Enclose clusters with gaps using functionals (ScalarPTreeSets or SPTSs):

Cp,d(x)=(x-p)od / (x-p)o(x-p) Conical Separating clusters by cone gaps

gap(Bp)

Oblique FAUST (Clustering Versions: Linear (the default), Spherical, Barrel, Conical )

Bp,d(x) = (x-p)o(x-p)-((x-p)od)2 Barrel: Separating clusters by barrel gaps.

Barrel: Search for Gap(Lp,d)>T and Gap(Bp)>T2.

Identify gaps in Bp,d SPTS after masking to between Lp,d gap1 and gap2

Spherical: Search Sp for spherical T gaps, {x | r2 Sp(x) < (r+T)2}= (using PINE masking?). The interior of the r-sphere about p encloses a sub-cluster.

Sp(x) = (x-p)o(x-p) Spherical: Separating clusters by spherical gaps.

The FAUST default is linear gaps wider than T (T=GapWidthMinimum) to isolate clusters. But there may be none. We can add barrel-shaped or cone-shaped gaps to limit the radial reach of our linear enclosures.

No gaps show on the red, blueor green projection lines

dprp

Lp,d(x) = (x-p)od Linear: Separating cluster, C, by

two (n-1)-dimensional hyperplanar gaps.

I.e., find a1<a2 such that

= LowerGap={x | a1<Lpd(x)<a1+T} and

= UpperGap={x | a2<Lpd(x)<a2+T} and

C = {x|a1+T<Lpd(x)<a2}

UpperGap

dp

LowerGap

a1

a2

Bpd

xx

Note: Bpd(x) = Sp(x) - L2pd(x)

Note: C2pd(x) = L2

pd(x) / Sp(x)

mn=p 140 122 594 3mx=q 540 228 993 365

d 0.58 0.15 0.58 0.53CONCRETE ST CM WA FA AG 8 140 192 807 3 8 168 122 780 3 9 190 162 803 3 10 310 192 851 3 20 230 195 759 14 20 238 187 847 3 21 212 180 779 14 21 191 162 804 14 22 166 176 780 28 22 234 198 852 14 22 230 195 758 14 23 234 198 852 28 23 190 162 803 14 23 363 165 756 7 24 168 122 780 28 24 338 175 756 3 24 286 145 804 3 24 222 189 870 14 24 230 195 759 28 25 319 156 880 3 25 222 189 870 28 25 230 195 758 28 25 195 166 906 14 25 212 180 779 28 25 166 176 780 14 25 250 187 861 14 26 191 162 804 28 26 195 166 906 28 26 238 228 594 7 26 238 187 847 14 26 213 159 904 14 28 190 162 803 28 28 389 158 926 3 28 234 198 852 56 28 199 192 826 28 28 140 192 807 28 28 324 184 660 3 29 380 154 605 3 29 375 127 993 3 29 313 176 612 3 29 250 187 861 28 29 166 176 780 56 29 222 189 870 56 40 214 182 786 28 40 190 162 803 100 40 469 138 841 3 40 238 187 847 56 40 333 228 594 270 40 212 180 779 100 41 333 228 594 365 41 390 146 756 3 41 222 189 870 100 41 191 162 804 100 41 531 142 894 3 41 190 228 670 28 41 380 228 594 90 41 380 228 594 270 41 380 228 594 180 41 230 195 758 100 41 402 147 852 3 42 475 228 594 270 42 190 228 670 90 42 428 228 594 90 42 475 228 594 90 42 475 228 594 365 42 199 192 826 180 42 428 228 594 180 42 250 187 861 100 43 213 159 904 56 43 475 228 594 180 43 313 176 612 7

43 428 228 594 270 43 213 159 904 100 44 428 228 594 365 44 238 187 847 100 44 199 192 826 360 44 140 192 807 180 44 380 228 594 365 45 140 192 807 360 46 375 127 993 7 46 375 127 993 7 46 266 228 670 28 46 374 170 757 7 46 214 182 785 28 47 190 228 670 180 47 214 182 786 56 47 425 151 804 7 47 266 228 670 90 47 531 142 894 7 47 380 154 605 7 48 304 228 670 28 49 304 228 670 90 49 425 154 887 7 49 425 154 887 7 49 266 228 670 180 49 425 154 887 7 60 425 154 887 28 60 375 127 993 56 60 425 154 887 28 60 425 154 887 28 61 374 170 757 28 62 540 162 676 28 62 425 151 804 28 63 374 170 757 56 63 375 127 993 91 64 425 154 887 56 64 425 154 887 56 64 425 154 887 56 65 425 151 804 56 65 374 170 757 91 65 425 154 887 91 65 313 176 612 56 65 425 154 887 91 65 425 154 887 91 66 439 186 708 28 66 319 156 880 56 67 469 138 841 28 67 313 176 612 91 67 425 151 804 91 68 286 145 804 28 68 475 181 782 28 68 319 156 880 91 68 402 147 852 28 68 338 175 756 91 69 469 138 841 56 71 363 165 756 28 71 363 165 756 28 71 363 165 756 28 71 363 165 756 28 71 469 138 841 91 72 475 181 782 56 72 439 186 708 56 73 286 145 804 56 73 439 186 708 91 74 390 146 756 28 74 475 181 782 91 74 402 147 852 56 75 402 147 852 91 75 324 184 660 28 77 363 165 756 56 77 286 145 804 91 77 363 165 756 56 77 363 165 756 56 77 363 165 756 56 79 390 146 756 56 79 363 165 756 91 79 363 165 756 91 79 363 165 756 91 79 363 165 756 91 80 324 184 660 56 83 390 146 756 91

