1. SL Gaps (SL) 1. Check SLp(y) ≡ (y-p)o(y-p) gaps (using a p grid).
Dot Gaps - Sparse Ends (DG-SE) 1. Check Dotpq(y) ≡ (y-p)o(p-q)/|p-q| gaps (using grids for p and d=(p-q)/|p-q|?) Check distances at sparse extremes.
SPD Gaps (SPD) 1. Check SPDpq(y) ≡ SLp(y) - Dotpq(y)2 gaps
Dot Gaps - KMeans (DG-KM) 1. Check Dotp,d(y) gaps (grids of p and d?) Check distances at sparse extremes After several rounds of 1, apply k-means to the resulting clusters (when k seems to be determined).
Dot Gaps - Density Analysis (DG-DA) 1. Check Dotp,d(y) gaps (grids of p and d?) against density of subcluster.
1.1 Check distances at sparse extremes against subcluster density. 2. Apply other methods once Dot ceases to be effective.
Dot Gaps - Square Length (DG-SL) 1. Check Dotp,d(y) (over grid of p,d) and SLp(y) (over grid of p) Check sparse ends distance with subcluster density. (Dotpd , SLp share construction steps!)
SL Gaps - Dot Gaps - Square Length Gaps (SL-DG-SPD) (Dotpq , SLp and SPDpq share construction steps!
SLp(y) ≡ (y-p)o(y-p) = yoy - 2 yop pop
Dotpq(y) ≡ (y-p)od=yod-pod= (1/|p-q|)yop - (1/|p-q|)yoq
Calc yoy, yop, yoq concurrently? Then constant multiplies 2*yop, (1/|p-q|)*yop concurrently. Then add | subtract.
Calculate Dotpq(y)2. Then subtract it from SLp(y)

2. Dot Gaps - Sparse Ends (DG-SE)
1. Check Dotp,d(y) gaps (grids of p and d?).
1.1 Check distances at sparse ends.
Analyzing the thinning at [8,9]:
e21i4 i8 i9 i17i24i26i27i28i38i50e2 e3 e12e5 e17e19e23e29e35e37i20i34 e i i i i i i i i i i e e e e e e e e e e i i
Actual dist from each F=7 to each F=10 is >=4. F-gap from F=6 to F=11 >=4. F-gap from F=6 to F=10>=4.
Separate at F=8.5 to CLUS2.1<8.5 (2 ver, 43 vir) and CLUS2.2>8.5 (44 ver, 4 vir)
Dot gp>=4
p=nnnn
q=xxxx
0 2
3 2
4 1
5 1
7 1
9 2
10 1
11 1
12 1
13 2
14 1
15 3
16 4
17 3
18 2
19 8
20 2
21 3
22 1
23 4
24 5
25 4
26 5
27 5
28 4
29 3
30 2
31 2
32 2
33 4
34 5
36 3
37 2
38 4
40 1
42 1
43 1
44 2
45 5
47 6
48 3
49 4
50 6
51 4
52 3
53 5
54 3
55 5
56 1
57 1
58 2
59 1
60 1
Dot gp>=4 CLUS2
p=aaan
q=aaax
0 3
1 3
2 8
3 2
4 6
5 5
6 5
7 11
8 2
9 4
10 12
11 8
12 13
13 5
14 3
15 7
Sparse lower end
i32 i18 i19 i23 i6 i36 F i i i i i i
i32 and i18 gap>=4 outliers
Thin area: ( )
e4 i7 e10 e31 e32 s14 i39 s16 s19 e49 s15 e44 e11 e8 s6 s34 F e i e e e s i s s e s e e e s s
So i39,s16,s19,s49,s15 are "thin area" outliers AND s14 is also.
Separate at 42, giving CLUS1<41 (50 Setosa, 4 Versicolor, e8,e11,e44,e49) and CLUS2>=41.
So, two rounds of Dotpd(y) gap analysis yields CLUS1 (50 Setosa, plus 4 Versicolor)
CLUS2.1 (43 Virginica, plus 2 Versicolor)
CLUS2.2 (44 Veriscolor, plus 4 Virginica)
and picks out 3 Virginica, 4 Setosa as outliers
(More outliers would result by applying 1.1 to the sparse ends of the 2nd round?).
Round1: p=nnnn (n=min) and q=xxxx (x=max)
Round2: p=aaan (a=avg) and q=aaax
Sparse Ends analysis should accomplish the same outlier detection that a few steps of SL accomplishes. If an outlier is surrounded at a fixed distance then those neighbors will show up as sparse end neighbors and the outlier-ness of the point will be detected by looking at pairwise distances of that sparse end.
Sparse upper end
s23 s43 s9 s39 s42 s14 F s s s s s s
no gap>4 outliers

