1. Machine Learning = moving data up some concept hierarchy (increasing information and/or/by reducing volume).
ML takes two forms: clustering and classification (unsupervised and supervised).
Clustering groups similar objects into a single higher level object or a cluster.
Classification does the same but is supervised by an existing class assignment function on a Training Set, f:TS{Classes}.
1. Clustering can be done for Anomaly Detection (detecting those object that are dissimilar from the rest; which boils down to finding singleton [and/or doubleton..] clusters.
2. Clustering can be done to develop a Training Set for classification of future unclassified objects.
Machine Learning moves data up a concept hierarchy, according to some criteria which we call a similarity.
A similarity is usually a function, s:XXCardinalSet s.t. xX s(x,x)s(x,y) yX (every x must be at least as similar to itself as it is to any other object) and s(x,y)=s(y,x). OrdinalSet is usually a subset of {0,1,...} (e.g., binary {0,1}={No,Yes}).
Classification is binary clustering: s(x,y)=1 iff f(x)=f(y). Using that part of f which is known to predict that which is unknown
NCL: (k is discovered, not specified. Assign each (object,class) a ClassWeight, CWReals (could be <0).
Classes "take next ticket" as they're discovered (tickets are 1,2,... Initially, all classes empty; All CWs=0.
Do for next d, compute Ld = Xod until after masking off new cluster, count is too high (doesn't drop enough)
For the next PCI in Ld (next-larger starting from smallest)
If followed by a PCD, declare next Classk and define it to be the set spherically gapped (or PCDed)
around centroid, Ck=Avg or VoMk over Ld-1[PCI, PCD]. Mask off this ticketed new Classk and contin
If followed by a PCI, declare next Classk and define it to be the set spherically gapped (or PCDed) around the centroid, Ck=Avg or VoMk over Ld-1[ (3PCI1+PCI2)/4, PCI2 ) Mask off this ticketed new Classk and continue.
For the next-smaller PCI (starting from largest) in Ld
If preceded by a PCD, declare next Classk and define it to be the set spherically gapped (or PCDed) around centroid Ck=Avg or VoMk over Ld-1[PCD, PCI]. Mask off this ticketed new Classk, contin.
If preceded by a PCI, declare next Classk and define it to be the set spherically gapped (or PCDed) around the centroid, Ck=Avg or VoMk over Ld-1( PCI2, (3PCI1+PCI2)/4] Mask off this ticketed new Classk and continue.

2. Pillar pk-means clusterer (k is not specified - it reveals itself.)
1a. Choose m1 maximizing D1=Dis(X,avgX).
1b. Check if m1 is outlier with Sm1 1c. Repeat until m1 is a non-outlier
2a. Choose m2 maximizing D2=Dis(X,m1)
2b. Check if m2 is outlier with Sm2 2c. Repeat until m2 is a non-outlier.
2d. Compute M1,2=Pm1m M2,1=Pm1<m2
3a. Choose m3 maximizing D3=D2+Dis(X,m2)
3b. Check if m3 is outlier with Sm3 3c. Repeat until m3 is a non-outlier.
3d. Compute Mi,3=Pmim M3,i=Pmi<m3 i<3
4a. Choose m4 maximizing D4=D3+Dis(X,m3)
4b. Check if m4 is outlier with Sm4 4c. Repeat until m4 is a non-outlier.
4d. Compute Mi,4=Pmim M3,i=Pmi<m4 i<4
...
Do until the MinDist(mh,mk)k<h < Threshold
Mj = &hj Mj,h are the mask pTrees of the k clusters for the k first round clusters.
Apply pk-means from here on. m1 m4 m3 m2

3. FAUST LinearSphericalRadial classifier
FAUST Oblique Analytics X(X1..Xn)Rn, |X|=N; Classes={C1..CK}; d=(d1..dn), |d|=1; p=(p1..pn)Rn; Functionals:
Ld,p (X-p)od= Xod-pod= Ld-pod Lmind,p,k= min(Ld,p&Ck) Lmaxd,p,k= max(Ld,p&Ck)
Sp  (X-p)o(X-p)= XoX+Xo(-2p)+pop Sminp,k = min(Sp&Ck) Smaxp,k = max(Sp&Ck)
Rd,p Sp-L2d,p= XoX+Xo(-2p)+pop-L2d-2pod*Xod+pod2= L-2p-(2pod)d+pop+pod2+XoX-L2d Rmind,pd,k = min(Rd,p&Ck) Rmaxaxd,pd,k = max(Rd,p&Ck)
FAUST CountChange clusterer If DensThres unrealized, cut C at PCCsLd,p&C with next (d,pd)dpSet
FAUST TopKOutliers Use D2NN=SqDist(x, X')=rank2Sx for TopKOutlier-slider.
FAUST Linear classifier yCk iff yLHk  {z | Lmind,p,k  (z-p)od  Lmmaxd,pd,k}  (d,p)dpSet
LHk is a hull around Ck. dpSet is a set of (d,p) pairs, e.g., (Diag,DiagStartPt).
RkiPtr(x,PtrRankiSx). RkiSD(x,rankiS2) ordered desc on rankiSx as constructed.
XoX+pop-2Xop -
[Xod-pod]2
(X-p)o(X-p) -
[(X-p)od]2 = p x d
(x-p)o(x-p)
(x-p)od = |x-p| cos
 (x-p)o(x-p) - (x-p)od2
Xod2 - 2pod Xod + pod2
+ pod2+XoX+pop or
Xod2 - 2pod Xod
- 2pod Xod + pod2
XoX+pop-2Xop
- Xod2
+ XoX
Pre-compute what? 1. col stats(min, avg, max, std,...) ; 2. XoX; Xop, p=class_Avg/Med); 3. Xod, d=interclass_Avg/Med_UnitVector; 4. Xox, d2(X,x), Rkid2(X,x), xX, i=2..; 5. Ld,p and Rd,p d,pdpSet
FAUST LinearSphericalRadial classifier
yCk iff yLSRHk{z | Tmind,p,k  (z-p)od  Tmaxd,p,k (d,p)dpSet, T=L|S|R }