0 1 7 7 1 411 1 112 2 315 2 116 2 117 1 118 3 119 1 120 2 121 2 122 3 123 4 124 5 125 1 126 3 127 2 229 1 130 1 232 2 133 1 134 4 135 1 136 2 137 2 138 4 139 3 140 4 141 1 142 4 143 6 144 2 246 9 147 1 148 3 149 3 150 1 151 7 152 2 153 1 154 1 155 1 156 2 157 1 158 4 462 5 163 2 265 1 166 1 167 2 168 4 169 1 170 6 272 1 173 1 174 5 175 1 176 1 278 1 179 3 281 1 283 2 184 1 286 1

L=1L=1

L=38 M=35 H=32

L=2 M=15 H=23

L=2 M=1

R Ct Gp11 3 112 7 113 6 114 10 115 4 116 9 117 1 No gaps! Also radii start at 11 (not 0)., which is a problem

l 0m 0h 22 0 =r1 14.8 =r2 Notice HI is separatable at r=14.8 but not by gap analysis. By what?

An Oblique FAUST Barrel algorithmon Concrete with barrel Gap Threshold, T=12

Look for L gaps in(x-p)od/4 Ct Gap>=3

If the minimum barrel radii >> 0, we have chosen a d-line far from the data. I will try choosing p and q to be actual points (e.g., the first and last).Here are the formulas from the spreadsheet:G=(B12-B$6)*B$9+(C12-C$6)*C$9+(D12-D$6)*D$9+(E12-E$6)*E$9 H=G12-$G$9 L=(x-p)od-min I=(B12-B$6)^2+(C12-C$6)^2+(D12-D$6)^2+(E12-E$6)^2 J=@SQRT(I12-G12^2) B=SQRT[(x-p)o(x-p)-(x-p)od^2]If you do not round, calculating pTree bitslices effectively truncates, which is fine.For fixed point, it all works fine. Here are the bislice formulas:@MOD(@INT(F/2^6),2)@MOD(@INT(F/2^5),2)@MOD(@INT(F/2^4),2)@MOD(@INT(F/2^3),2)@MOD(@INT(F/2^2),2)@MOD(@INT(F/2^1),2)@MOD(@INT(F/2^0),2) Keep going (take bitslices to the right of decimal pt)@MOD(@INT(F/2^-1),2)@MOD(@INT(F/2^-2),2) ...Floating point? Bitslice mantissa. Exponent shifts the slice name. E.g.,

.1011 25

.0010 23

.1010 2-1

24

100

23

000

10110. 10. .01010

22

100

21

110

20

000

2-1

000

2-2

001

2-3

000

2-4

001

mn=p 140 122 594 3mx=q 540 228 993 365

d 0.58 0.15 0.58 0.53CONCRETE ST CM WA FA AG 8 140 192 807 3 8 168 122 780 3 9 190 162 803 3 10 310 192 851 3 20 230 195 759 14 20 238 187 847 3 21 212 180 779 14 21 191 162 804 14 22 166 176 780 28 22 234 198 852 14 22 230 195 758 14 23 234 198 852 28 23 190 162 803 14 23 363 165 756 7 24 168 122 780 28 24 338 175 756 3 24 286 145 804 3 24 222 189 870 14 24 230 195 759 28 25 319 156 880 3 25 222 189 870 28 25 230 195 758 28 25 195 166 906 14 25 212 180 779 28 25 166 176 780 14 25 250 187 861 14 26 191 162 804 28 26 195 166 906 28 26 238 228 594 7 26 238 187 847 14 26 213 159 904 14 28 190 162 803 28 28 389 158 926 3 28 234 198 852 56 28 199 192 826 28 28 140 192 807 28 28 324 184 660 3 29 380 154 605 3 29 375 127 993 3 29 313 176 612 3 29 250 187 861 28 29 166 176 780 56 29 222 189 870 56 40 214 182 786 28 40 190 162 803 100 40 469 138 841 3 40 238 187 847 56 40 333 228 594 270 40 212 180 779 100 41 333 228 594 365 41 390 146 756 3 41 222 189 870 100 41 191 162 804 100 41 531 142 894 3 41 190 228 670 28 41 380 228 594 90 41 380 228 594 270 41 380 228 594 180 41 230 195 758 100 41 402 147 852 3 42 475 228 594 270 42 190 228 670 90 42 428 228 594 90 42 475 228 594 90 42 475 228 594 365 42 199 192 826 180 42 428 228 594 180 42 250 187 861 100 43 213 159 904 56 43 475 228 594 180 43 313 176 612 7