3. DG-SE (other corners) Check Dotp,d(y) gaps>=4 Check sparse ends.
CLUS1
p=nxnn
q=xnxx
0 1
2 1
4 1
6 2
9 1
10 1
11 2
12 2
13 3
14 3
15 2
16 2
17 4
18 3
19 3
20 2
21 5
22 6
23 5
24 2
25 7
26 3
27 2
28 2
29 1
30 3
31 3
32 7
33 4
34 1
35 1
36 2
37 2
39 1
41 1
42 1
43 1
Sparse low end (check [0,9]
i23 i6 i36 i8 i31 i3 i26 i i i i i i i
i3, i26, i36 >=4 singleton outliers
{i23,i6}, {i8,i31} doubleton ols
Sparse low end (checking [0,7]
i1 i18i19i10i37i5 i6 i23i32i44i45i49i25i8 i15i41i21i33i29i4 i3 i16 i i i i i i i i i i i i i i i i i i i i i i i i i
i1, i18, i19, i10, i37, i32 >=4 outliers
Dotgp>=4
p=xnnn
q=nxxx
0 1
1 1
2 1
3 2
4 7
5 1
6 7
7 5
8 9
9 3
10 7
11 3
12 5
13 4
14 5
15 4
16 8
17 4
18 7
19 3
20 5
21 1
22 4
23 1
24 1
31 2
33 2
34 12
35 8
36 17
37 6
38 2
39 2
Sparse hi end (checking [34,43]
e20e31e10e32e15e30e11e44e8 e49 e e e e e e e e e e
e30,e49,ei15,e11 >=4 singleton ols
{e44,e8} doubleton ols
gap:(24,31) CLUS1<27.5 (50 versi, 49 virg) CLUS2>27.5 (50 set, 1 virg)
Sparse hi end (checking [38,39]
s42 s36 s37 s1 s s s s
s37, s1 outliers
Thinning (8,13)
Split in middle=10.5
CLUS_1.1<10.5 (21 virg, 2 ver)
CLUS_1.2>10.5 (12 virg, 42 ver)
CLUS1
Dotgp>=4
p=nnnn
q=xxxx
0 1
1 2
2 2
3 1
4 2
5 1
6 6
7 2
8 3
9 1
10 2
11 2
12 2
13 6
14 6
15 7
16 2
17 2
18 3
19 3
20 2
21 2
22 3
23 4
24 2
25 1
26 2
27 3
28 1
29 1
Clus1
p=nnxn
q=xxnx
0 2
1 1
2 5
3 8
4 9
5 6
6 9
7 14
8 11
9 7
10 4
11 2
13 2
Thinning (7,9)
Split in middle=7.5
CLUS_1.2.1 < 7.5 (10 virg, 4 ver)
CLUS_1.2.2 > 7.5 ( 1 virg, 38 ver)
i15 gap>=4 outlier at F=0
Sparse hi end (checking [10,13]
e34i2 i14i43e41i20i7 i35 e i i i e i i i
i7, i35 >=4 singleton outliers
CLUS1
Dotgp>=4
p=nnnx
q=xxxn
0 1
4 1
5 3
6 5
7 4
8 3
9 6
10 7
11 3
12 4
13 8
14 4
15 4
16 3
17 8
18 5
19 3
20 1
21 1
22 3
23 1
CLUS1.2
Dotgp>=4
p=aaan
q=aaax
0 1
4 4
5 3
6 3
7 4
8 1
9 5
10 7
11 3
12 5
13 3
14 6
15 1
16 4
17 1
18 1
19 2
hi end gap outlier i30
CLUS1.2.1
Dotgp>=4
p=anaa
q=axaa
0 1
1 1
2 1
4 2
6 3
7 4
9 2
CLUS1.2.1
Dotgp>=4
p=aana
q=aaxa
0 5
1 2
2 3
3 2
4 1
6 1 C
i24e7 i34i47i27i28e34e36e21i50i2 i43i14i22 i e i i i i e e e i i i i i
CLUS1.2.1
p=naaa
q=xaaa
0 4
1 1
2 1
3 2
4 2
5 2
6 1
7 1

4. Check Dotp,d(y) for thinnings. Calc AVG of each side of thinning as p,q. redo.
p=nnnn
q=xxxx
0 2
3 2
4 1
5 1
7 1
9 2
10 1
11 1
12 1
13 2
14 1
15 3
16 4
17 3
18 2
19 8
20 2
21 3
22 1
23 4
24 5
25 4
26 5
27 5
28 4
29 3
30 2
31 2
32 2
33 4
34 5
36 3
37 2
38 4
40 1
42 1
43 1
44 2
45 5
47 6
48 3
49 4
50 6
51 4
52 3
53 5
54 3
55 5
56 1
57 1
58 2
59 1
60 1
Dot
p=AVG>22
q=AVG<22
0 1
1 1
2 2
3 2
4 3
5 6
6 7
7 10
8 9
9 3
10 3
11 2
12 1
19 1
23 1
24 2
26 1
29 1
30 1
31 2
32 2
33 2
34 4
35 2
36 5
37 1
38 3
39 2
40 3
41 4
42 4
43 2
44 3
45 6
46 7
47 2
48 3
49 1
50 3
52 7
53 1
54 4
55 1
56 3
57 3
58 2
59 2
61 1
62 1
63 1
64 1
66 1
67 1
68 1
69 1
70 1

5. Barrel Clustering: (This method attempts to build barrel-shaped gaps around clusters)
Furthest Point or Mean Point q
Allows for a better fit around convex clusters that are elongated in one direction (not round).
Gaps in dot product lengths [projections] on the line.
Exhaustive Search for all barrel gaps:
It takes two parameters for a pseudo- exhaustive search (exhaustive modulo a grid width).
1. A StartPoint, p (an n-vector, so n dimensional)
2. A UnitVector, d (a n-direction, so n-1 dimensional - grid on the surface of sphere in Rn).
Then for every choice of (p,d) (e.g., in a grid of points in R2n-1) two functionals are used to enclose subclusters in barrel shaped gaps.
a. SquareBarrelRadius functional, BR(y) = (y-p)o(y-p) - ((y-p)od)2
b. BarrelLength functional, BL(y) = (y-p)od y
barrel
cap gap
width
Given a p, do we need a full grid of ds (directions)? No! d and -d give the same BL-gaps.
Given d, do we need a full grid of p starting pts? No! All p' s.t. p'=p+cd give same gaps.
Hill climb gap width from a good starting point and direction.
MATH: Need dot product projection length and dot product projection distance (in red). p
barrel radius
gap width
squared is y -
(yof)
fof f
o y - f y
y - f
|f| yo
= y -
(yof)
fof f
squared = yoy - 2
(yof)2
fof
fof
(fof)2
Squared y on f Proj Dis = yoy -
(yof)2
fof
dot product projection distance
squared = yoy - 2
(yof)2
fof + yo
dot prod proj len f
|f|
Squared y-p on q-p Projection Distance = (y-p)o(y-p) -
( (y-p)o(q-p) )2
(q-p)o(q-p)
1st
= yoy -2yop + pop -
( yo(q-p) - p o(q-p
|q-p| 2
M-p
|M-p|
(y-p)o
For the dot product length projections (caps) we already needed:
= ( yo(M-p) -
po M-p )
That is, we needed to compute the green constants and the blue and red dot product functionals in an optimal way (and then do the PTreeSet additions/subtractions/multiplications). What is optimal? (minimizing PTreeSet functional creations and PTreeSet operations.)

