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Ideas

Ideas. f=distance dominated functional, avgGap=(f max -f min )/fct may be a good measurement for setting thresholds, e.g., x is an outlier=anomaly if gap around {x} > 3*avgGap?. If the minimum barrel radii >> 0, we have chosen a d-line far from the data.

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Ideas

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  1. Ideas f=distance dominated functional, avgGap=(fmax-fmin)/fct may be a good measurement for setting thresholds, e.g., x is an outlier=anomaly if gap around {x} > 3*avgGap? If the minimum barrel radii >> 0, we have chosen a d-line far from the data. It may be advisable to pick p to ba an actual data point. 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] Note we don't round, so we are calculating pTree bitslices by truncating. We don't even need to do that! For fixed piont, 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 the mantissa. The exponent shifts the slice name. E.g., .1011  25 .0010  23 .1010  2-1 If d and t are trained over DocTerm, DT, Gradient=G=(Gd, Gt). Instead of a LineSearch using F(s)=f +sG, always use 2dRectangleSearch, F(sd,st)=F(f + sd*Gd + st*Gt). Set F/sd =0 and F/st=0. Better to find dense cells (sphere, barrel, cone) then fuse them? It's difficult to position spheres, barrels, cones around clusters (bumps, protrusion etc.) For outlier clusters (singleton\doubleton) this does not apply. An algorithm?: start with small barrel radius, find the dense region between two consecutive gaps within this pipe. Should identify a portion of a dense cluster. How to go from there? Lots of possibilities. a. Use centroid of dense pipe piece as sphere|barrel center. b. Move to a better centroid for that cluster by a gradient asc/desc process c. In a "GA mutation" fashion, jump to a nearby centroid, governed by some fitness function (e.g., count in dense pipe piece). 24 1 0 0 22 1 0 0 21 1 1 0 20 0 0 0 2-1 0 0 0 2-2 0 0 1 2-3 0 0 0 2-4 0 0 1 23 0 0 0 SSPTS = set of all SPTSs (columns of reals); V = n-dim vector space. Code operations on SSPTS (both 1 level or multi-level): SSPTS  SSPTS  SSPTS (Binary Algebraic Operations): including: +, -, /,RWP =Row_Wise_Product 10110. 10. .01010 {SPTSk}k=1..n  SSPTS (Unary ops.Typically SPTSk=Vk) incl: SDv(Square Distance from a fixed vector, vV) DPv(Dot Product with a fixed vector, vV) ERa= FP's EinRings (n=1, rR) result masks rows s.t. row < a Oblique FAUST, Barrel (OFLB) Alternate Lpqx, Bpqx to produce a cluster dendogram (topdown). Take p=1st_TR pt? d=vomavg Defining Avg Density? AvD = count / k=1..dim(maxk-mink)? This is for the purpose of choosing good Thresholds. MinGapThresh=Tb,AvD≡ b*(1/ AvD)1/dim? (b=adjustable parameter If we're given a TrainingSet, TR, with K classes, is avgk=1..Kvomk a better mediod than VoM? Take p=MinCorner, q=MaxCorner of box circumscribing {VoMk}k=1..K better than not circ box of TR? SPTS  Rincludes AGa= YC's Aggregates and iceberg queies: count, sum, avg, max, min, median, rank_k, top_k, IceBergQueries. SSPTS  SSPTS (Unary Operations) including: SPc=Scalar_Product (Multiply each SPTS row by same constant, c. Use const SPTS? all rows=c, then RWP. More efficient? w/o forming const SPTS? Use c's bit pattern c only? (subset of previous with n = |SSPTS|?) Note, SSPTS includes SPTSs of all cardinalities (= depths = # of rows) It seems best to code on SSPTS rather than on SSPTSn (card(SPTS)=n). Of course, it is very important to know what the rows represent so as to avoid nonsense results, however, why restrict the operations themselves? When SPTS operands are of different depths, the result SPTS's depth = depth of the shallowest operand (operate from the top of each).

  2. Oblique FAUST (OF)Clustering:Linear (default) OFL,Spherical OFS, Barrel OFB, Conical OFC) a2 Define distance function ds(x,y):TBLTBLR ds(x,y)= kCRrk|xk-yk|2 + kCCck|xk-yk| where CR is the set of real columns, CC is the set of categorical columns (consider coded columns as real) and rk, ck are real coefficients. Each method uses a real valued functional from X to R and all methods are completely data parallel (data can be distributed over a cluster, processed in parallel (dot product), then the partial results sent home to be added. Bpdx a1 x r d p d No gaps show on the red, blue or green projection lines Note: Bpd(x) = Sp(x) - L2pd(x) Cp,d(x)=(x-p)od / (x-p)o(x-p) Oblique FAUST Cone (OFC)(Enclose clusters with cone gaps) GapLower GapUpper Note: C2pd(x) = L2pd(x) /Sp(x) p p gapBarrel Assume a real number table, TBL(C1..Cn), (= n-dim vector space; or categorical columns, either code to real numbers or bitmap, e.g., a Month column can be coded as {1,...,12} and a Color column can be bitmapped by Red(yes=1|no=0)...Violet(yes=1|no=0) ). TBL is converted to a PTreeSet. Lp,d:XR: Lp,d(x)=(x-p)od Oblique FAUST Linear (OFL) clustering (Enclose clusters between (n-1)-dimensional hyperplanar gaps) Find a1<a2 such that =GapLower={x | a1<Lpd(x)<a1+T}and =GapUpper={x | a2<Lpd(x)<a2+T}and C={x|a1+T<Lpd(x)<a2} Bp,d(x)=(x-p)o(x-p)-((x-p)od)2Oblique FAUST Barrel (OFB)(Enclose clusters with barrel gaps) Search for GapLower>T, GapUpper>T and GapBarrel>T2 (BR≡Barrel_Radius) Sp(x)=(x-p)o(x-p) Oblique FAUST Spherical (OFS)(Enclose clusters with spherical gaps) Search Sp for spherical gap, {x | r2  Sp(x) < (r+T)2}= so that the interior of the r-sphere about p encloses a sub-cluster.