43 428 228 594 270 43 213 159 904 100 44 428 228 594 365 44 238 187 847 100 44 199 192 826 360 44 140 192 807 180 44 380 228 594 365 45 140 192 807 360 46 375 127 993 7 46 375 127 993 7 46 266 228 670 28 46 374 170 757 7 46 214 182 785 28 47 190 228 670 180 47 214 182 786 56 47 425 151 804 7 47 266 228 670 90 47 531 142 894 7 47 380 154 605 7 48 304 228 670 28 49 304 228 670 90 49 425 154 887 7 49 425 154 887 7 49 266 228 670 180 49 425 154 887 7 60 425 154 887 28 60 375 127 993 56 60 425 154 887 28 60 425 154 887 28 61 374 170 757 28 62 540 162 676 28 62 425 151 804 28 63 374 170 757 56 63 375 127 993 91 64 425 154 887 56 64 425 154 887 56 64 425 154 887 56 65 425 151 804 56 65 374 170 757 91 65 425 154 887 91 65 313 176 612 56 65 425 154 887 91 65 425 154 887 91 66 439 186 708 28 66 319 156 880 56 67 469 138 841 28 67 313 176 612 91 67 425 151 804 91 68 286 145 804 28 68 475 181 782 28 68 319 156 880 91 68 402 147 852 28 68 338 175 756 91 69 469 138 841 56 71 363 165 756 28 71 363 165 756 28 71 363 165 756 28 71 363 165 756 28 71 469 138 841 91 72 475 181 782 56 72 439 186 708 56 73 286 145 804 56 73 439 186 708 91 74 390 146 756 28 74 475 181 782 91 74 402 147 852 56 75 402 147 852 91 75 324 184 660 28 77 363 165 756 56 77 286 145 804 91 77 363 165 756 56 77 363 165 756 56 77 363 165 756 56 79 390 146 756 56 79 363 165 756 91 79 363 165 756 91 79 363 165 756 91 79 363 165 756 91 80 324 184 660 56 83 390 146 756 91

0 1 9 9 1 5 14 1 1 15 1 1 16 1 3 19 1 1 20 3 3 23 1 1 24 3 1 25 1 1 26 2 1 27 1 1 28 1 1 29 3 1 30 3 1 31 3 1 32 4 2 34 2 1 35 1 1 36 2 3 39 2 3 42 2 2 44 2 1 45 3 1 46 1 1 47 1 1 48 1 1 49 2 1 50 2 1 51 3 1 52 3 1 53 3 1 54 1 1 55 1 1 56 8 1 57 1 1 58 1 1 59 1 1 60 1 1 61 8 2 63 3 1 64 1 1 65 3 1 66 1 1 67 5 1 68 3 1 69 1 1 70 1 2 72 1 1 73 1 1 74 2 1 75 1 1 76 1 1 77 3 5 82 3 1 83 2 1 84 2 3 87 1 1 88 2 1 89 1 1 90 4 1 91 1 1 92 1 1 93 5 2 95 1 2 97 1 1 98 4 1 99 2 1100 1 3103 1 2105 3 3108 1 2110 1 1111 2 3114 1

L=2Oblique FAUST Barrel alg Concrete, T=12Try p and q to be actual points (e.g., the first and last).

M=1

L=5 M=2

L=17 M=8 H=4

L=18 M=26 H=28

L=18 M=3 H=4

L=2 M=9 H=13

L=2 M=9 H=4

L=2 M=3 H=1 outliers H=1

r Ct Gp 0 1 14 14 1 12 26 1 1 27 1 30 57 1 29 86 2

L=4 M=1L=1 M=1

r Ct Gp 0 1 3 3 2 4 7 2 2 9 1 312 3 113 1 114 1 418 1 220 1 121 1 122 2 123 2 3356 1 662 2 365 1 166 1 167 1 976 1 177 1 481 1 1091 2

1 round L, 1 rund R. Rround:1. if the least radius is not 0, set p=1st point.2. Take [1st] large r-gap.3. If there are singleton gapped r-rings, outlier.4. All other gapped r-rings, analyzed further. Pts are separated out from the rest by mn gap

thresh but scattered around ring?

L=6 M=8 H=4

L=3 M=2 H=4

Distances 21 22 40 46 21 0.0 48.3 15.9 15.5 singleton L outlir 22 48.3 0.0 48.7 48.6 singleton L outlir 40 15.9 48.7 0.0 1.0 doubleton 46 15.5 48.6 1.0 0.0 M ouliers L=3 M=1 H=4

L=6 M=2 H=4L=3 M=1 H=4L=3 M=2 H=4

L=1 M=1 H=2

Distances 28 75 80 28 0.0 25.0 53.0 75 25.0 0.0 28.0 80 53.0 28.0 0.0 all singleton ouliers

L=1 M=1 H=1

Distances 47 65 47 0.0 154.9 65 154.9 0.0 all singleton ouliers

L=1 M=1 H=2L=1 M=1 H=1

Distances 29 47 29 0.0 4.0 47 4.0 0.0

r Ct Gp 0 1 14 14 1 2 16 1 1 17 2 1 18 1 1 19 1 1 20 2 1 21 1 2 23 1 3 26 1 2 28 4 1 29 1 2 31 1 2 33 1 3 36 2 1 37 1 1 38 6 1 39 2 1 40 3 1 41 6 1 42 1 2 44 4 2 46 5 1 47 2 1 48 1 1 49 2 1 50 1 1 51 1 13 64 1 1 65 2 3 68 1 2 70 1 10