6. 4 functionals in the dot product group of gap clusterers on a VectorSpace subset, Y (yY):
1. SLp(y) = (y-p)o(y-p), p a fixed vector. Square Length functional primarily for outlier identification and densities.
2. Dotd(y) = yod, (d is a unit vector) the Dot-product functional.
Using d=q-p/|q-p| and y-p
Dotp,q(y) = (y-p)o(q-p)/|q-p|
yod projection d y
y - (yod)d = projection. Squaring its length: (y-yodd)o(y-yodd)=yoy-(yod)2 
yod projection (neg) d y
y - (yod)d
so again yoy - (yod)2 = squared proj 
(y-p)o(y-p) -
( (y-p)o(q-p) )2
(q-p)o(q-p)
= yoy -2yop + pop - 2
3. SPDd(y) = yoy - (yod)2 (d a unit vector) is the Square Projection Distance functional
E.g., if d ≡ (q-p)/|q-p|, d = unit vector from vector p to vector q, then SPD(y)=
But to avoid creating an entirely new VectorPTreeSet(Y-p)
for the space (with origin shifted to p), we think it useful
to alter the expression to : SPDpq(y)
q-p
|q-p| - yo po
where we might:
1st compute the constant vector nd the ScalarPTreeSet
q-p
|q-p| yo po
q-p
|q-p| yo -
3rd the constant th the SPTreeSet
pop yo
q-p
|q-p| po -
5th the SPTreeSet th the constant
yoy, yop
= yoy -2yop + pop - 2
7th the SPTreeSets 8th the SPTreeSet
q-p
|M-p| - yo po
Is it better to leave all the additions and subtractions for one mega-step at the end? Other efficiency thoughts?
q-p
|q-p|
(y-p)o = - yo po
We note that Dot(y)=yod shares many construction steps with SPD.
4. CAd(y) = yod/|y|, (d unit vector) the Cone Angle functional. Using d=q-p/|q-p| and y=x-p CAp,q(y) = (y-p)od/|y-p|
SCAp,q(y) = (y-p)od2/|y-p|2 = (y-p)od2/(y-p)o(y-p), Squared Cone Angle functional

7. CLUS1.2 is pure Versicolor (45 of the 50).
SPD p
q e14
V Ct
2 10
3 12
4 12
5 12
6 8
7 11
8 9
9 5
10 9
11 4
12 4
13 2
14 1
17 2
18 3
19 10
20 5
21 6
22 5
23 6
24 6
25 3
27 2
29 2
30 1
SPD on CLUS1
p e11
q =MN
V Ct
2 3
3 4
4 5
5 7
6 2
7 2
8 6
9 6
10 3
11 4
12 2
13 4
14 4
15 3
16 2
17 1
18 5
19 1
20 2
22 2
23 1
24 1
25 1
26 1
29 1
SPD p
q e14
V Ct
1 6
2 4
3 8
4 4
5 10
6 2
7 2
8 2
9 7
10 2
11 2
12 2
13 1
15 2
17 1
18 4
19 2
20 4
22 1
24 1
25 1
26 1
29 1
31 2
32 2
33 3
i15 i36
i32
SPD p q
V Ct
2 8
3 10
4 10
5 10
6 5
7 10
8 6
9 8
10 6
11 1
mask: V<8.5
CTs SMs
CTe SMe
CTi SMi
CLUS1
mask: V<12.5
5 SMe
24 SMi
CLUS1.1
thin gap
mask: 8.5<V<15.5
CTs SMs
CTe SMe
CTi SMi
CLUS2
masking V>6:
Total_e Masked_e
Total_i Masked_i
However I cheated a bit.
I used p=MinVect(e) and q=MaxVect(e) which makes it somewhat supervised.
START OVER WITH THE FULL >
mask: V>12.5
45 SMe
0 SMi
CLUS1.2
mask: V>15.5:
CTs SMs
CTe SMe
CTi SMi
This tube contains 49 setosa
+ 2 virginica
CLUS3
CLUS1.2 is pure Versicolor (45 of the 50).
CLUS3 is almost pure Setosa (49 of the 50, plus 2 virginica)
CLUS2 is almost purely [1/2 of] viriginica (24 of 50, plus 1 setosa).
CLUS1.1 is the other 24 virginicas, plus the other 5 versicolors.
So this method clusters IRIS quite well (albeit into 4 clusters, not three).
Note that caps were not put on these tubes.
Also, this was NOT unsupervised clustering! I took advantage of my knowledge of the classes to carefully chose the unit vector points, p and q E.g., p = MinVector(Versicolor) and q = MaxVector(Versicolor. True, if one sequenced thru a fine enough d-grid of all unit vectors [directions], one would happen upon a unit vector closely aligned to d=q-p/|q-p| but that would be a whole lot more work that I did here (would take much longer).
In worst case though, for totally unsupervised clustering. there would be no other way than to sequence through a grid of unit vectors.
However, a good heuristic might be to try all unit vectors "corner-to-corner" and "middle-of-face-TO-middle-of-opposite-face" first, etc.
Another thought would be to try to introduce some sort of hill climbing to "work our way" toward a good combination of a radial gap plus two good linear cap gaps for that radial gap.