4. FAUST Oblique LSR Classification IRIS150
Ld d=1000 p=origin S E I
Ld d=0100 p=origin S E I
Ld d=0010 p=origin S E I
Ld d=0001 p=origin
S 1 6 E I 16
26 32 12 1
15 3
2 1 6 48
50 15
2 1 34 50
22 16 34 28 99
393
1096
1217
1826
270
1558
2568 66
310
1750
4104 55
3139
3850
4954
8134
9809
3000
6120
6251
p=AvgS
279 5
171
186
748
998 1
517, 4 79
633 3
234
793
1417
712
636 9, 3
983
1369
2.8
158
199 5
3617 7
152
611
p=AvgE 24
132
730
1622
2281
388
1369 5
1403
929
1892
2824 96
273
1776
2747
5.9
319
453
454
1397
p=AvgI

5. APPENDIX
LSR IRIS150-.
Consider all 3 functionals, L, S and R. What's the most efficient way to calculate all 3?\
o=origin; pRn; dRn, |d|=1; {Ck}k=1..K are the classes;
An operation enclosed in a parallelogram, , means it is a pTree op, not a scalar operation (on just numeric operands)
Lp,d  (X - p) o d = Lo,d - [pod] minLp,d,k = min[Lp,d & Ck] maxLp,d,k = max[Lp,d & Ck[
= [minLo,d,k] - pod = [maxLo,d,k] - pod
= min(Xod & Ck) - pod = max(Xod & Ck) - pod OR
= min(X&Ck) o d - pod = max(X&Ck) o d - pod
Sp = (X - p)o(X - p) = -2Xop+So+pop =
Lo,-2p + (So+pop) minSp,k=minSp&Ck maxSp,k = maxSp&Ck
= min[(X o (-2p) &Ck)] + (XoX+pop)
=max[(X o (-2p) &Ck)] + (XoX+pop)
OR = min[(X&Ck)o-2p] + (XoX+pop)
=max[(X&Ck)o-2p] + (XoX+pop)
Rp,d  Sp, - Lp,d2 minRp,d,k=min[Rp,d&Ck] maxRp,d,k=max[Rp,d&Ck]
I suggest that we use each of the functionals with each of the pairs, (p,d) that we select for application (since, to get R we need to compute L and S anyway).
So it would make sense to develop an optimal (minimum work and time) procedure to create L, S and R for any (p,d) in the set.

6. LSR on IRIS150
C13
Dse S E I
L H
y isa OTHER if yoDse (-,495)(802,1061)(2725,)
y isa OTHER or S if yoDse  C1,1  [ 495 , 802]
y isa OTHER or I if yoDse  C1,2  [1061 ,1270]
y isa OTHER or E or I if yoDse  C1,3  [1270 ,2010
C1,3:
0 s
49 e
11 i
y isa OTHER or I if yoDse  C1,4  [2010 ,2725]
Dei E y isa O if yoDei (-,-117)(-3,)
I y isa O or E or I if yoDei  C2,1  [-62 ,-44]
L H y isa O or I if yoDei  C2,2  [-44 , -3]
C2,1:
2 e
4 i
Dei E y isa O if yoDei (-,420)(459,480)(501,)
I y isa O or E if yoDei  C3,1  [420 ,459]
L H y isa O or I if yoDei  C3,2  [480 ,501]
Continue this on clusters with OTHER + one class, so the hull fits tightely (reducing false positives), using diagonals?
C1,1:
D=1000
y isa O if yoD(-,43)(58,)
L H y isa O|S if yoD C2,3  [43,58]
C2,3:
D=0100
y isa O if yoD(-,23)(44,)
L H y isa O|S if yoD C3,3  [23,44]
C3,3:
D=0010
y isa O if yoD(-,10)(19,)
L H y isa O|S if yoD C4,1  [10,19]
C4,1:
D=0001
1 6 y isa O if yoD(-,1)(6,)
L H y isa O|S if yoD C5,1  [1,6]
C5,1:
D=1100
y isa O if yoD(-,68)(117,)
L H y isa O|S if yoD C6,1  [68,117]
C6,1:
D=1010
y isa O if yoD(-,54)(146,)
L H y isa O|S if yoD C7,1  [54,146]
C7,1:
D=1001
y isa O if yoD(-,44)(100,)
L H y isa O|S if yoD C8,1  [44,100]
C8,1:
D=0110
y isa O if yoD(-,36)(105,)
L H y isa O|S if yoD C9,1  [36,105]
C9,1:
D=0101
y isa O if yoD(-,26)(61,)
L H y isa O|S if yoD Ca,1  [26,61]
Ca,1:
D=0011
y isa O if yoD(-,12)(91,)
L H y isa O|S if yoD Cb,1  [12,91]
Cb,1:
D=1110
y isa O if yoD(-,81)(182,)
L H y isa O|S if yoD Cc,1  [81,182]
Cc,1:
D=1101
y isa O if yoD(-,71)(137,)
L H y isa O|S if yoD Cd,1  [71,137]
Cd,1:
D=1011
y isa O if yoD(-,55)(169,)
L H y isa O|S if yoD Ce,1  [55,169]
Ce,1:
D=0111
y isa O if yoD(-,39)(127,)
L H y isa O|S if yoD Cf,1  [39,127]
Cf,1:
D=1111
y isa O if yoD(-,84)(204,)
L H y isa O|S if yoD Cg,1  [84,204]
Cg,1:
D=1-100
y isa O if yoD(-,10)(22,)
L H y isa O|S if yoD Ch,1  [10,22]
Ch,1:
D=10-10
y isa O if yoD(-,3)(46,)
L H y isa O|S if yoD Ci,1  [3,46]
The amount of work yet to be done., even for only 4 attributes, is immense..
For each D, we should fit boundaries for each class, not just one class.
For 4 attributes, I count 77 diagonals*3 classes = 231 cases. How many in the Enron case with 10,000 columns? Too many for sure!!
D, not only cut at minCoD, maxCoD but also limit the radial reach for each class (barrel analytics)? Note, limiting the radial reach limits all other directions [other than the D direction] in one step and therefore by the same amount. I.e., it limits all directions assuming perfectly round clusters).
Think about Enron, some words (columns) have high count and others have low count. Our radial reach threshold would be based on the highest count and therefore admit many false positives. We can cluster directions (words) by count and limit radial reach differently for different clusters??