80 1 5 85 1 7 92 1 2 94 1 5 99 1 7106 1 1107 1 4111 1 5116 1 8124 1 1125 1

L=1

L8 H2

both Houtliers

L=5 M=2 H=1

2Ms, Houtliers

L=4 M=12 H=21

L=4 M=2 H=2

Dist 42 49 66 72 7342 0.0 211.6 307.7 294.8 281.949 211.6 0.0 159.0 150.3 146.466 307.7 159.0 0.0 28.0 63.072 294.8 150.3 28.0 0.0 35.073 281.9 146.4 63.0 35.0 0.0 all outliers

outliers

outliersoutlier

M=2

outliers

M=2

r Ct Gp 0 1 6 6 2 7 13 1 30 43 1 2 45 1 1 46 1 6 52 1 11 63 4 2 65 2 10 75 1 4 79 2 3 82 1 1 83 1 1 84 2 2 86 1 2 88 1 1

88 1 1 89 4 2 91 1 1 92 1 9101 1 2103 1 9112 1 10122 4 2124 1 10134 1

outliers

L=1 M=3

L=1 M=5

M8 H7

M8 H9

r/2 Ct Gp 0 2 18 18 1 16 34 2 5 39 1 4 43 1 8 51 1 41 92 4 5 97 1 12109 1 13122 1 all

outliers Dist 41 43 65 68 68 68

41 0.0 134.4 392.4 413.9 402.8 437.643 134.4 0.0 309.6 313.6 361.8 363.865 392.4 309.6 0.0 68.3 135.1 60.368 413.9 313.6 68.3 0.0 196.3 106.768 402.8 361.8 135.1 196.3 0.0 108.368 437.6 363.8 60.3 106.7 108.3 0.0 all outliers

r/5 Ct Gp 0 1 1010 3 111 3 213 2 114 3 418 1 523 2 124 1 125 1 530 1 535 1 338 1 4482 1 688 1 1199 2

L=1 M=9 H=13

M=3 H=3 (21 between M's and H's)

L=1 M=1 H=3 (M and H's separate but d(L,M)=4outlier

L=1 M=1 H=3 M=1 outlier

all outlier

So 2 errors out of 150 = 99% accuracy

p=vom .5 .39 0.13 0.18 q=mean .48 .38 0.12 0.21 ABALONE b=.5ring len diam heig Shell 4 0.17 0.13 0.09 0.01 4 0.24 0.19 0.06 0.02 4 0.21 0.15 0.05 0.01 5 0.17 0.13 0.05 0.01 5 0.20 0.15 0.05 0.01 5 0.27 0.19 0.06 0.03 5 0.24 0.17 0.04 0.02 6 0.36 0.28 0.08 0.07 6 0.37 0.27 0.09 0.07 6 0.37 0.26 0.07 0.07 6 0.32 0.24 0.07 0.04 6 0.31 0.23 0.07 0.04 7 0.30 0.23 0.08 0.04 7 0.38 0.29 0.08 0.08 7 0.36 0.26 0.09 0.07 7 0.40 0.31 0.1 0.11 7 0.26 0.2 0.06 0.02 7 0.4 0.32 0.09 0.1 7 0.35 0.26 0.09 0.06 7 0.33 0.25 0.08 0.05 7 0.42 0.32 0.09 0.1 7 0.46 0.37 0.12 0.15 7 0.50 0.4 0.12 0.17 7 0.39 0.29 0.09 0.07 7 0.35 0.28 0.08 0.11 7 0.32 0.26 0.09 0.06 7 0.45 0.34 0.10 0.13 7 0.36 0.29 0.08 0.1 7 0.47 0.37 0.13 0.16 7 0.35 0.26 0.09 0.07 8 0.47 0.35 0.12 0.13 8 0.42 0.33 0.11 0.13 8 0.52 0.41 0.12 0.19 8 0.4 0.32 0.11 0.1 8 0.42 0.3 0.09 0.12 8 0.52 0.4 0.12 0.18 8 0.46 0.35 0.10 0.12 8 0.47 0.38 0.13 0.17 8 0.45 0.35 0.10 0.14 8 0.44 0.35 0.12 0.13 8 0.40 0.30 0.08 0.08 8 0.38 0.29 0.08 0.08 8 0.47 0.37 0.13 0.14 8 0.5 0.4 0.14 0.22 9 0.57 0.44 0.14 0.28 9 0.43 0.39 0.10 0.13 9 0.35 0.29 0.09 0.09 9 0.51 0.39 0.13 0.2 9 0.53 0.42 0.13 0.21 9 0.53 0.43 0.15 0.25 9 0.52 0.42 0.16 0.24 9 0.50 0.40 0.11 0.17 9 0.47 0.36 0.10 0.14 9 0.46 0.35 0.13 0.16 9 0.42 0.35 0.10 0.16 9 0.45 0.32 0.1 0.11 9 0.43 0.32 0.08 0.10 9 0.45 0.33 0.10 0.11 9 0.49 0.39 0.12 0.15 9 0.44 0.35 0.12 0.12 9 0.47 0.37 0.12 0.16 9 0.34 0.25 0.09 0.06 9 0.56 0.43 0.15 0.31 9 0.40 0.32 0.11 0.11 9 0.46 0.36 0.10 0.17 9 0.59 0.47 0.14 0.31 9 0.28 0.20 0.08 0.04 9 0.47 0.37 0.12 0.14 9 0.55 0.41 0.13 0.2 9 0.37 0.28 0.09 0.09