8. SPD on CLUS1
p C1US1axxx
q C1US1aaaa
V Ct
1 3
2 5
3 9
4 13
5 18
6 12
7 4
8 1
9 2
no thinnings
SPD on CLUS1
p C1US1xaxx
q C1US1aaaa
V Ct
1 4
2 13
3 7
4 19
5 9
6 7
7 9
8 2
SPD on CLUS1
p C1US1xxax
q C1US1aaaa
V Ct
1 1
2 4
3 3
4 9
5 9
6 14
7 9
8 4
9 6
10 3
11 3
12 1
14 2
15 1
no thinnings
SPD on CLUS1
p C1US1xxxa
q C1US1aaaa
V Ct
1 1
2 3
3 10
4 15
5 16
6 12
7 7
8 3
9 1
10 1
no thinnings
SPD
p axxx
q aaaa
V Ct
2 1
3 5
4 6
5 6
6 8
7 6
8 8
9 15
10 7
11 8
12 13
13 8
14 14
15 9
16 13
17 6
18 4
19 4
20 3
21 4
23 1
25 1
mask: V<3.5
14 SM versi
10 SM virgi CL1.1?
mask: V<11.5
0 SM setosa
46 SM versicolor
24 SM virginica
CLUS1
mask: V>3.5
0 SM setosa
32 SM versi
14 SM virgi
CLUS1.2?
mask: V>11.5
50 SM setosa
4 SM versicolor
26 SM virginica
CLUS2
SPD on CLUS2
p C1US2axxx
q C1US2aaaa
V Ct
6 2
7 2
8 6
9 13
10 7
11 7
12 4
13 5
14 11
15 9
16 2
18 4
21 2
22 1
23 3
25 1
26 1
SPD on CLUS1
p C1US1axax
q C1US1aaaa
V Ct
1 1
2 3
3 4
4 2
5 12
6 13
7 9
8 7
9 2
10 7
11 4
13 2
14 1
17 2
18 1
SPD on CLUS1
p C11aaxx
q C11aaaa
V Ct
1 1
2 7
3 10
4 13
5 13
6 13
7 6
8 2
9 2
11 1
no thinnings
SPD on CLUS1
p C1US1axxa
q C1US1aaaa
V Ct
1 1
2 2
3 6
4 9
5 12
6 17
7 8
8 6
9 5
10 1
11 1
no thinnings
mask: V<13.5
44 SM setosa
0 SM versicolor
02 SM virginica
CLUS2.1
mask: V<9.5
37 SM vers
16 SM virg CL1.1?
mask: 100>V>13.5
6 SM setosa
4 SM versicolor
24 SM virginica
CLUS2.2
mask: V>9.5
9 SM vers
8 SM virg CL1.2?
SPD on CLUS1
C11xaax
C11aaaa
V Ct
1 2
2 3
3 4
4 8
5 8
6 14
7 8
8 4
9 5
10 6
11 1
12 3
14 1
15 2
no thins
C11axaa
C11aaaa
V Ct
1 2
2 2
3 2
4 10
5 3
6 13
7 8
8 7
9 4
10 3
11 6
12 2
13 2
14 2
17 2
18 1
19 1
SPD on CLUS1
C11xxaa
C11aaaa
V Ct
1 1
2 4
3 6
4 9
5 10
6 7
7 9
8 5
9 3
10 4
11 2
12 4
13 1
14 3
17 2
SPD on C1
C11aaax
C11aaaa
V Ct
1 3
2 1
3 3
4 4
5 12
6 15
7 4
8 5
9 4
10 7
11 4
12 2
13 1
14 1
15 1
17 1
18 1
19 1
SPD on CLUS1
C11xaxa
C11aaaa
V Ct
1 2
2 3
3 12
4 12
5 10
6 15
7 7
8 4
9 1
10 2
11 1
no thins
C11aaxa
C11aaaa
V Ct
1 2
2 3
3 6
4 12
5 11
6 9
7 11
8 5
9 5
10 1
11 3
13 2
C11xaaa
C11aaaa
V Ct
1 2
2 4
3 5
4 9
5 10
6 9
7 5
8 6
9 2
10 6
11 3
12 1
13 2
14 2
15 2
17 2
mask: V<5.5
16 ver
3 virCL1.1?
mask: V<5.5
26 ver
4 vir CL1.1?
mask: V>5.5
30 ver
21 virCL1.1?
mask: V>5.5
20 ver
20 vir CL1.1?

9. 95 remaining versicolor and virginica=SubClus1.
i p
max
V Ct
0 2
1 2
2 2
3 5
4 3
5 3
6 4
7 4
8 7
9 2
10 3
11 1
12 4
13 5
14 4
15 7
16 2
17 5
18 3
19 1
20 1
21 4
23 2
24 2
25 4
26 1
27 2
28 1
29 2
30 1
32 1
{e4, e40} form a doubleton outlier set
i7 and e10 are singleton outliers
x=s (58=avg(y1) )
V Ct
0 3 s15, s17, s34
1 12 s 6,11,16,19,20,22,28,32,3337,47,49
2 12 s 1,10 13,18,21,27,29,40,41,44,45,50
3 7 s 2,12,23,24,35,36,38
4 10 s 2,3,7,13,25,26,30,31,46,48
5 2 s4, s43
6 2 s9,s39
7 1 s14
8 1 i39
9 1 s32 ^^all 50 setosa + i39
e49
16 2
17 2
19 1
20 2
21 5
22 4
23 3
24 4
25 1
27 8
28 2
29 2
30 4
31 1
32 4
34 2
35 2
36 2
37 3
38 2
39 2
40 4
41 1
43 2
44 4
45 2
46 1
47 2
48 1
50 4
52 2
53 2
54 2
56 2
57 1 i1
i31 vv 9 virginica
i10 i8
i36
i32
i16
i18
i23
i19
But here I mistakenly used the mean rather than the max corner.
So I will redo - but note the high level of cluster and outlier revelation?????
i p
max
V Ct
0 2
2 6
3 3
4 4
5 4
6 2
7 6
8 9
9 2
10 2
11 2
12 5
13 7
14 2
15 6
16 2
17 5
19 3
20 2
22 3
23 2
24 3
25 2
26 1
27 1
28 1
29 3
30 1
31 2
e32
e11
e8,44
e49
i39
60 1
61 1
62 1
63 1
64 1
65 1
66 1
67 3
68 4
69 4
70 3
71 3
72 4
73 2
74 5
75 1
76 2
77 1
78 3
79 1 s3 s9
s39,43
s42
s23
s14
2 actual gap-ouliers,
checking distances reveals 4 e-outlier (versicolor),
5 s-outliers (setosa).
i p
max
V Ct
0 2
1 1
2 3
3 3
4 4
5 2
6 6
7 3
8 5
9 4
10 4
11 2
12 3
13 4
14 6
15 4
16 1
17 7
18 2
19 3
20 2
22 2
23 1
24 2
25 4
26 4
27 1
28 2
29 2
30 1
32 2
33 1
34 1
35 1
No new outliers reviealed
95 remaining versicolor and virginica=SubClus1.
Continue outlier id rounds on SC1 (maxSL, maxSW, max PW) then do "capped tube" (further subclusters.)
1. (y-p)o(y-p) remove edge outliers ( thr>2*50)
2. lthin gaps in SPD: d, from an edge point to MN.
3 For each thin PL, do len gap anal of pts in " tube".
e13 i7 e40 e4 e10 F e i e e e
e32 e11 e8 e44 e49 e e e e e
45 remaining setosa.
This is SubCluster 2
(may have additional outliers or sub-subclusters but we will not analyse further (would be done in practice tho
SPD(y) =(y-p)o(y-p)-(y-p)od2 d: mn-mx
V Ct
Next slide
i p
max
V Ct
0 2
1 10
2 11
3 6
4 15
5 4
6 8
7 9
8 4
9 5
10 2
11 7
13 4
14 2
15 2
16 1
17 1
18 1
19 1
e30, e15 outliers
e20,e31,e32 form SC12
Declared tripleton outlier set? (But they are not singleton outliers.)
s3 s9 s39 s43 s42 s23 s s s s s s
e13 e20 e15 e31 e32 e30 F e e e e e e