7. Dot Product SPTS computation: XoD = k=1..nXkDk(
= 22
+ 21 (1 p1,0
+ 1 p11
+ 20 (1 p1,0
1 p1,1
+ 1 p2,1 )
+ 1 p2,0
+ 1 p2,0 ) 1
3 3 D
D1,1
D1,0
1 1
D2,1
D2,0 1 3 2 X
X1 X2
p11 p10
p21 p20 6 9
XoD 1
pXoD,3
pXoD,2
pXoD,1
pXoD,0
/*Calc PXoD,i after PXoD,i-1 CarrySet=CARi-1,i RawSet=RSi */ INPUT: CARi-1,i, RSi
ROUTINE: PXoD,i=RSiCARi-1,i CARi,i+1=RSi&CARi-1,i OUTPUT: PXoD,i, CARi,i+1 1
CAR12,3 
& 1 
& 1
CAR22,3
PXoD,2 1 1
CAR11,2 
& 
& 1
CAR21,2
PXoD,1 1 
& 1 1 1 1
PXoD,0
CAR10,1 
& 
& 1
PXoD,3
CAR13,4
Different data.
3 3 D
D1,1
D1,0
1 1
D2,1
D2,0
1 1 (
= 22
+ 21 (1 p1,0
+ 1 p11
+ 20 (1 p1,0
1 p1,1
+ 1 p2,1 )
+ 1 p2,0
+ 1 p2,0 ) 1 3 2 X
pTrees 6 18 9
XoD 1 1 
& 1 1 1 
& 1 
& 1 1 1 1 
& 1 1 
& 1 
& 1 
&
PXoD,2 1 
&
PXoD,1 1 
& 1
PXoD,0
CAR10,1 
& 1 
&
PXoD,3
PXoD,4
We have extended the Galois field, GF(2)={0,1}, XOR=add, AND=mult to pTrees.
SPTS multiplication:
(Note, pTree multiplication = &)
= (21
p1,1
+20
p1,0)
(21
p2,1
p2,0)
= 22
+21(
p2,0+
+ 20
p1,0
p2,0
X1*X2 1
& 1
& 1
& 1 1
&
pX1*X2,0 1 
&
pX1*X2,2 1 
&
pX1*X2,1 1 3 2 X
X1 X2
p11 p10
p21 p20 9
X1*X2
pX1*X2,3
pX1*X2,2
pX1*X2,1
pX1*X2,0 1
pX1*X2,3

8. FAUST Oblique: XoD used in CCC, TKO, PLC and LARC) and (x-X)o(x-X)
Example:
FAUST Oblique: XoD used in CCC, TKO, PLC and LARC) and (x-X)o(x-X)
= -2Xox+xox+XoX is used in TKO.
So in FAUST, we need to construct lots of SPTSs of the type, X dotted with a fixed vector, a costly pTree calculation
(Note that XoX is costly too, but it is a 1-time calculation (a pre-calculation?). xox is calculated for each individual x but it's a scalar calculation and just a read-off of a row of XoX, once XoX is calculated.. Thus, we should optimize the living he__ out of the XoD calculation!!! The methods on the previous seem efficient. Is there a better method? Then for TKO we need to computer ranks:
RankK: p is what's left of K yet to be counted, initially p=K V is the RankKvalue, initially 0.
For i=bitwidth+1 to 0 if Count(P&Pi)  p { KVal=KVal+2i; P=P&Pi };
else /* < p */ { p=p-Count(P&Pi); P=P&P'i };
RankN-1(XoD)=Rank2(XoD)
1 1
D=x1
D1,1
D1,0
0 1
D2,1
D2,0 1
P=P&p1
32
1*21+ P p1
n=1
p=2 1
P=p0&P
22
1*21+1*20=3 so -2x1oX = -6 P
&p0
n=0
p=2 1 3 2 X
X1 X2
p11 p10
p21 p20 2 3
XoD 1 p3 p2 p1
p,0
RankN-1(XoD)=Rank2(XoD)
3 0
D=x2
D1,1
D1,0
1 1
D2,1
D2,0
0 0 3 9 2
XoD 1 p3 p2 p1
p,0 1
P=P&p'3
1<2
2-1=1
0*23+ P p3
n=3
p=2 1
P=p'2&P
0<1
1-0=1
0*23+0*22 P
&p2
n=2
p=1 1
P=p1&P
21
0*23+0*22+1*21+ P
&p1
n=1
p=1 1
P=p0&P
11
0*23+0*22+1*21+1*20=3
so -2x2oX= -6 P
&p0
n=0
p=1
RankN-1(XoD)=Rank2(XoD)
2 1
D=x3
D1,1
D1,0
1 0
D2,1
D2,0
0 1 3 6 5
XoD 1 p3 p2 p1
p,0 1
P=P&p2
22
1*22+ P p2
n=2
p=2 1
P=p'1&P
1<2
2-1=1
1*22+0*21 P
&p1
n=1
p=2 1
P=p0&P
11
1*22+0*21+1*20=5
so -2x3oX= -10 P
&p0
n=0
p=1