Oblique Barrel FAUST on Abalone:Alternate Lpqx, Bpqx for cluster dendogram topdownAvg Density: AD = count / k=1..dim(maxk-mink) T = MinGapThreshold=b/(dim*AD), b=parameterIf we're given a TrainingSet, TS, with K classes, is

avgk=1..Kvomk a better mediod than VoM?Take p=MinCorner, q=MaxCorner of box

circumscribing {VoMk}k=1..K (not circTS)And then take p=1st_TS pt? First, p=vom, q=avg

10 0.56 0.44 0.16 0.26 10 0.53 0.40 0.14 0.20 10 0.47 0.36 0.12 0.15 10 0.58 0.47 0.16 0.33 10 0.47 0.37 0.13 0.17 10 0.38 0.27 0.1 0.08 10 0.44 0.34 0.1 0.13 10 0.5 0.4 0.12 0.22 10 0.6 0.47 0.15 0.30 10 0.53 0.41 0.14 0.20 10 0.61 0.48 0.16 0.30 10 0.53 0.41 0.13 0.2 10 0.48 0.36 0.13 0.16 10 0.43 0.35 0.11 0.13 10 0.55 0.45 0.14 0.26 10 0.56 0.44 0.15 0.24 10 0.47 0.39 0.12 0.17 10 0.47 0.37 0.12 0.15 10 0.49 0.38 0.12 0.17 10 0.47 0.35 0.1 0.18 10 0.44 0.36 0.12 0.15 10 0.44 0.34 0.10 0.16 10 0.46 0.35 0.12 0.15 10 0.62 0.49 0.16 0.39 11 0.49 0.38 0.13 0.19 11 0.57 0.42 0.14 0.2 11 0.55 0.42 0.14 0.28 11 0.35 0.28 0.09 0.07 11 0.49 0.37 0.13 0.22 11 0.58 0.45 0.18 0.28 11 0.53 0.41 0.11 0.18 11 0.56 0.45 0.15 0.24 11 0.54 0.41 0.12 0.21 11 0.57 0.44 0.13 0.26 11 0.56 0.44 0.14 0.3 11 0.57 0.48 0.17 0.38 11 0.54 0.43 0.16 0.28 12 0.56 0.44 0.16 0.32 12 0.57 0.46 0.18 0.44 12 0.62 0.51 0.17 0.67 12 0.52 0.41 0.15 0.23 12 0.55 0.44 0.15 0.26 12 0.60 0.47 0.16 0.34 12 0.57 0.45 0.16 0.33 12 0.5 0.4 0.13 0.24 12 0.56 0.44 0.15 0.27 12 0.59 0.44 0.14 0.28 13 0.70 0.55 0.2 0.49 13 0.60 0.45 0.19 0.31 13 0.59 0.49 0.18 0.48 13 0.59 0.46 0.17 0.39 13 0.62 0.46 0.14 0.4 13 0.55 0.42 0.13 0.27 14 0.58 0.45 0.14 0.36 14 0.55 0.42 0.13 0.27 14 0.66 0.53 0.19 0.48 14 0.52 0.38 0.14 0.21 14 0.51 0.42 0.14 0.25 14 0.61 0.47 0.17 0.34 15 0.6 0.47 0.15 0.28 15 0.45 0.36 0.09 0.15 15 0.55 0.43 0.14 0.25 15 0.69 0.56 0.19 0.48 15 0.60 0.47 0.18 0.29 15 0.68 0.56 0.16 0.46 15 0.53 0.43 0.16 0.33 15 0.56 0.42 0.13 0.25 16 0.52 0.42 0.16 0.32 16 0.54 0.47 0.15 0.34 16 0.65 0.52 0.19 0.44 16 0.54 0.42 0.12 0.26 17 0.59 0.48 0.16 0.41 18 0.71 0.54 0.16 0.78 18 0.66 0.52 0.16 0.35 19 0.55 0.44 0.15 0.32 19 0.68 0.55 0.17 0.45 19 0.7 0.53 0.16 0.54 20 0.59 0.47 0.17 0.42 20 0.53 0.41 0.15 0.33 21 0.59 0.47 0.16 0.58

p=vom, q=meanL*300 Ct Gap>6,7 0 1 1 1 2 1 2 1 1 3 3 1 4 4 1 5 3 1 6 9 1 7 5 1 8 9 1 9 10 1 10 4 1 11 6 1 12 6 1 13 10 1 14 7 1 15 6 1 16 7 1 17 4 1 18 3 2 20 5 1 21 6 1 22 1 1 23 2 1 24 2 1 25 2 1 26 1 2 28 3 2 30 3 2 32 2 2 34 1 1 35 3 1 36 3 1 37 1 1 38 1 1 39 3 1 40 2 2 42 1 2 44 1 2 46 1 8 54 1 2 56 1 3 59 1 26 85 1 17102 1 12114 1 gaps only at end

p=vomK, q=mnKL*100 Ct Gap>2.3 0 2 2 2 1 1 3 1 3 6 1 1 7 1 2 9 2 2 11 1 3 14 1 1 15 1 1 16 1 2 18 2 1 19 1 1 20 2 1 21 3 1 22 2 1 23 1 1 24 6 1 25 1 1 26 1 2 28 3 1 29 1 1 30 3 2 32 3 1 33 2 1 34 3 1 35 5 1 36 3 1 37 5 1 38 3 1 39 5 1 40 3 1 41 2 1 42 1 1 43 2 1 44 2 1 45 3 1 46 3 1 47 4 1 48 3 1 49 1 1 50 3 2 52 2 1 53 3 1 54 8 1 55 3 1 56 3 1 57 4 1 58 2 1 59 1 1 60 2 1 61 5 1 62 2 2 64 1 1 65 2 1 66 1 2 68 3 1 69 2 1 70 1 4 74 1 2 76 1 2 78 1 1 79 2 1 80 1 3 83 2 2 85 1 3 88 1 13101 1