  3. M1 H1 L18 M26 H28 C2 M3 L1 M3 H3 C31 L1 M1 H4 C32 H1 M1 H5 C33 H1 H2 M1 M1 M2 M1 H3 C26 L1 L1 M1 d=4 M2 L1 M1 H1 M3 H1 C4 H4 L20 M9 H4 C1 M1 L2 M3 H4C25 L1 L9 M1C21 M1 M3 . H3 L2 M4 H3C23 L1 M1 d=4 . H4 L1 M1 L2 M3 H16C24 H1 M3 M1 L1 M1 L2 M12 H 17 C3 L2 M1 C0 L4 M3 H1C22 L6 L3 M1 C211 H1 L1 M1 ' H3 M2 H2 M1 H1C251 H1 L1 L1 M2 H16 C241 M1 H5 M3 . H1 L1 M1 C231 M1 M1 . H5 M1 . H5 M3 L1 H1 L1 H1 H5 M2 H1 M3 H1 C27 M1 L1 L2 L11 M3 C11 L4 M1 M2 L1 M2 H1 C12 L3 M1 H3 L1 M1 H5 C2411 M2 . H1 L11 M3 L1 M1 L1 L1 H1 M2 if 1st B radius>>0, use p=min_radius_pt OF LB...LBClustering on Concrete(STrength,ConcreteMix,WAter,FineAggregate, AGgregate). Assess STerror L<40M<60H 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 (x-p)od/4 gp3 C11 15 1 2 17 1 3 20 1 1 21 1 1 22 3 1 23 1 1 24 2 1 25 1 4 29 3 (x-p)od/4 gp3 C21 35 1 1 36 1 1 37 2 1 38 1 1 39 1 3 42 3 1 43 1 (x-p)od/4 Ct Gp3 C p 140 192 807 3 T=MGW=12 d=x-n=.58 .15 .58 .53 CONCRETE 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 Br/4 gp3 C211 0 1 4 4 1 3 7 1 3 10 1 Br/4 gp3 C0 19 2 46 65 1 0 1 7 7 1 4 11 2 1 12 1 3 15 4 2 17 1 1 18 3 1 19 2 1 20 2 1 21 1 1 22 6 1 23 3 1 24 4 1 25 2 1 26 1 1 27 2 2 29 2 3 32 2 1 33 3 1 34 2 1 35 1 1 36 2 1 37 3 1 38 3 1 39 5 1 40 2 1 41 2 1 42 8 1 43 1 1 44 2 1 45 3 1 46 6 1 47 2 1 48 2 1 49 3 1 50 2 1 51 8 2 53 1 1 54 1 1 55 1 1 56 3 1 57 1 1 58 3 3 61 2 1 62 3 1 63 2 2 65 1 1 66 2 1 67 5 1 68 1 1 69 1 1 70 5 1 71 1 2 73 1 1 74 6 1 75 1 3 78 4 3 81 1 1 82 1 1 83 2 3 86 1 Br/4 gp3 C1 0 1 5... 7 1 3... 21 1 3 24 4 19 43 1 4 47 1 4... 53 1 3... 59 1 7... 68 1 10... 79 1 (x-p)od/4 gp3 C23 30 3 4 34 1 3 37 1 4 41 1 4 45 1 5 50 1 1 51 1 (x-p)od/4 gp3 C12 20 1 3 23 1 4 27 2 Br/4 gp3 C231 0 1 35 35 1 (x-p)od/4 gp3 C22 38 1 1 39 1 1 40 1 5 45 1 3 48 1 3 51 1 1 52 1 5 57 1 Br/4 gp3 C241 0 1 2 ... 4 4 5 9 1 1 10 4 6 16 1 1 17 4 3 20 1 41 61 1 (x-p)od/4 g3 C411 13 1 9 21 5 Br/4 gp3 C2 0 1 3 3 1 1 ... 8 1 5 13 1 3 16 2 1 18 3 3 21 1 1 ... 26 2 3 29 1 7 36 1 2 ... 43 2 3 46 2 1 ... 48 2 3 51 1 9 60 1 2 62 3 13 75 1 7 82 1 1 83 1 4 87 1 1 ... 91 1 3 94 2 Br/4 gp3 C251 0 1 25 25 1 (x-p)od/4 gp3 C24 36 1 3 39 1 2 ... 53 1 5 58 1 (x-p)od/4 gp3 C25 38 1 4 42 1 3 45 1 4 49 1 5 54 1 1 55 1 3 58 1 (x-p)od/4 gp3 C26 46 1 3 49 1 2 51 1 5 56 1 (x-p)od/4 gp3 C27 46 1 3 49 1 2 51 1 5 56 1 (x-p)od/4 gp3 C33 0 1 30 30 4 2 32 1 Br/4 ct gp3 C3 0 1 13 13 3 1 14 3 3 17 5 2 19 1 4 23 1 6 29 2 1 30 2 2 32 1 2 34 1 4 38 1 4 42 1 2 44 1 3 47 1 57 104 1 4 108 1 2 110 1 3 113 1 11 124 3 (x-p)od/4 gp3 C31 67 3 3 70 3 (x-p)od/4 gp3 C32 0 2 32 32 3 2 34 1 c(Clust dendogram w/o purity) c0 c1 c2 c3 c4 c31 c32 c33 c11 c12 c21 c22 c23 c24 c25 c26 c27 c251 c211 c231 c241 c2411 Br/4 ct gp3 C4 40 1 33 73 1 1 74 1 42 116 1