9. (n=3) c=Count(P&P4,3)= 3 < 6
RankN-1(XoD)=Rank2(XoD)
3 3 D
D1,1
D1,0
0 1
D2,1
D2,0
n=3
p=2
n=2
p=2
n=1
p=2
n=0
p=2 1 3 2 X
X1 X2
p11 p10
p21 p20 2 3
XoD 1 p3 p2 p1
p,0 P p3 P
&p2 P
&p1 P
&p0 1 1
22
1*23+ 1 1
0<2
2-0=2
1*23+0*22+ 1 1
0<2
2-0=2
1*23+0*22+0*21+ 1 1
22
1*23+0*22+0*21+1*20=9
pTree Rank(K) computation: (Rank(N-1) gives 2nd smallest which is very useful in outlier analysis?) 1 1 1 1
P=P&p3
P=p'2&P
P=p'1&P
P=p0&P
Cross out the 0-positions of P each step.
(n=3) c=Count(P&P4,3)= < 6
p=6–3=3; P=P&P’4,3 masks off highest (val 8)
{0}
X P4, P4, P4, P4,0 10 5 6 7 11 9 3 1 1 1 1
(n=2) c=Count(P&P4,2)= >= 3
P=P&P4,2 masks off lowest (val 4)
{1}
(n=1) c=Count(P&P4,1)= < 3
p=3-2=1; P=P&P'4,1 masks off highest (val8-2=6 )
{0}
(n=0) c=Count(P&P4,0 )= >= 1
P=P&P4,0
{1}
RankKval=0; p=K; c=0; P=Pure1; /*Note: n=bitwidth-1. The RankK Points are returned as the resulting pTree, P*/
For i=n to 0 {c=Count(P&Pi); If (c>=p) {RankVal=RankVal+2i; P=P&Pi };
else {p=p-c; P=P&P'i };
return RankKval, P; /* Above K=7-1=6 (looking for the Rank6 or 6th highest vaue (which is also the 2nd lowest value) */
23 * * * * =
RankKval=
P=MapRankKPts= ListRankKPts={2} 1
{0}
{1}
{0}
{1}

10. UDR Univariate Distribution Revealer (on Spaeth:)15
UDR Univariate Distribution Revealer (on Spaeth:)
applied to S, a column of numbers in bistlice format (an SpTS), will produce the DistributionTree of S DT(S)
depth=h=0
depth=h=1
Y y1 y2 y y y y y y y y y
ya 13 4
pb 10 9
yc 11 10
yd 9 11
ye 11 11
yf 7 8
yofM 11 27 23 34 53 80
118
114
125
110
121
109 83 p6 1 p5 1 p4 1 p3 1 p2 1 p1 1 p0 1
p6' 1
p5' 1
p4' 1
p3' 1
p2' 1
p1' 1
p0' 1
node2,3
[96.128) f=
depthDT(S)b≡BitWidth(S) h=depth of a node k=node offset
Nodeh,k has a ptr to pTree{xS | F(x)[k2b-h+1, (k+1)2b-h+1)} and its 1count
p6' 1
5/64 [0,64) p6
10/64 [64,128)
p5' 1
3/32[0,32)
2/32[64,96) p5
2/32[32,64)
¼[96,128)
p3' 1
0[0,8) p3
1[8,16)
1[16,24)
1[24,32)
1[32,40)
0[40,48)
1[48,56)
0[56,64)
2[80,88)
0[88,96)
0[96,104)
2[194,112)
3[112,120)
3[120,128)
p4' 1
1/16[0,16) p4
2/16[16,32)
1[32,48)
1[48,64)
0[64,80)
2[80,96)
2[96,112)
6[112,128)
Pre-compute and enter into the ToC, all DT(Yk) plus those for selected Linear Functionals (e.g., d=main diagonals, ModeVector .
Suggestion: In our pTree-base, every pTree (basic, mask,...) should be referenced in ToC( pTree, pTreeLocationPointer, pTreeOneCount ).and these
OneCts should be repeated everywhere (e.g., in every DT). The reason is that these OneCts help us in selecting the pertinent pTrees to access - and in fact are often all we need to know about the pTree to get the answers we are after.).