C6=4

C6=3 C7=1 C9=1

6 7 8 9 10 11 12 13 14 15 16 =C5 17 14 25 24 13 9 4 5 6 8

13 14 15 16 =C 1 1 1 313 14 15 16 =C 1 1 312 outlier

16 outlier

0 1 8 8 1 3 11 1 3 14 1 1 15 1 3 18 1 2 20 1 2 22 1 2 24 1 1 25 1 1 26 1 1 27 1 5 32 1 1 33 3 2 35 1 2 37 1 1 38 2 1 39 2 1 40 3 1 41 2 1 42 3 1 43 2 1 44 4 1 45 1 1 46 1 3 49 2 1 50 3 1 51 1 1 52 1 2 54 3 1 55 2 1 56 6 2 58 1 1 59 1 1 60 2 2 62 3 1 63 2 1 64 3 1 65 3 1 66 2 1 67 4 1 68 1 1 69 4 1 70 4 1 71 2 1 72 3 1

73 2 1 74 1 1 75 3 1 76 2 1 77 2 1 78 1 1 79 2 1 80 1 1 81 1 1 82 1 1 83 2 1 84 2 1 85 3 1 86 2 2 88 4 1 89 4 3 92 1 3 95 1 1 96 1 2 98 1 5103 1 11114 1

16 outlier

13 outlier

15=1 16=1

outlier outlier

Some Thoughts

1. Defining outliers: Given a distance dominated functional, f, avgGap = (fmax-fmin) / count. y is an outlier if there is a gap around y > 3*avgGap.

2. If d and t are trained over DT and let Gradient=G=(Gd, Gt). Instead of line searching using F(s) = f +sG, always use a 2D rectangle search, sse(sd,st) = sse(f + sd*Gd + st*Gt) Set sse/sd =0 and sse/st = 0.

3. I am tending to go toward identifying "dense cells" (be they spherical, barrel or cone) and then fusing them together using some fusion algorithm. It is difficult to position spheres, barrels and cones properly around convex clusters. Also if the cluster is has some small parts that protrude, it will not enclose well. For singleton cluster identification (outliers) and doubleton too, this doesn't apply.

4. I am tending to go toward identifying "dense cells" (be they spherical, barrel or cone) and then fusing them together using some fusion algorithm. It is difficult to position spheres, barrels and cones properly around convex clusters. Also if the cluster is has some small parts that protrude, it will not enclose well. For singleton cluster identification (outliers) and doubleton too, this doesn't apply.

5. On approach to try is to start with a small barrel radius and find the most dense region between two consecutive gaps. That should identify a portion of a dense cluster. The question of how to go from there brings to mind lots of possibilities

1. take the "center" of that dense pipe piece as the center of a spherical or barrel cluster and look for gaps.2. try to move to a better centroid for that cluster by some sort of gradient asc/desc process.3. In a "GA mutation" fashion, jump to a nearby centroid, governed by some fitness function (e.g., count in

pipe).

ABALONE ring len diam heig Shell 4 0.17 0.13 0.09 0.01 4 0.24 0.19 0.06 0.02 4 0.21 0.15 0.05 0.01 5 0.17 0.13 0.05 0.01 5 0.20 0.15 0.05 0.01 5 0.27 0.19 0.06 0.03 5 0.24 0.17 0.04 0.02 6 0.36 0.28 0.08 0.07 6 0.37 0.27 0.09 0.07 6 0.37 0.26 0.07 0.07 6 0.32 0.24 0.07 0.04 6 0.31 0.23 0.07 0.04 7 0.30 0.23 0.08 0.04 7 0.38 0.29 0.08 0.08 7 0.36 0.26 0.09 0.07 7 0.40 0.31 0.1 0.11 7 0.26 0.2 0.06 0.02 7 0.4 0.32 0.09 0.1 7 0.35 0.26 0.09 0.06 7 0.33 0.25 0.08 0.05 7 0.42 0.32 0.09 0.1 7 0.46 0.37 0.12 0.15 7 0.50 0.4 0.12 0.17 7 0.39 0.29 0.09 0.07 7 0.35 0.28 0.08 0.11 7 0.32 0.26 0.09 0.06 7 0.45 0.34 0.10 0.13 7 0.36 0.29 0.08 0.1 7 0.47 0.37 0.13 0.16 7 0.35 0.26 0.09 0.07 8 0.47 0.35 0.12 0.13 8 0.42 0.33 0.11 0.13 8 0.52 0.41 0.12 0.19 8 0.4 0.32 0.11 0.1 8 0.42 0.3 0.09 0.12 8 0.52 0.4 0.12 0.18 8 0.46 0.35 0.10 0.12 8 0.47 0.38 0.13 0.17 8 0.45 0.35 0.10 0.14 8 0.44 0.35 0.12 0.13 8 0.40 0.30 0.08 0.08 8 0.38 0.29 0.08 0.08 8 0.47 0.37 0.13 0.14 8 0.5 0.4 0.14 0.22 9 0.57 0.44 0.14 0.28 9 0.43 0.39 0.10 0.13 9 0.35 0.29 0.09 0.09 9 0.51 0.39 0.13 0.2 9 0.53 0.42 0.13 0.21 9 0.53 0.43 0.15 0.25 9 0.52 0.42 0.16 0.24 9 0.50 0.40 0.11 0.17 9 0.47 0.36 0.10 0.14 9 0.46 0.35 0.13 0.16 9 0.42 0.35 0.10 0.16 9 0.45 0.32 0.1 0.11 9 0.43 0.32 0.08 0.10 9 0.45 0.33 0.10 0.11 9 0.49 0.39 0.12 0.15 9 0.44 0.35 0.12 0.12 9 0.47 0.37 0.12 0.16 9 0.34 0.25 0.09 0.06 9 0.56 0.43 0.15 0.31 9 0.40 0.32 0.11 0.11 9 0.46 0.36 0.10 0.17 9 0.59 0.47 0.14 0.31 9 0.28 0.20 0.08 0.04 9 0.47 0.37 0.12 0.14 9 0.55 0.41 0.13 0.2 9 0.37 0.28 0.09 0.09