  4. (x-p)od/4 Ct Gap>=3 p=mn. q=mx OF_LBL: Clustering on Concrete(STrength,ConcreteMix,WAter,FineAggregate, AGgregate) (Cluster CM,WA,FA,AG. Assess error w L:ST<40 M:41<ST<60) (x-p)o(xop)/4 Ct Gap>=3, p=1st (x-p)od/4 Ct Gap>=3 0 1 5 5 1 1 6 3 2 8 1 1 9 2 1 10 2 1 11 3 1 12 7 1 13 1 1 14 3 1 15 1 1 16 3 1 17 4 1 18 3 1 19 1 1 20 1 1 21 1 1 22 1 1 23 5 1 24 1 2 26 1 2 28 2 1 29 2 1 30 1 1 31 1 2 33 1 6 39 1 3 42 1 1 43 2 3 46 9 1 47 7 1 48 2 1 49 1 1 50 1 2 52 4 1 53 1 1 54 2 1 55 1 1 56 1 1 57 3 2 59 1 6 65 2 1 66 3 1 67 2 10 77 1 1 78 1 2 80 1 5 85 1 1 86 1 4 90 1 1 91 2 2 93 1 1 94 1 2 96 1 0 1 7 7 1 4 11 1 1 12 2 3 15 2 1 16 2 1 17 1 1 18 3 1 19 1 1 20 2 1 21 2 1 22 3 1 23 4 1 24 5 1 25 1 1 26 3 1 27 2 2 29 1 1 30 1 2 32 2 1 33 1 1 34 4 1 35 1 1 36 2 1 37 2 1 38 4 1 39 3 1 40 4 1 41 1 1 42 4 1 43 6 1 44 2 2 46 9 1 47 1 1 48 3 1 49 3 1 50 1 1 51 7 1 52 2 1 53 1 1 54 1 1 55 1 1 56 2 1 57 1 1 58 4 4 62 5 1 63 2 2 65 1 1 66 1 1 67 2 1 68 4 1 69 1 1 70 6 2 72 1 1 73 1 1 74 5 1 75 1 1 76 1 2 78 1 1 79 3 2 81 1 2 83 2 1 84 1 2 86 1 L=0 M=2 H=1 L=1 M=1 L=2 M=12 H=21 L=0 M=3 H=1 L=1 L=1 L=2 M=10 H=21C2 L=1 M=1 H=1 L=1 M=1 H=5 L=1 M=1 H=5 L=1 M=2 H=5 L=1 M=2 H=5 L=1 M=5 H=1 M=1 L=34 M=14 H=4 C1 L=1 M=4 H=6 L=0 M=2 H=8 H=1 H=4 M=1 H=6 H=1 H=2 M=3 M=2 M=1 M=1 L=1 M=1 H=4 L=5 L=12 M=2 L=17 M=12 H=3 C11 H=1 L=1 M=1 H=1 L=1 M=1 H=6 M=3 H=3 L=40 M=36 H=32 (x-p)o(xop)/4 Ct Gap>=3, p=1st 0 1 13 13 3 1 14 3 3 17 5 1 18 1 4 22 1 1 23 3 5 28 1 1 29 1 1 30 2 2 32 1 1 33 1 2 35 1 3 38 1 4 42 1 2 44 1 3 47 1 56 103 1 5 108 1 2 110 1 3 113 1 11 124 3 0 1 2 2 1 1 3 1 1 4 2 4 8 2 1 9 1 1 10 1 1 11 3 1 12 3 1 13 1 1 14 3 3 17 2 2 19 1 1 20 1 1 21 1 1 22 1 1 23 1 1 24 2 1 25 3 1 26 1 1 27 4 2 29 2 1 30 4 2 32 1 2 34 2 1 35 2 2 37 2 1 38 1 2 40 1 6 46 1 OF_LBL: 75% accurate Next OFL..L (just linears) for comparison First: LBLBL on C11 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 min=p 140 122 594 3 max=q 540 228 993 365 T=12 d 0.58 0.15 0.58 0.53 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 (x-p)od/4 Ct Gap>=3 p=mn. q=mx 0 1 2 2 1 1 3 1 1 4 4 3 7 2 1 8 1 1 9 1 1 10 4 1 11 2 6 17 1 1 18 7 3 21 1 4 25 1 1 26 1 2 28 1 7 35 1 3 38 1 18 56 1 8 64 1