10. Cone Clustering: (finding cone-shaped clusters)
x=s2
cone=.1
39 2
40 1
41 1
44 1
45 1
46 1
47 1
i39
59 2
60 4
61 3
62 6
63 10
64 10
65 5
66 4
67 4
69 1
70 1 59
w maxs-to-mins
cone=.939
i25
i40
i16 i42
i17 i38
i11 i48
22 2
23 1
i34 i50
i24 i28
i27
27 5
28 3
29 2
30 2
31 3
32 4
34 3
35 4
36 2
37 2
38 2
39 3
40 1
41 2
46 1
47 2
48 1
i39
53 1
54 2
55 1
56 1
57 8
58 5
59 4
60 7
61 4
62 5
63 5
64 1
65 3
66 1
67 1
68 1
114
14 i and 100 s/e.
So picks i as 0
w naaa-xaaa
cone=.95
12 1
13 2
14 1
15 2
16 1
17 1
18 4
19 3
20 2
21 3
22 5
i21
24 5
25 1
27 1
28 1
29 2 i7
41/43 e
so picks e
Corner points
Gap in dot product projections onto the cornerpoints line.
x=s1
cone=1/√2
60 3
61 4
62 3
63 10
64 15
65 9
66 3
67 1
69 2 50
x=s2
cone=1/√2
47 1
59 2
60 4
61 3
62 6
63 10
64 10
65 5
66 4
67 4
69 1
70 1 51
x=s2
cone=.9
59 2
60 3
61 3
62 5
63 9
64 10
65 5
66 4
67 4
69 1
70 1 47
Cosine cone gap (over some  angle)
w maxs
cone=.707
0 2
8 1
10 3
12 2
13 1
14 3
15 1
16 3
17 5
18 3
19 5
20 6
21 2
22 4
23 3
24 3
25 9
26 3
27 3
28 3
29 5
30 3
31 4
32 3
33 2
34 2
35 2
36 4
37 1
38 1
40 1
41 4
42 5
43 5
44 7
45 3
46 1
47 6
48 6
49 2
51 1
52 2
53 1
55 1
137
w maxs
cone=.93
8 1 i10
13 1
14 3
16 2
17 2
18 1
19 3
20 4
21 1
24 1
25 4
e21 e34
27 2
29 2 i7
27/29 are i's
F=(y-M)o(x-M)/|x-M|-mn restricted to a cosine cone on IRIS
w aaan-aaax
cone=.54
7 3 i27 i28
8 1
9 3
i20 i34
11 7
12 13
13 5
14 3
15 7
19 1
20 1
21 7
22 7
23 28
24 6
100/104 s or e
so 0 picks i
x=i1
cone=.707
34 1
35 1
36 2
37 2
38 3
39 5
40 4
42 6
43 2
44 7
45 5
47 2
48 3
49 3
50 3
51 4
52 3
53 2
54 2
55 4
56 2
57 1
58 1
59 1
60 1
61 1
62 1
63 1
64 1
66 1 75
x=e1
cone=.707
33 1
36 2
37 2
38 3
39 1
40 5
41 4
42 2
43 1
44 1
45 6
46 4
47 5
48 1
49 2
50 5
51 1
52 2
54 2
55 1
57 2
58 1
60 1
62 1
63 1
64 1
65 2 60
Cosine conical gapping seems quick and easy (cosine = dot product divided by both lengths.
Length of the fixed vector, x-M, is a one-time calculation.
Length y-M changes with y so build the PTreeSet.
w maxs
cone=.925
8 1 i10
13 1
14 3
16 3
17 2
18 2
19 3
20 4
21 1
24 1
25 5
e21 e34
27 2
28 1
29 2
e35 i7
31/34 are i's
w xnnn-nxxx
cone=.95
8 2 i22 i50
10 2
i28
i24 i27 i34
13 2
14 4
15 3
16 8
17 4
18 7
19 3
20 5
21 1
22 1
23 1
i39
43/50 e so picks out e

11. FxM(x,y)=yo(x-M)/|x-M|-min on XX≡{(x,y)|x,yX}, where X(x,y) is a Spaeth image table Cluster by splitting at all F_gaps > 2
APPENDIX
The 15 Value_Arrays (one for each x) z z z z z z z z z z z z z z z X
x y \y= a b
1 1 x=1 1 f
M d
a b
b c e c
d a
e 8
f 7 9
x y FxM
z1 z1 14
z1 z2 12
z1 z3 12
z1 z4 11
z1 z5 10
z1 z6 6
z1 z7 1
z1 z8 2
z1 z9 0
z1 z10 2
z1 z11 2
z1 z12 1
z1 z13 2
z1 z14 0
z1 z15 5
9 5  M
(=MeanVector)
The 15 Count_Arrays z z z z z z z z z z z z z z z
Level0, stride=z1 PointSet (as a pTree mask) z1 z2 z3 z4 z5 z6 z7 z8 z9
z10
z11
z12
z13
z14
z15
gap: 10-6
gap: 5-2
pTree masks of the 3 z1_clusters (obtained by ORing)
z11 1
z12 1
z13 1
The FAUST algorithm:
1. project onto each Mx line using the dot product with the unit vector from M to x. (only x=z1 is shown)
2. Generate each Value Array, F[x0]|(y), xX (also generate the Count_Arrays and the mask pTrees).
3. Analyze all gaps and create sub-cluster pTree Masks.

12. Cluster by splitting at gaps > 2
yo(z7-M)/|z7-M| ValueArrays z z z z z z z z z z z z z z z
yo(z7-M)/|z7-M| CountArrays z z z z z z z z z z z z z z z
Cluster by splitting at gaps > 2
x y x\y a b
3 3 4
9 3 6 f
14 2 8
M d
13 4 a b
10 9 b c e
1110 c
9 11 d a
1111 e 8
7 8 f
x y F
z1 z1 14
z1 z2 12
z1 z3 12
z1 z4 11
z1 z5 10
z1 z6 6
z1 z7 1
z1 z8 2
z1 z9 0
z1 z10 2
z1 z11 2
z1 z12 1
z1 z13 2
z1 z14 0
z1 z15 5
9 5 Mean
z11 1
z12 1
z13 1
gap: 6-9
z71 1
z72 1
In Step_3 of the algorithm we can:
Analyze one of the gap arrays (e.g., As done for z1. Subclusters is shown above)
then start over on each subcluster.
Or we can analyze all gap arrays concurrently (in parallel using the same F - saving the [substantial?] re-compute costs?) and then intersect the subcluster partitions we get from each x_ValueArray gap analysis, forthe final subclustering.
Here we use the second alternative, judiciously choosing only the x's that are likely to be productive (choosing z7 next).
Many are likely to produce redundant partitions - e.g., z1, z2, z3, z4, z6 - as their projection lines will be nearly coincident.
How should we choose the sequence of "productive" strides? One way would be to always choose the remaining stride with the shortest ValueArray, so that the chances of decent sized gaps is maximized. Other ways of choosing?