11. So let us look at ways of doing the work to calculate As we recall from the below, the task is to ADD bitslices giving a result bitslice and a set of carry bitslices to carry forward
XoD = k=1..nXk*Dk X
pTrees
XoD 1 3 2 1 1 1 1 6 9 1 1 1 1
3 3 D
D1,1
D1,0
1 1
D2,1
D2,0
1 1 (
= 22
+ 21
1 p1,0
p11
+ 20
1 p1,1
p2,1 )
p2,0
+ 1 p2,1 )
p2,0 ) 1 1 1 1 1 1 1 1 1
I believe we add by successive XORs and the carry set is the raw set with one 1-bit turned off iff the sum at that bit is a 1-bit
Or we can characterize the carry as the raw set minus the result (always carry forward a set of pTrees plus one negative one).
We want a routine that constructs the result pTree from a positive set of pTrees plus a negative set always consisting of 1 pTree.
The routine is: successive XORs across the positive set then XOR with the negative set pTree (because the successive pset XOR gives us the odd values and if you subtract one pTree, the 1-bits of it change odd to even and vice versa.):
/*For PXoD,i (after PXoD,i-1). CarrySetPos=CSPi-1,i CarrySetNeg=CSNi-1,i RawSet=RSi CSP-1=CSN-1=*/
INPUT: CSPi-1, CSNi-1, RSi
ROUTINE: PXoD,i=RSiCSPi-1,iCSNi-1,i CSNi,i+1=CSNi-1,iPXoD,i; CSPi,i+1=CSPi-1,iRSi-1;
OUTPUT: PXoD,i, CSNi,i CSPi,i+1 (
= 22
+ 21
1 p1,0
p11
+ 20
1 p1,1
p2,1 )
p2,0
+ 1 p2,1 )
p2,0 ) 1 1 1 1 1 1
RS1
CSN0,1=
CSN-1.0PXoD,0
CSP0,1=
CSP-1,0RS0
RS0
CSP-1,0=CSN-1,0=
PXoD,0
PXoD,1 1 1 1 1 1 1 1 1 1  =      = 1 1 1 

12. XoD=k=1,2Xk*Dk with pTrees: qN..q0, N=22B+roof(log2n)+2B+1
CCC Clusterer If DT (and/or DUT) not exceeded at C, partition C further by cutting at each gap and PCC in CoD
For a table X(X1...Xn), the SPTS, Xk*Dk is the column of numbers, xk*Dk. XoD is the sum of those SPTSs, k=1..nXk*Dk
Xk*Dk = Dkb2bpk,b
= 2BDkpk,B
Dkpk,0
= Dk(2Bpk,B
+..+20pk,0) =
(2Bpk,B
+..+20pk,0)
(2BDk,B+..+20Dk,0)
+ 22B-1(Dk,B-1pk,B
+..+20Dk,0pk,0
= 22B(
Dk,Bpk,B)
+Dk,Bpk,B-1)
XoD = k=1..nXk*Dk
k=1..n (
= 22B
+ 22B-1
Dk,B pk,B-1
+ Dk,B-1 pk,B
+ 22B-2
Dk,B pk,B-2
+ Dk,B-1 pk,B-1
+ Dk,B-2 pk,B
+ 22B-3
Dk,B pk,B-3
+ Dk,B-1 pk,B-2
+ Dk,B-2 pk,B-1
+Dk,B-3 pk,B
+ 23
Dk,B pk,0
+ Dk,2 pk,1
+ Dk,1 pk,2
+Dk,0 pk,3
+ 22
Dk,2 pk,0
+ Dk,1 pk,1
+ Dk,0 pk,2
+ 21
Dk,1 pk,0
+ Dk,0 pk,1
+ 20
Dk,0 pk,0
Dk,B pk,B
. . . 1 3 2 X
pTrees
1 2 D
D1,1
D1,0
0 1
D2,1
D2,0
1 0
B=1
XoD=k=1,2Xk*Dk with pTrees: qN..q0,
N=22B+roof(log2n)+2B+1
k=1..2 (
= 22
+ 21
Dk,1 pk,0
+ Dk,0 pk,1
+ 20
Dk,0 pk,0
Dk,1 pk,1
q0 = p1,0 = no carry 1
q1= carry1= 1 (
= 22
+ 21
D1,1 p1,0
+ D1,0 p11
+ 20
D1,0 p1,0
D1,1 p1,1
+ D2,1 p2,1 )
+ D2,1 p2,0
+ D2,0 p2,1 )
+ D2,0 p2,0 )
q2=carry1= no carry 1 (
= 22
+ 21
D1,1 p1,0
+ D1,0 p11
+ 20
D1,0 p1,0
D1,1 p1,1
+ D2,1 p2,1 )
+ D2,1 p2,0
+ D2,0 p2,1 )
+ D2,0 p2,0 ) 1
So, DotProduct involves just multi-operand pTree
addition. (no SPTSs and no multiplications)
Engineering shortcut tricka would be huge!!!
q0 = carry0= 1
3 3 D
D1,1
D1,0
1 1
D2,1
D2,0
1 1
q1=carry0+raw1= carry1= 1 2 (
= 22
+ 21
1 p1,0
p11
+ 20
1 p1,1
p2,1 )
p2,0
+ 1 p2,1 )
p2,0 ) 1
A carryTree is a valueTree or vTree, as is the rawTree at each level (rawTree = valueTree before carry is incl.).
In what form is it best to carry the carryTree over? (for speediest of processing?)
1. multiple pTrees added at next level? (since the pTrees at the next level are in that form and need to be added)
2. carryTree as a SPTS, s1? (next level rawTree=SPTS, s2, then s10& s20 = qnext_level and carrynext_level ?
q2=carry1+raw2= carry2= 1
q3=carry2 = carry3= 1