  5. (x-p)od/4 Ct Gap>=3 OFL..L: Clustering on Concrete(STrength,ConcreteMix,WAter,FineAggregate, AGgregate) (Cluster CM,WA,FA,AG. Assess error w L:ST<40 M:41<ST<60) OFLL: doesn't look promising. Also tried staying with p=nnnn q=xxxx without promising results. L=1 M=7 H=6 L=1 H=12 L=1 M=1 L=2 M=12 H=21 C2 L=0 M=3 H=1 L=0 M=2 H=1 L=0 M=3 H=4 L=5 L=8 M=5 L=17 M=11 L=17 M=3 L=1 H=3 L=5 M=4 H=6 L=2 M=1 L=40 C1 M=36 H=32 0 1 6 6 1 5 11 2 1 12 1 2 14 1 1 15 3 2 17 1 1 18 3 1 19 2 1 20 2 1 21 2 1 22 5 1 23 3 1 24 4 1 25 2 1 26 1 1 27 2 2 29 2 2 31 2 2 33 3 1 34 2 1 35 2 1 36 1 1 37 4 1 38 3 1 39 4 1 40 2 1 41 2 1 42 8 1 43 2 1 44 1 1 45 4 1 46 5 1 47 3 1 48 1 1 49 4 1 50 5 1 51 4 1 52 1 2 54 1 1 55 3 1 56 1 1 57 2 1 58 2 3 61 3 1 62 2 1 63 2 2 65 1 1 66 2 1 67 5 1 68 1 1 69 2 1 70 4 1 71 1 2 73 2 1 74 5 1 75 1 2 77 1 1 78 3 3 81 1 1 82 1 1 83 2 3 86 1 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 (x-p)od/4 Ct Gap>=3 p=naaa q=xaaa min=p 140 122 594 3 max=q 540 228 993 365 T=12 d 0.58 0.15 0.58 0.53 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 0 3 6 6 3 1 7 2 5 12 9 1 13 2 1 14 2 4 18 9 2 20 4 2 22 5 1 23 3 1 24 4 3 27 3 4 31 3 5 36 4 5 41 2 1 42 1 1 43 4 1 44 2 2 46 3 2 48 1 1 49 2 6 55 13 3 58 4 2 60 5 2 62 4 3 65 1 6 71 2 1 72 2 2 74 3 9 83 2 17 100 1

  6. (x-p)od/4 Ct Gap>=3 Slide=OF_LS: Clustering on Concrete(STrength,ConcreteMix,WAter,FineAggregate, AGgregate) (Cluster CM,WA,FA,AG. Assess error w L:ST<40 M:41<ST<60) OF_LS: in the Spherical round, mask off at the first radial thining, then try the last point. Repeat alternating first, last, middle.... At any S-round, if p has no nbrs within T move up (if last, or down if 1st) and redo with that p. Note: Should probably pick p randomly? M=7 L=1 M=1 L=2 M=12 H=21 C2 L=0 M=3 H=1 L=5 M=3 L=0 M=1 H=3 L=18 M=4 H=4 H=19 L=1 M=8 L=1 M=0 H=3 L=1 M=10 L=12 L=40 C1 M=36 H=32 0 1 6 6 1 5 11 2 1 12 1 2 14 1 1 15 3 2 17 1 1 18 3 1 19 2 1 20 2 1 21 2 1 22 5 1 23 3 1 24 4 1 25 2 1 26 1 1 27 2 2 29 2 2 31 2 2 33 3 1 34 2 1 35 2 1 36 1 1 37 4 1 38 3 1 39 4 1 40 2 1 41 2 1 42 8 1 43 2 1 44 1 1 45 4 1 46 5 1 47 3 1 48 1 1 49 4 1 50 5 1 51 4 1 52 1 2 54 1 1 55 3 1 56 1 1 57 2 1 58 2 3 61 3 1 62 2 1 63 2 2 65 1 1 66 2 1 67 5 1 68 1 1 69 2 1 70 4 1 71 1 2 73 2 1 74 5 1 75 1 2 77 1 1 78 3 3 81 1 1 82 1 1 83 2 3 86 1 (x-p)o(x-p)/4 Ct Gap>=3 p=first 0 1 6 6 1 4 10 1 1 11 1 3 14 2 1 15 2 1 16 3 3 19 1 1 20 5 3 ... 109 1 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 min=p 140 122 594 3 max=q 540 228 993 365 T=12 d 0.58 0.15 0.58 0.53 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 (x-p)o(x-p)/4 Ct Gp>=3 p=3RD last 0 4 3 3 1 3 6 1 2 8 5 1 9 1 3 12 1 3 15 4 1 16 1 1 17 1 4 21 2 2 (x-p)o(x-p)/4 Ct Gp>=3 p=last 0 1 7 7 1 5 12 1 1 13 1 1 14 2 1 (x-p)o(x-p)/4 Ct Gp>=3 p=mid 0 2 9 9 1 8 17 2 2 19 1 1 20 1 1 21 2 1 22 2 2 (x-p)o(x-p)/4 Ct Gp>=3 p=middle 0 1 12 12 1 10 22 2 1 23 1 2 25 2 6 31 1 1 32 1 2 (x-p)o(x-p)/4 Ct Gp>=3 p=1st 0 1 11 11 1 4 15 1 1 16 1 1 17 2 1 18 3 1 19 1 1 20 1 2 22 2 1 23 1 2 25 2 1 26 1 2 28 4 3 31 3 1 32 2 1 33 3 4 (x-p)o(x-p)/4 Ct Gp>=3 p=1st 0 1 6 6 1 13 19 3 4 23 1 1 24 2 1 25 2 2 (x-p)o(x-p)/4 Ct Gp>=3 p=last 0 1 8 8 1 7 15 1 14 29 1 1 313 176 612 91 390 146 756 3 425 151 804 28 540 162 676 28 266 228 670 180 last=outliers (x-p)o(x-p)/4 Ct Gp>=3 p=last 19 1 3 22 1 2 24 1 5 29 1 4 33 1 5 38 1 1 39 1 4 43 1 4 47 1 1 48 1 1 49 1 8 57 1 1 Other 5 are outliers

