks vhftaeeksvitglwgkvnveetggeavgrllvvypwtqrffdsfgnmss mvhftaeekaavtslwskmnveetggealgrllvvypwtqrffdsfgnls

49. 0 122 2 3 129 2 3 129 129 5 135 130 132 132 3 132 132 131 143 144 273 153 152 275 282 122 0 122 123 109 122 122 109 108 123 163 88 110 110 122 107 111 180 184 138 304 176 176 306 324 2 122 0 3 129 2 3 129 129 5 137 131 132 132 3 132 132 133 145 144 275 153 152 277 284 3 123 3 0 130 3 4 130 130 6 138 132 133 133 4 133 133 134 146 144 274 153 152 276 283 129 109 129 130 0 130 130 3 2 131 158 100 41 40 130 30 42 174 180 91 300 154 154 301 318 2 122 2 3 130 0 3 130 130 5 136 131 133 133 2 133 133 131 144 144 274 153 152 276 284 3 122 3 4 130 3 0 130 130 6 136 131 133 133 4 133 133 132 144 144 274 153 152 276 284 129 109 129 130 3 130 130 0 3 131 159 102 42 41 130 29 43 175 180 94 300 153 153 301 318 129 108 129 130 2 130 130 3 0 131 158 101 41 40 130 30 42 174 180 92 300 153 153 301 318 5 123 5 6 131 5 6 131 131 0 135 132 134 134 6 134 134 132 144 145 273 155 154 275 282 135 163 137 138 158 136 136 159 158 135 0 155 159 159 137 159 159 146 158 173 302 196 196 303 331 130 88 131 132 100 131 131 102 101 132 155 0 92 92 131 89 93 178 184 127 303 164 164 305 318 132 110 132 133 41 133 133 42 41 134 159 92 0 3 133 16 5 177 183 61 302 137 141 304 320 132 110 132 133 40 133 133 41 40 134 159 92 3 0 133 15 2 177 183 60 301 137 141 303 320 3 122 3 4 130 2 4 130 130 6 137 131 133 133 0 133 133 133 144 144 275 153 152 277 285 132 107 132 133 30 133 133 29 30 134 159 89 16 15 133 0 17 177 185 71 302 147 147 304 319 132 111 132 133 42 133 133 43 42 134 159 93 5 2 133 17 0 177 183 62 301 138 142 303 320 131 180 133 134 174 131 132 175 174 132 146 178 177 177 133 177 177 0 162 175 304 196 197 308 329 143 184 145 146 180 144 144 180 180 144 158 184 183 183 144 185 183 162 0 183 306 207 207 308 333 144 138 144 144 91 144 144 94 92 145 173 127 61 60 144 71 62 175 183 0 302 83 85 303 325 273 304 275 274 300 274 274 300 300 273 302 303 302 301 275 302 301 304 306 302 0 305 307 36 219 153 176 153 153 154 153 153 153 153 155 196 164 137 137 153 147 138 196 207 83 305 0 6 303 319 152 176 152 152 154 152 152 153 153 154 196 164 141 141 152 147 142 197 207 85 307 6 0 303 319 275 306 277 276 301 276 276 301 301 275 303 305 304 303 277 304 303 308 308 303 36 303 303 0 216 282 324 284 283 318 284 284 318 318 282 331 318 320 320 285 319 320 329 333 325 219 319 319 216 0

50. Initial