13. Should we pre-compute all pk,i*pk,j p'k,i*p'k,j pk,i*p'k,j
Question:
Which primitives are needed
and how do we compute them?
X(X1...Xn) D2NN yields a 1.a-type outlier detector (top k objects, x, dissimilarity from X-{x}).
D2NN = each min[D2NN(x)]
(x-X)o(x-X)= k=1..n(xk-Xk)(xk-Xk)=k=1..n(b=B..02bxk,b-2bpk,b)(
(b=B..02bxk,b-2bpk,b)
=k=1..n( b=B..02b(xk,b-pk,b) ) (
----ak,b---
b=B..02b(xk,b-pk,b) )
( 22Bak,Bak,B +
22B-1( ak,Bak,B-1 + ak,B-1ak,B ) +
{ 22Bak,Bak,B }
22B-2( ak,Bak,B-2 + ak,B-1ak,B-1 + ak,B-2ak,B ) +
{2B-1ak,Bak,B B-2ak,B-12
22B-3( ak,Bak,B-3 + ak,B-1ak,B-2 + ak,B-2ak,B-1 + ak,B-3ak,B ) +
{ 22B-2( ak,Bak,B-3 + ak,B-1ak,B-2 ) }
22B-4(ak,Bak,B-4+ak,B-1ak,B-3+ak,B-2ak,B-2+ak,B-3ak,B-1+ak,B-4ak,B)...
{22B-3( ak,Bak,B-4+ak,B-1ak,B-3)+22B-4ak,B-22}
(2Bak,B+
2B-1ak,B-1+..+
21ak, 1+
20ak, 0)
=k
D2NN=multi-op pTree adds?
When xk,b=1, ak,b=p'k,b and
when xk,b=0, ak,b= -pk.b
So D2NN just multi-op pTree mults/adds/subtrs?
Each D2NN row (each xX) is separate calc.
=22B ( ak,B2 + ak,Bak,B-1 ) +
22B-1( ak,Bak,B-2 ) +
22B-2( ak,B-12
22B-3( ak,Bak,B-4+ak,B-1ak,B-3) +
22B-4ak,B
+ ak,Bak,B-3 + ak,B-1ak,B-2 ) +
Should we pre-compute all pk,i*pk,j
p'k,i*p'k,j
pk,i*p'k,j
ANOTHER TRY!
X(X1...Xn) RKN (Rank K Nbr), K=|X|-1, yields1.a_outlier_detector (top y dissimilarity from X-{x}).
Install in RKN, each RankK(D2NN(x)) (1-time construct but for. e.g., 1 trillion xs? |X|=N=1T, slow. Parallelization?)
xX, the square distance from x to its neighbors (near and far) is the column of number (vTree or SPTS)
d2(x,X)= (x-X)o(x-X)= k=1..n|xk-Xk|2= k=1..n(xk-Xk)(xk-Xk)= k=1..n(xk2-2xkXk+Xk2)
= -2 kxkXk kxk kXk2
3. Pick this from XoX for each x and add to 2.
= xoX xox XoX
5. Add 3 to this
k=1..n i=B..0,j=B..02i+jpk,ipk,j
1. precompute pTree products within each k
i,j 2i+j kpk,ipk,j
2. Calculate this sum one time (independent of the x)
-2xoX cost is linear in |X|=N. xox cost is ~zero XoX is 1-time -amortized over xX (i.e., =1/N) or precomputed
The addition cost, -2xoX + xox + XoX, is linear in |X|=N So, overall, the cost is linear in |X|=n.
Data parallelization? No! (Need all of X at each site.) Code parallelization? Yes! (After replicating X to all sites,
Each site creates/saves D2NN for its partition of X, then sends requested number(s) (e.g., RKN(x) ) back.

14. LSR on IRIS150-3 Here we use the diagonals.
Here we try using other p points for the R step (other than the one used for the L step).
d=e1 p=AVGs, L=(X-p)od S E I
d=e1 p=AvgS, L=Xod
S&L
E&L
I&L
R & L
[43,49)
S(16)
128
[49,58)
E(24)I(6)
0 S(34) 99
393
1096
1217
1825
[58,70)
E(26)
I(32)
270
792
1558
2567
[70,79]
I(12)
2081
3444
R(p,d,X)
S E I
128
270
393
1558
3444
d=e1 p=AS L=(X-p)od (-pod=-50.06)
S&L
-1; E&L
I&L
-8,-2 16
[-2,8)
34, 24, 6 99
393
1096
1217
1825
[20,29] 12
[8,20)
26, 32
270
792
1558
2567
E=26
I=5
p=AvgS
30ambigs, 5 errs
d=e4 p=AvgS, L=(X-p)od
S&L
E&L
I&L
-2,4) 50
[7,11) 28
[16,23]
I=34
[11,16)
22,
127.5
648.7
1554.7
2892
E=22
I=7
p=AvgS
I(1)
E(50) I(7)
(36,7) 63
70 (11)
Only overlap L=[58,70), R[792,1557] (E(26), I(5))
With just d=e1, we get good hulls using LARC:
While  Ip,d containing >1class, for next (d,p)
create L(p,d)Xod-pod, R(p,d)XoX+pop-2Xop-L2
1.  MnCls(L), MxCls(L), create a linear boundary.
2.  MnCls(R), MxCls(R).create a radial boundary.
3. Use R&Ck to create intra-Ck radial boundaries
Hk = {I | Lp,d includes Ck}
I(42)
d=e1 p=AS L=(X-p)od (-pod=-50.06)
S&L
-1; E&L
I&L
-8,-2 16
[-2,8)
34, 24, 6 99
393
1096
1217
1825
[20,29] 12
[8,20) w
p=AvgE
26, 32
1.9
51.8
78.6
633
<--E=6
I=4
d=e4 p=AvgS, L=(X-p)od
S&L
E&L
I&L
-2,4) 50
[7,11) 28
[16,23]
I=34
[11,16)
22,
5.7
36.2
151.06
611
E=17
I=7
p=AvgE
d=e1 p=AS L=(X-p)od (-pod=-50.06)
S&L
-1; E&L
I&L
-8,-2 16
[-2,8)
34, 24, 6 99
393
1096
1217
1825
[20,29] 12
[8,20) w
p=AvgI
26, 32
0.62
34.9
387.8
1369
<--E=25
I=10
d=e4 p=AvgS, L=(X-p)od
S&L
E&L
I&L
-2,4) 50
[7,11) 28
[16,23]
I=34
[11,16)
22,
127.5
1555
2892
E=22
I=8
p=AvgI
For e4, the best choice of p for the R step is also p=AvgE.
(There are mistakes in this column on the previous slide!)
There is a best choice of p for the R step (p=AvgE) but how would we decide that ahead of time?