  7. OFLS: on Conc(STrength,ConcreteMix,WAter,FineAggregate, AGgregate) (Cluster CM,WA,FA,AG. Assess error w L:ST<40 M:41<ST<60) L=1 ol L3 H10 L43 M59 H55 M3 (x-p)od Ct T=8 0 1 3 3 1 2 5 1 2 7 1 1 8 2 1 9 1 1 10 3 2 12 2 1 13 1 1 14 4 1 15 4 1 16 3 1 17 4 1 18 1 1 19 3 3 22 4 1 23 1 1 24 5 1 25 3 1 26 4 1 27 1 1 28 2 1 29 3 1 30 1 1 31 5 1 32 4 1 33 6 1 34 2 1 35 1 3 38 6 1 39 2 1 40 3 1 41 3 1 42 3 1 43 1 1 44 6 1 45 1 1 46 3 1 47 2 1 48 2 1 49 4 1 50 5 1 51 2 1 52 1 1 53 3 1 54 1 2 56 3 2 58 4 1 59 1 2 61 2 1 62 3 1 63 1 2 65 4 2 67 2 4 71 1 2 73 1 5 78 1 1 79 1 2 81 1 3 84 1 7 91 1 6 97 1 No gaps Next step do a Sp to firn a dense cell at p (x-p)o(x-p)/4 Ct T=2 DIS 20 22 23 24 25 25 26 29 20 0 17 28 30 38 21 11 31 22 17 0 14 23 27 21 13 26 23 28 14 0 27 23 26 19 21 24 30 23 27 0 14 29 28 33 25 38 27 23 14 0 33 31 29 25 21 21 26 29 33 0 18 14 26 11 13 19 28 31 18 0 23 29 31 26 21 33 29 14 23 0 0 1 4 4 1 4 8 1 2 10 1 1 11 1 1 12 1 2 14 2 2 16 1 1 17 4 1 18 1 1 19 5 1 20 3 1 21 14 1 22 16 1 p(rand) 24 23190 162 803 14 min 140 122 594 3 max 540 228 993 365 T=Dim*AvgGap = 4* 1.58 = 6.3 d=|x-n| .58 .15 .58 .53 ROW ST CM WA FA AG 12 8 140 192 807 3 13 8 168 122 780 3 14 9 190 162 803 3 15 10 310 192 851 3 16 20 230 195 759 14 17 20 238 187 847 3 18 21 212 180 779 14 19 21 191 162 804 14 20 22 166 176 780 28 21 22 234 198 852 14 22 22 230 195 758 14 23 23 234 198 852 28 24 23 190 162 803 14 25 23 363 165 756 7 26 24 168 122 780 28 27 24 338 175 756 3 28 24 286 145 804 3 29 24 222 189 870 14 30 24 230 195 759 28 31 25 319 156 880 3 32 25 222 189 870 28 33 25 230 195 758 28 34 25 195 166 906 14 35 25 212 180 779 28 36 25 166 176 780 14 37 25 250 187 861 14 38 26 191 162 804 28 39 26 195 166 906 28 40 26 238 228 594 7 41 26 238 187 847 14 42 26 213 159 904 14 43 28 190 162 803 28 44 28 389 158 926 3 45 28 234 198 852 56 46 28 199 192 826 28 47 28 140 192 807 28 48 28 324 184 660 3 49 29 380 154 605 3 50 29 375 127 993 3 51 29 313 176 612 3 52 29 250 187 861 28 53 29 166 176 780 56 54 29 222 189 870 56 55 40 214 182 786 28 56 40 190 162 803 100 57 40 469 138 841 3 58 40 238 187 847 56 59 40 333 228 594 270 60 40 212 180 779 100 61 41 333 228 594 365 62 41 390 146 756 3 63 41 222 189 870 100 64 41 191 162 804 100 65 41 531 142 894 3 66 41 190 228 670 28 67 41 380 228 594 90 68 41 380 228 594 270 69 41 380 228 594 180 70 41 230 195 758 100 71 41 402 147 852 3 72 42 475 228 594 270 73 42 190 228 670 90 74 42 428 228 594 90 75 42 475 228 594 90 76 42 475 228 594 365 77 42 199 192 826 180 78 42 428 228 594 180 79 42 250 187 861 100 80 43 213 159 904 56 81 43 475 228 594 180 82 43 313 176 612 7 83 43 428 228 594 270 84 43 213 159 904 100 85 44 428 228 594 365 86 44 238 187 847 100 87 44 199 192 826 360 88 44 140 192 807 180 89 44 380 228 594 365 90 45 140 192 807 360 91 46 375 127 993 7 92 46 266 228 670 28 93 46 374 170 757 7 94 46 214 182 785 28 95 47 190 228 670 180 96 47 214 182 786 56 97 47 425 151 804 7 98 47 266 228 670 90 99 47 531 142 894 7 100 47 380 154 605 7 101 48 304 228 670 28 102 49 304 228 670 90 103 49 425 154 887 7 104 49 425 154 887 7 105 49 266 228 670 180 106 49 425 154 887 7 107 60 425 154 887 28 108 60 375 127 993 56 109 60 425 154 887 28 110 60 425 154 887 28 111 61 374 170 757 28 112 62 540 162 676 28 113 62 425 151 804 28 114 63 374 170 757 56 115 63 375 127 993 91 116 64 425 154 887 56 117 64 425 154 887 56 118 64 425 154 887 56 119 65 425 151 804 56 120 65 374 170 757 91 121 65 425 154 887 91 122 65 313 176 612 56 123 65 425 154 887 91 124 65 425 154 887 91 125 66 439 186 708 28 126 66 319 156 880 56 127 67 469 138 841 28 128 67 313 176 612 91 129 67 425 151 804 91 130 68 286 145 804 28 131 68 475 181 782 28 132 68 319 156 880 91 133 68 402 147 852 28 134 68 338 175 756 91 135 69 469 138 841 56 136 71 363 165 756 28 137 71 363 165 756 28 138 71 363 165 756 28 139 71 363 165 756 28 140 71 469 138 841 91 141 72 475 181 782 56 142 72 439 186 708 56 143 73 286 145 804 56 144 73 439 186 708 91 145 74 390 146 756 28 146 74 475 181 782 91 147 74 402 147 852 56 148 75 402 147 852 91 149 75 324 184 660 28 150 77 363 165 756 56 151 77 286 145 804 91 152 77 363 165 756 56 153 77 363 165 756 56 154 77 363 165 756 56 155 79 390 146 756 56 156 79 363 165 756 91 157 79 363 165 756 91 158 79 363 165 756 91 159 79 363 165 756 91 160 80 324 184 660 56 161 83 390 146 756 91