13. Cluster by splitting at gaps > 2
yo(x-M)/|x-M| Value Arrays z z z z z z z z z z z z z z z
Cluster by splitting at gaps > 2
yo(x-M)/|x-M| Count Arrays z z z z z z z z z z z z z z z
x y x\y a b
3 3 4
9 3 6 f
14 2 8
M d
13 4 a b
10 9 b c e
1110 c
9 11 d a
1111 e 8
7 8 f
x y F
z1 z1 14
z1 z2 12
z1 z3 12
z1 z4 11
z1 z5 10
z1 z6 6
z1 z7 1
z1 z8 2
z1 z9 0
z1 z10 2
z1 z11 2
z1 z12 1
z1 z13 2
z1 z14 0
z1 z15 5
9 5 Mean
z11 1
z12 1
z13 1
gap: 3-7
z71 1
z72 1
zd1 1
zd2 1
We choose zd=z13 next (Should have been first? Since it's ValueArray is shortest?)
Note, z8, z9, za projection lines will be nearly coincident with that of z7.

14. Cluster by splitting at gaps > 2
yo(x-M)/|x-M| Value Arrays z z z z z z z z z z z z z z z
Cluster by splitting at gaps > 2
yo(x-M)/|x-M| Count Arrays z z z z z z z z z z z z z z z
x y x\y a b
3 3 4
9 3 6 f
14 2 8
M d
13 4 a b
10 9 b c e
1110 c
9 11 d a
1111 e 8
7 8 f
x y F
z1 z1 14
z1 z2 12
z1 z3 12
z1 z4 11
z1 z5 10
z1 z6 6
z1 z7 1
z1 z8 2
z1 z9 0
z1 z10 2
z1 z11 2
z1 z12 1
z1 z13 2
z1 z14 0
z1 z15 5
9 5 Mean
z11 1
z12 1
z13 1
z71 1
z72 1
zd1 1
zd2 1
AND each red with each blue with each green, to get the subcluster masks (12 ANDs producing 5 sub-clusters.

15. F1(x,y) = L1Distance(x,y) = (|x1-y1|+|x2-y2|) on XX≡{(x,y)|x,yX},
Cluster by splitting at all F1_gaps
L1(x,y) Value Array z z z z z z z z z z z z z z z
L1(x,y) Count Array z z z z z z z z z z z z z z z
x y x\y a b
3 3 4
9 3 6 f
14 2 8 d
13 4 a b
10 9 b c e
1110 c
9 11 d a
1111 e 8
7 8 f
(redundant subclustering)
gap: 10-5
There is a z1-gap, but it produces a subclustering that was already discovered by a previous round.
Which z values will give new subclusterings?

16. Re-confirms zf an anomaly.
L1(x,y) Value Array z z z z z z z z z z z z z z z
L1(x,y) Count Array z z z z z z z z z z z z z z z
This re-confirms z6 as an anomaly or outlier, since it was already declared so during the linear gap analysis.
x y x\y a b
3 3 4
9 3 6 f
14 2 8
M d
13 4 a b
10 9 b c e
1110 c
9 11 d a
1111 e 8
7 8 f
Re-confirms zf an anomaly.
After having subclustered with linear gap analysis, which is best for determining larger subclusters, we run this round gap algorithm out only 2 steps to determine if there are any singleFvalue gaps>2 (the points in the singleFvalueGapped set are then declared anomalies). So we run it out two steps only, then find those points for which the one initial gap determined by those first two values is sufficient to declare outlierness. Doing that here, we reconfirm the outlierness of z6 and zf, while finding new outliers, z5 and za.

17. Using F=yo(x-M)/|x-M|-MIN on IRIS, one stride at a time (s1=setosa1 first)
For virginica1
Val Ct
0 1
1 1
2 2
3 5
4 6
5 11
6 12
7 4
8 2
9 5
10 1
17 1
22 1
24 2
25 1
27 1
28 1
29 2
30 1
31 3
32 4
33 1
34 4
35 2
36 2
37 4
38 4
39 5
40 4
42 6
43 2
44 7
45 5
47 2
48 3
49 3
50 3
51 4
52 3
53 2
54 2
55 4
56 2
57 1
58 1
59 1
60 1
61 1
62 1
63 1
64 1
66 1
F(i39)=17
F<17 (50
Setosa)
vers1
Val Ct
0 1
2 4
3 1
4 1
5 3
6 3
7 8
8 3
9 7
10 6
11 4
12 4
13 3
15 2
19 2
20 2
21 1
26 2
27 3
28 4
30 2
31 5
32 4
33 3
34 1
36 3
37 5
38 4
39 5
40 7
41 4
42 2
43 2
44 1
45 6
46 4
47 5
48 1
49 2
50 5
51 1
52 2
54 2
55 1
57 2
58 1
60 1
62 1
63 1
64 1
65 2
F<19 (50 setosa)
19<F<22 {vers8,12,39,44,49}
22<F
yo(s1-M)/|s1-M|-69)
Val Ct
0 1
3 1
4 2
7 1
8 1
9 2
10 1
12 4
14 5
15 2
16 4
17 1
18 4
19 5
20 1
21 2
22 2
23 8
24 4
25 3
26 2
27 5
28 3
29 4
30 4
31 3
32 2
33 2
34 4
35 5
36 2
37 2
38 1
39 1
40 1
41 1
43 1
44 1
45 1
52 1
60 3
61 4
62 3
63 10
64 15
65 9
66 3
67 1
69 2
F(i39)=52 virginica39 is an outlier. 2 clusters, F<52 (ct=99) and F>52 (50 Setosa)
virgini39
Val Ct
0 1
1 2
2 1
4 2
6 1
7 1
8 7
9 2
10 2
11 7
12 2
13 3
14 7
15 4
16 10
17 4
18 6
19 9
20 3
21 6
22 3
23 6
24 3
25 1
27 3
28 2
32 1
39 1
40 1
41 1
42 8
43 13
44 17
45 4
46 5
47 1
F=32 vers49 outlier.
32<F (50 Setosa, vir39)
AVG(ver8,12,39,44,49)
Val Ct
0 1
1 1
7 5
10 3
12 2
13 2
14 3
15 5
16 2
17 5
18 8
19 4
20 3
21 4
22 3
23 8
24 4
25 4
26 3
27 7
28 7
29 4
30 5
31 4
32 5
33 8
34 2
35 6
36 5
37 3
38 2
39 8
40 6
41 3
43 1
44 2
45 1
47 1
F=0 vir32 outlier
F=1 vir18 outlier
F=7 vir6,10,19,23,36 subcluster?