15. LSR on IRIS150
SRR(AVGs,dse) on C1,1 S
y isa O if yoD (-,-184)(123,381)(2046,)
y isa O or S(50) if yoD  C1,1  [-184 , 123]
y isa O if y isa C1,1 AND SRR(AVGs,Dse)(154,)
y isa O or S(50) if y isa C1,1 AND SRR(AVGs,DSE)[0,154]
y isa O or I(1) if yoD  C1,2  [ 381 , 590]
Dse ; xoDes: S E I
y isa O or E(50) or I(11) if yoD  C1,3  [ 590 ,1331]
y isa O or I(38) if yoD  C1,4  [1331 ,2046]
SRR(AVGs,dse) on C1,2only one such I
SRR(AVGs,dse) onC1,3 E I
y isa O if y isa C1,3 AND SRR(AVGs,Dse)(-,2)U(143,)
y isa O or E(10) if y isa C1,3 AND SRR in [2,7)
y isa O or E(40) or I(10) if y isa C1,3 AND SRR in [7,137) = C2,1
y isa O or I(1) if y isa C1,3 AND SRR in [137,143]
etc.
y isa O if yoD (-,-2) (19,)
y isa O or I(8) if yoD  [ -2 , 1.4]
y isa O or E(40) or I(2) if yoD  C3,1 [ 1.4 ,19]
Dei ; xoDei on C2,1: E I
SRR(AVGe,dei) onC3,1 E I
y isa O if y isa C3,1 AND SRR(AVGs,Dei)[0,2)(370,)
y isa O or E(4) if y isa C3,1 AND SRR(AVGs,Dei)[2,8)
y isa O or E(27) or I(2) if y isa C3,1 AND SRR(AVGs,Dei)[8,106)
y isa O or E(9) if y isa C3,1 AND SRR(AVGs,Dei)[106,370]
We use the Radial steps to remove false positives from gaps and ends. We are effectively projecting onto a 2-dim range, generated by the Dline and the Dline (which measures the perpendicular radial reach from the D-line). In the D projections, we can attempt to cluster directions into "similar" clusters in some way and limit the domain of our projections to one of these clusters at a time, accommodating "oval" shaped or elongated clusters giving a better hull fit. E.g., in the Enron case the dimensions would be words that have about the same count, reducing false positives.
d=e1=1000; The xod limits: S E I
y isa O if yoD(-,43)(79,)
y isa O or S( 9) if yoD[43,47]
y isa O or S(41) or E(26) or I( 7) if yoD(47,60) (yC1,2)
y isa O or E(24) or I(32) if yoD[60,72] (yC1,3)
y isa O if yoD[43,47]&SRR(-,52)(60,)
y isa O or I(11) if yoD(72,79]
y isa O if yoD[72,79]&SRR(-,49)(78,)
LSR on IRIS150-2
We use the diagonals.
Also we set a MinGapThres=2
which will mean we stay 2 units
away from any cut
y isa O or E( 3) if yoD[18,23)
y isa O if yoD(-,18)(46,)
y isa O or E(13) or I( 4) if yoD[23,28) (yC2,1)
y isa O or S(13) or E(10) or I( 3) if yoD[28,34) (yC2,2)
y isa O or S(28) if yoD[34,46]
y isa O if yoD[18,23)&SRR[0,21)
y isa O if yoD[34,46]&SRR[0,32][46,)
d=e2=0100 on C1,2 xod lims: S E I
d=e2=0100 on C1,3 xod lims: E
I zero differentiation!
y isa O or E(17) if yoD[60,72]&SRR[1.2,20]
y isa O or I(25)if yoD[60,72]&SRR[66,799]
y isa O or E( 7) or I( 7)if yoD[60,72]&SRR[20, 66]
y isa O if yoD[0,1.2)(799,)
d=e3=0010 on C2,2 xod lims: S E I
y isa O if yoD(-,28)(33,)
y isa O or S(13) or E(10) or I(3) if yoD[28,33]
d=e3=0001 xod lims: E I
y isa O or S(13) if yoD[1,5]
y isa O if yoD(-,1)(5,12)(24,)
y isa O or E( 9) if yoD[12,16)
y isa O or E( 1) or I( 3) if yoD[16,24)
y isa O if yoD[12,16)&SRR[0,208)(558,)
y isa O if yoD[16,24)&SRR[0,1198)(1199,1254)1424,)
y isa O or E(1) if yoD[16,24)&SRR[1198,1199]
y isa O or I(3) if yoD[16,24)&SRR[1254,1424]