  8. p=vom, q=mean L/4 Ct Gap>4 0 2 8 8 1 2 10 1 1 11 1 1 12 3 1 13 1 1 14 1 1 15 2 1 16 1 1 17 3 1 18 1 2 20 4 2 22 1 1 23 4 1 24 2 1 25 2 1 26 3 1 27 3 1 28 1 1 29 3 1 30 5 1 31 5 1 32 1 1 33 1 1 34 5 1 35 3 1 36 6 1 37 4 1 38 1 1 39 2 1 40 6 1 41 1 1 42 8 1 43 5 1 44 3 1 45 7 1 46 8 1 47 1 1 48 3 1 49 1 2 51 1 1 52 2 1 53 1 1 54 2 1 55 3 1 56 2 4 60 2 1 61 4 5 66 1 2 68 1 3 71 1 7 78 1 2 80 1 1 81 1 4 85 1 1 86 1 1 87 1 4 91 1 5 96 1 6 102 1 5 107 1 4 111 1 5 116 1 1 117 1 7 124 1 Oblique FAUST Barrel (OFB) Clustering on Concrete(STrength,ConcreteMix,WAter,FineAggregate, AGgregate) 0 3 2 0 2 0 0 1 0 0 1 0 1 2 0 0 0 1 1 1 5 4 7 27 0 5 12 10 3 5 24 5 2 0 0 3 0 23 0 43 27 55 p 210 178 780 19 q 329 170 781 85 L/4 Ct Gap>4 0 1 4 4 1 4 8 1 8 16 1 3 19 2 7 26 1 1 27 2 4 31 1 1 32 3 1 33 2 1 34 1 4 38 1 1 39 2 3 42 1 1 43 2 1 44 3 1 45 4 1 46 4 1 47 5 1 48 4 1 49 9 1 50 1 2 52 1 2 54 1 4 58 1 1 59 3 2 61 9 1 62 3 2 64 1 6 70 1 1 71 3 1 72 1 1 73 3 2 75 2 3 78 2 1 79 2 1 80 4 6 86 2 1 87 6 1 88 12 1 89 4 1 90 1 3 93 2 1 94 2 3 97 2 4 101 2 1 102 1 3 105 2 STrength is the class label so we assess error with L:ST<40 M:41<ST<60. Here we try a "maximized dense cell" approach. Find dense regions between consecutive pipe gaps (p=1st, q=last pt. The pipe is around the pq=line of radius r0 as follows If pts evenly spaced, how far apart each adjacent pair? N=num of equiwidth subints on a side=count1/n =150.25=3.5 L = equiwidth of a side = Avgk=1..n(maxk-mink) / N M= length main diagonal of a singleton cube (= dis between evenly spaced points) = (n*L2)1/2 = 144 Set pipe radius r0=M (furthest nbr) or L=72 (nearest nbr Let pipe center to be C0 GWT = Tb,AvD≡ b * (1/ AvD)1/dim (b is an adjustable parameter, e.g., b=n; AvD= count / k=1..dim (maxk-mink) ) It may be that the pipe is at the edge of a cluster so we try to find the center of the cluster as follows: a. increase barrel stave radius to r1, where Count 1st falls precipitously (reached one edge of the cluster). b. Increase barrel stave radius further until it changes precipitously again at r1 (If it falls, we are leaving our cluster. If it rises, we are entering another cluster on the "found edge" side of our cluster). c. Let C1 = VoM of that barrel stave (between r1-T and r1). d. Find center, C2, of the dense region of r0-pipe through C0 and C1 that contains C0 and C1. e. Find the first spherical gap with center, C3= Avg(e1, e2), where e1, e2 are the pts at the ends of this pipe and on its center line; and radius at least, r3=|e1-e2|/2 76% accuracy. Alg: L(mn-vom) R(find 1st 2 dense radii) L(AVG(means of densities)