Oblique Barrel FAUST on Abalone:Use a pipe to look for densest region (between gaps Look for spherical gaps from center of that region.

10 0.56 0.44 0.16 0.26 10 0.53 0.40 0.14 0.20 10 0.47 0.36 0.12 0.15 10 0.58 0.47 0.16 0.33 10 0.47 0.37 0.13 0.17 10 0.38 0.27 0.1 0.08 10 0.44 0.34 0.1 0.13 10 0.5 0.4 0.12 0.22 10 0.6 0.47 0.15 0.30 10 0.53 0.41 0.14 0.20 10 0.61 0.48 0.16 0.30 10 0.53 0.41 0.13 0.2 10 0.48 0.36 0.13 0.16 10 0.43 0.35 0.11 0.13 10 0.55 0.45 0.14 0.26 10 0.56 0.44 0.15 0.24 10 0.47 0.39 0.12 0.17 10 0.47 0.37 0.12 0.15 10 0.49 0.38 0.12 0.17 10 0.47 0.35 0.1 0.18 10 0.44 0.36 0.12 0.15 10 0.44 0.34 0.10 0.16 10 0.46 0.35 0.12 0.15 10 0.62 0.49 0.16 0.39 11 0.49 0.38 0.13 0.19 11 0.57 0.42 0.14 0.2 11 0.55 0.42 0.14 0.28 11 0.35 0.28 0.09 0.07 11 0.49 0.37 0.13 0.22 11 0.58 0.45 0.18 0.28 11 0.53 0.41 0.11 0.18 11 0.56 0.45 0.15 0.24 11 0.54 0.41 0.12 0.21 11 0.57 0.44 0.13 0.26 11 0.56 0.44 0.14 0.3 11 0.57 0.48 0.17 0.38 11 0.54 0.43 0.16 0.28 12 0.56 0.44 0.16 0.32 12 0.57 0.46 0.18 0.44 12 0.62 0.51 0.17 0.67 12 0.52 0.41 0.15 0.23 12 0.55 0.44 0.15 0.26 12 0.60 0.47 0.16 0.34 12 0.57 0.45 0.16 0.33 12 0.5 0.4 0.13 0.24 12 0.56 0.44 0.15 0.27 12 0.59 0.44 0.14 0.28 13 0.70 0.55 0.2 0.49 13 0.60 0.45 0.19 0.31 13 0.59 0.49 0.18 0.48 13 0.59 0.46 0.17 0.39 13 0.62 0.46 0.14 0.4 13 0.55 0.42 0.13 0.27 14 0.58 0.45 0.14 0.36 14 0.55 0.42 0.13 0.27 14 0.66 0.53 0.19 0.48 14 0.52 0.38 0.14 0.21 14 0.51 0.42 0.14 0.25 14 0.61 0.47 0.17 0.34 15 0.6 0.47 0.15 0.28 15 0.45 0.36 0.09 0.15 15 0.55 0.43 0.14 0.25 15 0.69 0.56 0.19 0.48 15 0.60 0.47 0.18 0.29 15 0.68 0.56 0.16 0.46 15 0.53 0.43 0.16 0.33 15 0.56 0.42 0.13 0.25 16 0.52 0.42 0.16 0.32 16 0.54 0.47 0.15 0.34 16 0.65 0.52 0.19 0.44 16 0.54 0.42 0.12 0.26 17 0.59 0.48 0.16 0.41 18 0.71 0.54 0.16 0.78 18 0.66 0.52 0.16 0.35 19 0.55 0.44 0.15 0.32 19 0.68 0.55 0.17 0.45 19 0.7 0.53 0.16 0.54 20 0.59 0.47 0.17 0.42 20 0.53 0.41 0.15 0.33 21 0.59 0.47 0.16 0.58

p=vom, q=mean 200R Ct Gp 1 1 1 2 2 1 3 1 1 4 5 1 5 2 1 6 3 1 7 4 1 8 5 1 9 3 1 10 4 1 11 5 2 13 4 1