18. F=yo(x-M)/|x-M|-MIN on IRIS, subclustering as we go.
On Clus(F<52) ver1
F(virg7)=0 outlier
F(virg32)=25 outlier
Val Ct
0 1
4 1
5 5
6 3
7 5
8 3
9 8
10 11
11 14
12 8
13 8
14 5
15 3
16 7
17 5
18 6
19 2
20 1
21 1
22 1
25 1
F=yo(x-M)/|x-M|-MIN on IRIS, subclustering as we go.
On Remaining, mx mn mx mx
Val Ct
0 3
1 4
2 11
3 14
4 14
5 9
6 10
7 2
8 6
9 2
11 2
For s1 (i.e.,
yo(s1-M)/|s1-M|-69)
Val Ct
0 1
3 1
4 2
7 1
8 1
9 2
10 1
12 4
14 5
15 2
16 4
17 1
18 4
19 5
20 1
21 2
22 2
23 8
24 4
25 3
26 2
27 5
28 3
29 4
30 4
31 3
32 2
33 2
34 4
35 5
36 2
37 2
38 1
39 1
40 1
41 1
43 1
44 1
45 1
outlier
60 3
61 4
62 3
63 10
64 15
65 9
66 3
67 1
69 2
F(i39)=52 i39=virgi39 outlier. Clusters, F<52 (ct=99) and F>52 (50 Setosa)
On Remaining, max's
Val Ct
0 2 e8 outlier
1 2 e11 outlier
7 2
8 1
9 4
10 1
11 2
12 2
13 4
14 3
15 1
16 4
17 2
18 2
19 3
20 4
21 6
22 5
23 5
24 4
25 2
26 2
27 1
28 2
29 4
30 5
31 1
32 3
33 2
34 2
35 3
36 2
37 1
38 1 i8
i10 i36
i6 i23
i19
i18
i6 i8 i10 i19 i23 i35 i i i i i i
i6 i10 i18 i19 i23 i35
all declared outliers
e4 e38 e19 i20 F e e
e outlier
i outlier
On Remaining, max's
Val Ct
e44 outlier
6 1
7 2
8 1
9 3
10 1
11 3
12 5
13 2
14 2
15 3
17 3
18 3
19 5
20 1
21 9
22 5
23 4
24 2
26 4
27 2
28 2
29 4
30 2
31 3
32 3
33 2
34 3
35 2
36 1
37 1
38 1
39 1
e36
outlier?
On Remaining,
mx mx mx mn
Val Ct
0 1
1 2
2 3
3 1
5 5
6 4
7 5
8 2
9 3
10 5
11 4
12 7
13 5
14 2
15 4
16 4
17 7
18 4
19 4
20 2
21 2
22 1
24 1
25 1
27 2
29 2
On Remaining,
mn mn mx mx
Val Ct
0 1
1 3
2 3
3 7
4 7
5 7
6 5
7 5
8 3
9 8
10 4
11 4
12 11
13 4
14 8
15 4
16 1
18 1
On Remaining, mn mx mx mx
Val Ct
0 1
2 1
3 4
4 3
5 5
6 4
7 5
8 7
9 8
10 3
11 5
12 2
13 4
14 5
15 7
16 5
17 4
18 1
20 1
On Remaining w e35
Val Ct
0 1 i26 outlier
3 2
On remaining vir1
Val Ct
0 1
1 2
2 1
4 1
5 1
6 2
7 2
8 2
9 4
10 1
11 4
12 3
13 4
14 2
15 6
16 4
17 6
19 4
20 5
21 5
22 2
23 1
24 2
25 5
26 4
27 4
28 1
29 2
30 6
31 2
32 1
33 1
34 1
35 2
36 1
38 1
39 1
e35 e10 e
e outlier
i44 i3 i i
^^outlier
i3 i30 i31 i26 i8 i36 i
i outlier
i outlier
i outlier
i outlier
i outlier
Rem mn mx mn mx
Val Ct
0 1
1 1
2 1
3 1
4 1
5 1
6 1
8 1
9 3
10 5
11 5
12 3
13 7
14 6
15 4
16 6
17 7
18 5
19 4
20 2
21 3
22 7
23 4
24 3
25 1
26 1
27 2
e49 outlier
On Remaining, mn mx mx mx
Val Ct
0 1
1 1
2 1
3 5
4 6
5 5
6 4
7 9
8 4
9 4
10 4
11 3
12 5
13 6
14 6
15 7
16 5
17 4
18 4
20 1
22 1
Could look at distances for 0,1 and 20,22?
e13 e30 e32
e outlier
e outlier e
i44 i45 i49 i5 i37 i1 i i i i
i not outlier
i outlier