  9. p=vom, q=mean L/4 Ct Gap>4 0 2 8 8 1 2 10 1 1 11 1 1 12 3 1 13 1 1 14 1 1 15 2 1 16 1 1 17 3 1 18 1 2 20 4 2 22 1 1 23 4 1 24 2 1 25 2 1 26 3 1 27 3 1 28 1 1 29 3 1 30 5 1 31 5 1 32 1 1 33 1 1 34 5 1 35 3 1 36 6 1 37 4 1 38 1 1 39 2 1 40 6 1 41 1 1 42 8 1 43 5 1 44 3 1 45 7 1 46 8 1 47 1 1 48 3 1 49 1 2 51 1 1 52 2 1 53 1 1 54 2 1 55 3 1 56 2 4 60 2 1 61 4 5 66 1 2 68 1 3 71 1 7 78 1 2 80 1 1 81 1 4 85 1 1 86 1 1 87 1 4 91 1 5 96 1 6 102 1 5 107 1 4 111 1 5 116 1 1 117 1 7 124 1 Oblique FAUST Barrel (OFB) Clustering on Concrete(STrength,ConcreteMix,WAter,FineAggregate, AGgregate) 0 1 0 0 1 0 0 0 1 1 2 0 0 0 3 0 2 0 4 7 27 0 5 12 10 3 5 24 5 2 1 1 5 0 3 2 0 23 0 43 27 55 C1^^^ Very Simple MDC alg, do L(mn-vom) R(find 1st 2 dense annulii) L(2 means of those annulii) p 210 178 780 19 q 329 170 781 85 L/4 Ct Gap>4 0 1 4 4 1 4 8 1 8 16 1 3 19 2 7 26 1 1 27 2 4 31 1 1 32 3 1 33 2 1 34 1 4 38 1 1 39 2 3 42 1 1 43 2 1 44 3 1 45 4 1 46 4 1 47 5 1 48 4 1 49 9 1 50 1 2 52 1 2 54 1 4 58 1 1 59 3 2 61 9 1 62 3 2 64 1 6 70 1 1 71 3 1 72 1 1 73 3 2 75 2 3 78 2 1 79 2 1 80 4 6 86 2 1 87 6 1 88 12 1 89 4 1 90 1 3 93 2 1 94 2 3 97 2 4 101 2 1 102 1 3 105 2 ST=class. Assess error with L={ST<40}, M, H={ST>59} "Maximized Dense Cell" or MDC approach: Find dense regions between consecutive pipe gaps (e.g., p=1st, q=last pt. (pipe is around pq-line w radius r0) What r0 defines a pipe? (count >>0 but not too thick) If pts are evenly spaced, how far apart is each adjacent pair? N=# equiwidth subintervals on a side=count1/n =150.25=3.5 L = equiwidth of a side = Avgk=1..n(maxk-mink) / N M= length main diagonal of a singleton cube = (n*L2)1/n Set pipe radius r0=M (furthest nbr) or L=72 (nearest nbr Let C0 = center of dense region of the pipe. GWT=L or M? (If more dense than uniform spacing, dense cell.) 76% accuracy. This is the same accuracy as GV but without and gradient optimizations. GWT = Tb,AvD≡ b * (1/ AvD)1/dim (b=parameter, e.g., b=n; AvD= count / k=1..dim (maxk-mink) ) It may be that the pipe is at the edge of a cluster so we try to find the center of the cluster as follows: a. increase the radius of the annular gap to r1, where Count 1st falls precipitously (reached an edge of cluster). b. Increase radius further until it changes precipitously again at r2 (If it falls, we are leaving our cluster. If it rises, we are entering another cluster on the "found edge" side of our cluster). c. Let C1 = VoM of the annulus between r1-T and r1. d. Find center, C2 = (C0+C1)/2 e. Find the first spherical gap with center, C3= Avg(e1, e2), where e1, e2 are the pts at the ends of theC0C1C2 pipe of radius at least, r3=|e1-e2|/2 Next we compare the full MDC alg, do L(mn-vom) On the L/6 pipe, find C2 as left. L(2 means of those annulii)

  10. f=p1 and xofM-GT=23. First round of finding Lp gaps 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 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 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 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 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 p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 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 p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 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 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 p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 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 p4' 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 p5' 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 p3' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 1 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 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 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 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 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 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 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' 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 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' 0 0 1 1 1 1 1 1 0 1 0 0 0 0 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 p6 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 p3 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 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' 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 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 FAUST CLUSTER-fmg:O(logn) pTree method for finding P-gaps: P ≡ ScalarPTreeSet( c ofM ) xofM 11 27 23 34 53 80 118 114 125 114 110 121 109 125 83 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 X x1 x2 p1 1 1 p2 3 1 p3 2 2 p4 3 3 p5 6 2 p6 9 3 p7 15 1 p8 14 2 p9 15 3 pa 13 4 pb 10 9 pc 11 10 pd 9 11 pe 11 11 pf 7 8 f= OR between gap 2 and 3 for cluster C2={p5} width=23 =8 gap: [010 1000, 010 1111] =[40,48) width=23=8 gap: [000 0000, 000 0111]=[0,8) width=23 =8 gap: [011 1000, 011 1111] =[56,64) width = 24 =16 gap: [100 0000, 100 1111]= [64,80) width= 24 =16 gap: [101 1000, 110 0111]=[88,104) OR between gap 1 & 2 for cluster C1={p1,p3,p2,p4} between 3,4 cluster C3={p6,pf} Or for cluster C4={p7,p8,p9,pa,pb,pc,pd,pe} No zero counts yet (=gaps)