1000L Ct Gp 3 1 2 5 1 3 8 1 1 9 1 3 12 1 1 13 1 2 15 1 2 17 1 5 22 1 8 30 1 2 32 1 1 33 1 2 35 2 1 36 1 2 38 1 2 40 1 4 44 1 10 54 1 2 56 1 2 58 3 8 66 1 2 68 1 1 69 1 3 72 1 1 73 1 2 75 1 2 77 2 8 85 1 4 89 1 2 91 1 4 95 1

outliers

6 7 8 9 10 11 12 13 14 15 16 Den=6/10

5 17 2 3 2 1 4 5 6 8outlier

6 7 8 9 10 11 12 13 14 15 16 Den=8/11

1 2 2 1 1 1outlier6 7 8 9 10 11 12 13 14 15 16

Den=8/10 1 2 2 6 7 8 9 10 11 12 13 14 15 16 Den=3/4 1 2

6 7 8 9 10 11 12 13 14 15 16 Den=5/6 2 2 1outlier

2 9's1 10

Fval Ct Gp rnd(F-minF)/4 0 1 7 7 1 411 1 112 2 315 2 116 2 117 1 118 3 119 1 120 2 121 2 122 3 123 4 124 5 125 1 126 3 127 2 229 1 130 1 232 2 133 1 134 4 135 1 136 2 137 2 138 4 139 3 140 4 141 1 142 4 143 6 144 2 246 9 147 1 148 3 149 3 150 1 151 7 152 2 153 1 154 1 155 1 156 2 157 1 158 4 462 5 163 2 265 1 166 1 167 2 168 4 169 1 170 6 272 1 173 1 174 5 175 1 176 1 278 1 179 3 281 1 283 2 184 1 286 1

L=1L=1

L=2 M=1 1 error

Barrel Oblique FAUST1. One round of L-gapping2. One round of R-gapping

L=38 M=35 H=32

L=2 M=15 H=23

Fval Ct G [60,87) rnd(F-minF)/1000 0 2 1 1 1 5 6 3 2 8 2 3 11 2 5 16 4 3 19 1 1 20 1 5 25 3 2 27 1 2 29 2 3 32 3 1 33 1 4 37 1 1 38 1 1 39 1 2 41 1 11 52 1 1 53 1 4 57 1 1 58 1 1 59 2 1 60 1 1 61 1 4 65 1 8 73 1

L=1 M=5 H=3

L=1 M=4 H=22

L=1 M=5 H=7

L=1 M=5 H=6

L=1 M=5 H=6

L=1 M=2 H=1

L=1 M=5 H=1

L=1 M=2 H=22

L=1 M=1H=22L=1 M=1H=22 3 errors

Fval Ct G (13,60) rnd(F-minF)/500 0 1 2 2 4 1 3 1 4 7 1 1 8 1 1 9 2 1 10 5 2 12 2 2 14 2 1 15 2 1 16 2 2 18 5 1 19 4 2 21 5 2 23 2 2 25 3 1 26 1 1 27 1 1 28 2 1 29 1 1 30 1 3 33 4 1 34 1 1 35 1 1 36 1 1 37 3 1 38 1 1 39 3 1 40 1 2 42 2 1 43 1 1 44 2 1 45 1 4 49 2 1 50 2 1 51 2 1 52 1 2 54 2 2 56 4 4 60 3 1 61 3 4 65 1 2 67 1 1 68 1 1 69 1 1 70 1 3 73 1 1 74 1 4 78 1 8 86 2 2 88 1 1 89 1 3 92 1 7 99 1 3102 1 5107 2

Overall accuracy = 69.33%

L=1 M=2 H=6

L=1 M=3 H=6L=1 M=3 H=2

L=2 M=2 H=5

L=1 M=2 H=4

L=3 M=1 H=1L=2 M=1 H=1

L=4 M=3 H=2

L=10 M=7 H=4

L=3 M=4 H=4L=3 M=1 H=4L=3 M=1 H=4

L=5 M=1 H=4L=1 M=1 H=4

L=2 M=2H=4

L=1 M=1H=4L=1 M=1H=4L=1 M=2H=4

L=1 M=2H=4L=1 M=1H=4L=1 M=1 H=4L=1 M=1 H=4L=1 M=3 H=1 42 errors

X x1 x2p1 1 1p2 3 1p3 2 2p4 3 3p5 6 2p6 9 3p7 15 1p8 14 2p9 15 3pa 13 4pb 10 9pc 11 10pd 9 11pe 11 11pf 7 8

xofM 11 27 23 34 53 80118114125114110121109125 83

No zero counts yet (=gaps)

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p2 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0

p1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1

p0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p2' 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1

p1' 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0

p0' 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1

p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p4 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p5 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p6' 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

f=p1 and xofM-GT=23. First round of finding Lp gaps

width = 24 =16 gap: [100 0000, 100 1111]= [64,80)

width=23 =8 gap:[010 1000, 010 1111]=[40,48)

width=23 =8 gap:[011 1000, 011 1111]=[56,64)

width= 24 =16 gap: [101 1000, 110 0111]=[88,104)

width=23=8 gap:[000 0000, 000 0111]=[0,8)

OR between gap 1 & 2 for cluster C1={p1,p3,p2,p4}

OR between gap 2 and 3 for cluster C2={p5}

between 3,4 cluster C3={p6,pf} Or for cluster C4={p7,p8,p9,pa,pb,pc,pd,pe}

f=FAUST CLUSTER-fmg: O(logn)

pTree method for finding P-gaps:

P ≡ ScalarPTreeSet( c o fM )