19. outliers
gap>L1=32.1 s6
s14
s15
s16
s17
s19
s21
s23
s24
s32
s33
s34
s37
s42
s45 e1 e2 e3 e5 e6 e7 e9
e10
e11
e12
e13
e15
e18
e19
e21
e22
e23
e27
e28
e29
e30
e34
e36
e37
e38
e41
e49 i1 i3 i4 i5 i6 i7 i8 i9
i10
i12
i14
i15
i16
i18
i19
i20
i22
i23
i25
i26
i28
i30
i31
i32
i34
i35
i36
i37
i39
i41
i42
i45
i46
i47
i49
i50
outliers
gap>L1=42.8
s15
s16
s19
s23
s33
s34
s37
s42
s45 e1 e2 e7
e10
e11
e12
e13
e15
e19
e21
e22
e23
e27
e28
e30
e34
e36
e38
e41
e49 i1 i3 i5 i6 i7 i8 i9
i10
i12
i14
i15
i16
i18
i19
i20
i22
i23
i26
i30
i31
i32
i34
i35
i36
i39
outliers gp>L1=53.5
s15
s16
s23
s33
s34
s42
e10
e13
e15
e27
e28
e30
e36
e49 i1 i3 i7 i9
i10
i12
i15
i18
i19
i20
i26
i30
i32
i35
i36
i39
F=L1(x,y) on IRIS, masking to subclusters (go right down the table). Two rounds only
If we use L1gap=6, remove those outliers, then use linear gap analysis for larger subcluster revalation, let's see if we can separate Versicolor (e) from virginica (i).
outliers gap>L1=64.3
s15
s16
s23
s42
e10
e13
e49 i3 i7 i9
i10
i18
i19
i20
i32
i35
i36
i39
outliers gap>L1=74.95
L1gap
s42 9
e13 8
i7 10
i9 12
i10 12
i35 9
i36 9
i39 26

20. Val=0;p=K;c=0;P=Pure1; For i=n to 0 {c=Ct(P&Pi); If (c>=p){Val=Val+2i; P=P&Pi }; else{p=p-c; P=P&P'i }; return Val, P;
IDX z1 z2 : ze zf
IDY z1 z2 z3 z4 z5 z6 z7 z8 z9 za zb zc zd ze zf : X1 1 3 : 11 7 X2 1 : 11 8 X3 1 3 2 6 9 15 14 13 10 11 7 : 1 2 3 4 9 10 11 8 X4 : P3 1 : P2 1 P1 1 : P0 1 :
d(xy) 2 1 3 4 8 14 13 12 9 6 11 10 : 7 5
P'3 1 :
P'2 1 :
P'1 1 :
P'0 1 :
Need Rank(n-1) applied to each stride instead of the entire pTree. The result from stride=j gives the jth entry of SpS(X,d(x,X-x))
Parallelize over a large cluster?
Ct(P&Pi): revise the Count proc to kick out count for each stride (involves loop down pTree by register-lengths?
What does P represent after each step??
How does alg go on 2pDoop (w 2 level pTrees) where each stride is separate
Note: using d, not d2 (fewer pTrees). Can we estimate d? (using truncated McClarin series)
23 * * * *
1 = 1
n=3: c=Ct(P&P3)=10< 14, p=14–10=4; P=P&P' (elim 10 val8)
n=2: c=Ct(P&P2)= 1 < 4, p=4-1=3; P=P&P' (elim 1 val4)
n=1: c=Ct(P&P1)=2 < 3, p=3-2=1; P=P&P' (elim 2 val2)
n=0: c=Ct(P&P0 )=2>= P=P&P0 (elim 1 val<1)
23 * * * *
1 = 1
n=3: c=Ct(P&P3)=9< 14, p=14–9=5; P=P&P' (elim 9 val8)
n=2: c=Ct(P&P2)= 0 < 5, p=5-0=5; P=P&P' (elim 0 val4)
n=1: c=Ct(P&P1)=4 < 5, p=5-4=1; P=P&P' (elim 4 val2)
n=0: c=Ct(P&P0 )=1>= P=P&P0 (elim 1 val<1
23 * * * *
1 = 1
n=3: c=Ct(P&P3)= 9 < 14, p=14–9=5; P=P&P' (elim 9 val8)
n=2: c=Ct(P&P2)= 2 < 5, p=5-2=3; P=P&P' (elim 2 val4)2
n=1: c=Ct(P&P1)=2 < 3, p=3-2=1; P=P&P' (elim 2 val2)
n=0: c=Ct(P&P0 )=2>= P=P&P0 (elim 1 val<1)
23 * * * * 1
1 = 3
n=3: c=Ct(P&P3)= 6 < 14, p=14–6=8; P=P&P' (elim 6 val8)
n=2: c=Ct(P&P2)= 7 < 8, p=8-7=1; P=P&P' (elim 7 val4)2
n=1: c=Ct(P&P1)=11, p=1-1=0; P=P&P (elim 0 val2)
n=0: c=Ct(P&P0 )=1 P=P&P0 (elim 0)

21. Level-1 key map Red=pure stride (so no Level-0) e f g h
i a
j b c
k d m
0 0 13 12 11 10 23 22 21 20 33 32 31 30 43 42 41 40
a b c d e f g h i j k m 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Level-0:
key map 13 12 11 10 23 22 21 20
(6-e) = e
else pur0
(6-e) = f
else pur0
(6-e) = g
else pur0
(6-e) = h
else pur0
In this 2pDoop KEY-VALUE DB, we list keys. Should we bitmap? Each bitmap is a pTree in the KVDB. Each of these is existing, e.g., e here
5,7-a,f=f
else pur0
5,7-a,f=g
else pur0
5,7-a,f=h
else pur0
234789bcefg els pr0
234789bcefh else pr0
124-79c-f
h else pr0
(b-f) = i
else pur0
(b-f) = j
else pur0
(b-f) = k
else pur0
(b-f) = m
else pur0
(a) = j
else pur0
(a) = k
else pur0
(a) = m
else pur0
=SpS(XX,
-27(
p13p33 +
p13p32 +
p23p43
p23p42
(3-6,8,9)
k, els pr0
(3-6,8,9)
m els pr0 +
p13p31 +
26(
p13+p23+p33+p43
+p13p12+ p23p22+
p33p32 +
+p43p42 )
-26(
p23p41
124679bd
m els pr0
25(
p13p11+
p23p21 +
p33p31 +
p43p41 )
-25(
p13p30
+p23p40
+p12p31
+p22p41
+p12p32
+p22p42 e
f 5 6
g 7 h
i a
j b c
k d m 33 32 31 30 43 42 41 40
24(
p12+p22+p32+p42
+p13p10+
+p23p20
+p33p30
+p43p40
-24(p12p30
+p22p40
+p12p11+
+p22p21
+p32p31
+p42p41 )
23(
p12p10+
p22p20 +
p32p30 +
p42p40 )
-23(p11p31
+p11p30
+p21p41
+p21p40
p11+p21+p31+p41
+p11p10 +
+p21p20 +
+p31p30
+p41p40 )
-22(p10p30 +p20p40
p10+p20+p30+p40 )
22(