  11. Oblique FAUST Barrel (OFB) Clustering on Concrete(STrength,ConcreteMix,WAter,FineAggregate, AGgregate) (Cluster CM,WA,FA,AG. Assess error w L:ST<40 M:41<ST<60) 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 1 100 1 3 103 1 2 105 3 3 108 1 2 110 1 1 111 2 3 114 1 Alg: One OFL round, then one OFB but In OFB, if least radius is not 0 set p=first pt. Separate at all large r-gaps. 3. If an R-ring has 1 point outlier, else analyze R-ring further. r Ct Gp 0 1 3 3 2 4 7 2 2 9 1 3 12 3 1 13 1 1 14 1 4 18 1 2 20 1 1 21 1 1 22 2 1 23 2 33 56 1 6 62 2 3 65 1 1 66 1 1 67 1 9 76 1 1 77 1 4 81 1 10 91 2 r Ct Gp 0 1 14 14 1 12 26 1 1 27 1 30 57 1 29 86 2 L=3 M=1 H=4 L=17 M=8 H=4 L=1 L=1 L=2 M=15 H=23 L=2 M=1 L=1 M=1 H=3 M=1 outlier L=2 L=5 M=2 L=6 M=8 H=4 L=18 M=3 H=4 L=3 M=2 H=4 L=1 M=9 H=13 L=4 L=1 M=1 M=1 H=1 L=2 M=9 H=13 L=2 M=9 H=4 L=2 M=3 H=1 outliers L=18 M=26 H=28 L=1 M=1 H=3 (M and H's separate but d(L,M)=4 M=1 L=1 M=5 L=3 M=1 H=4 L=6 M=2 H=4 L=3 M=2 H=4 L=1 M=1 H=2 all outlier L=1 M=1 H=2 L=1 M=1 H=1 outlier L=5 M=2 H=1 L=1 M=1 H=1 L=4 M=12 H=21 M=3 H=3 (21 between M's and H's) outliers outliers outlier M=2 outliers M=2 outliers L=1 M=3 L=4 M=2 H=2 Distances 21 22 40 46 21 0.0 48.3 15.9 15.5 L outlir 22 48.3 0.0 48.7 48.6 L outlir 40 15.9 48.7 0.0 1.0 doubleton 46 15.5 48.6 1.0 0.0 M ouliers 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 singleton ouliers 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 L=1 L8 H2 Distances 29 47 29 0.0 4.0 47 4.0 0.0 Distances47 65 47 0.0 154.9 65 154.9 0.0 singleton ouliers 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 80 1 5 85 1 7 92 1 2 94 1 5 99 1 7 106 1 1 107 1 4 111 1 5 116 1 8 124 1 1 125 1 88 1 1 89 4 2 91 1 1 92 1 9 101 1 2 103 1 9 112 1 10 122 4 2 124 1 10 134 1 M8 H7 H9 Dist 42 49 66 72 73 42 0.0 211.6 307.7 294.8 281.9 49 211.6 0.0 159.0 150.3 146.4 66 307.7 159.0 0.0 28.0 63.0 72 294.8 150.3 28.0 0.0 35.0 73 281.9 146.4 63.0 35.0 0.0 all outliers L=38 M=35 H=32 Dist 41 43 65 68 68 68 41 0.0 134.4 392.4 413.9 402.8 437.6 43 134.4 0.0 309.6 313.6 361.8 363.8 65 392.4 309.6 0.0 68.3 135.1 60.3 68 413.9 313.6 68.3 0.0 196.3 106.7 68 402.8 361.8 135.1 196.3 0.0 108.3 68 437.6 363.8 60.3 106.7 108.3 0.0 all outliers (x-p)od/4 Ct Gap>=3 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 4 11 1 1 12 2 3 15 2 1 16 2 1 17 1 1 18 3 1 19 1 1 20 2 1 21 2 1 22 3 1 23 4 1 24 5 1 25 1 1 26 3 1 27 2 2 29 1 1 30 1 2 32 2 1 33 1 1 34 4 1 35 1 1 36 2 1 37 2 1 38 4 1 39 3 1 40 4 1 41 1 1 42 4 1 43 6 1 44 2 2 46 9 1 47 1 1 48 3 1 49 3 1 50 1 1 51 7 1 52 2 1 53 1 1 54 1 1 55 1 1 56 2 1 57 1 1 58 4 4 62 5 1 63 2 2 65 1 1 66 1 1 67 2 1 68 4 1 69 1 1 70 6 2 72 1 1 73 1 1 74 5 1 75 1 1 76 1 2 78 1 1 79 3 2 81 1 2 83 2 1 84 1 2 86 1 min=p 140 122 594 3 max=q 540 228 993 365 T=12 d 0.58 0.15 0.58 0.53 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 both H outliers 2Ms, H outliers 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 12 109 1 13 122 1 all outliers r/5 Ct Gp 0 1 10 10 3 1 11 3 2 13 2 1 14 3 4 18 1 5 23 2 1 24 1 1 25 1 5 30 1 5 35 1 3 38 1 44 82 1 6 88 1 11 99 2 R CtGp 11 3 1 12 7 1 13 6 1 1410 1 15 4 1 16 9 1 17 1 No gaps So 4 errors out of 150 = 98% accuracy. But those "errors" are pts close together, not separable.

  12. OFBon Abalone(Rings,Length,diameter,height,Shell) outlier 15=1 16=1 13 outlier 16 outlier C6=4 C6=3 C7=1 C9=1 outlier 6 7 8 9 10 11 12 13 14 15 16 =C 5 17 14 25 24 13 9 4 5 6 8 13 14 15 16 =C 1 1 1 3 13 14 15 16 =C 1 1 3 16 outlier 12 outlier p=vomK, q=mnK L*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 13 101 1 p=vom, q=mean L*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 17 102 1 12 114 1 gaps only at end 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 .5 .39 0.13 0.18 q=mean .48 .38 0.12 0.21 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 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 5 103 1 11 114 1

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