16. LSR IRIS150. Next, we examine:
d=AvgEAvgI p=AvgE, L=(X-p)od S E I
R(p,d,X)
S E I 2 32 76
357
514
[-17,-14)]
I(1)
[-14,11)
(50, 13)
2.8
134
[11,33]
I(36)
E=47
I=12
R(p,d,X)
S E I .3 .9
4.7
150
204
213
[12,17.5)]
I(1)
d=AvgSAvgI p=AvgS, L=(X-p)od S E I
[17.5,42)
(50,12) 6
192
205
[11,33]
I(37)
E=45
I=12
d=AvgSAvgE p=AvgS, L=(X-p)od S E I
R(p,d,X)
S E I 2 6
137
154
393
[11,18)]
I(1)
[18,42)
(50,11)
6.92
133
[42,64] 38
E=39
I=11
d=e1 p=AvgS, L=Xod
S&L
E&L
I&L
Note that each L=(X-p)od is just a shift
of Xod by -pod (for a given d).
Next, we examine:
For a fixed d, the SPTS, Lp,d. is just a shift of LdLorigin,d by -pod we get the same intervals to apply R to, independent of p (shifted by -pod).
Thus, we calculate once, lld=minXod hld=maxXod, then for each different p we shift these interval limit numbers by -pod since these numbers are really all we need for our hulls (Rather than going thru the SPTS calculation of (X-p)od anew  new p).
There is no reason we have to use the same p on each of those intervals either.
d=e1 p=AS L=(X-p)od (-pod=-50.06)
S&L
-1; E&L
I&L
-8,-2 16
[-2,8)
34, 24, 6 99
393
1096
1217
1825
[20,29] 12
[8,20)
26, 32
270
792
1558
2567
E=26
I=5
30ambigs, 5 errs
d=e2 p=AvgS, L=(X-p)od
S&L
E&L
I&L
,-13) 1
-13,-11
0, 2, 1
all=-11
[0,4) [4,
-11,0
29,47,46 66
310
352
1749
4104
1, 1
46,11
2, 1
9, 3
d=e3 p=AvgS, L=(X-p)od
S&L
E&L
I&L
-5,4) 47
[4,15)
[37,55]
I=34
[15,37)
50, 15
157
297
536
792
E=18
I=12
3, 1
d=e4 p=AvgS, L=(X-p)od
S&L
E&L
I&L
-2,4) 50
[7,11) 28
[16,23]
I=34
[11,16)
22, 16
11 16
E=22
I=16
38ambigs 16errs
d=e1 p=AE L=(X-p)od (-pod=-59.36)
S&L
E&L
I&L
-17-11 16
[-11,-1)
33, 21, 3 27
107
172
748
1150
[11,20]
I12
[-1,11)
26, 32 1 51 79
633
E=7
I=4
E=5
I=3
d=e2 p=AvgE, L=(X-p)od
-5 `17 S&L
E&L
I&L
,-6) 1
[-6, -5)
0, 2, 1 15 18 58 59
[7,11) [11,
1 err
[-5,7)
29,47, 46 3
234
793
1103
1417
13, 21
21,
d=e3 p=AvgE, L=(X-p)od
S&L
E&L
I&L
,-25) 48
-25,-12
[9,27]
I=34
[-12,9)
49, 15
2(17) 16
158
199
E=32
I=14
d=e4 p=AvgE, L=(X-p)od
S&L
E&L
I&L
-7] 50
[-3,1) 21
[5,12] 34
[1,5)
22, 16
E=22
I=16
d=e1 p=AI L=(X-p)od (-pod=-65.88)
S&L
E&L
I&L
[-17,-8)
33, 21, 3 38
126
132
730
1622
2181
[-8,4)
26, 32 34
1368
E=26
I=11
E=2
I=1
d=e2 p=AvgI, L=(X-p)od
-7 `15 S&L
E&L
I&L
,-6) 1
[6,11) [11,
[-7, 4)
29,46,46 5 36
929
1403
1893
2823
[-8, -7)
2, 1
allsame
E=2
I=1
E=47
I=22
[5, 9]
9, 2, 1
S=9
d=e3 p=AvgI, L=(X-p)od
S&L
E&L
I&L
,-25) 48
-25,-12
[9,27]
I=34
[-25,-4)
50, 15 5 11
318
453
E=32
I=14
E=46
d=e4 p=AvgI, L=(X-p)od
S&L
E&L
I&L
[5,12] 34
[-6,-3)
22, 16
same range
E=22
I=16
So on the next slide, we consider all 3 functionals, L, S and R. E.g., Why not apply S first to limit the spherical reach (eliminate FPs). S is calc'ed anyway?

17. LSR IRIS150 e2 Setosa 23 44 vErsicolor 20 34 -11 10 vIrginica 22 38
Ld d=0100 p=origin
Setosa
vErsicolor
vIrginica
d=0100 p=AS=( )
d=0100 p=AE=( )
d=0100 p=AI=( ) 1
2 1
all
-7.7 5
929
1403
1892
2823
15 3 6

18. p=AvgS p=AvgE p=AvgI d=e1 d=e2 d=e3 d=e4 FAUST Oblique, LSR Lp,d
Linear, Spherical, Radial classifier
Form Class Hulls using linear d boundaries thru min and max of Lk.d,p=(Ck&(X-p))od 
On every Ik,p,d{[epi,epi+1) | epj=minLk,p,d or maxLk,p,d for some k,p,d} interval add spherical and barrel boundaries with Sk,p and Rk,p,d similarly (use enough (p,d) pairs so that no 2 class hulls overlap)
Points outside all hulls are declared as "other".
all p,ddis(y,Ik,p,d) = unfitness of y being classed in k. Fitness of y in k is f(y,k) = 1/(1-uf(y,k))
XoX
4026
3501
3406
3306
3996
4742
3477
3885
2977
3588
4514
3720
3401
2871
5112
5426
4622
4031
4991
4279
4365
4211
3516
4004
3825
3660
3928
4158
4060
3493
3525
4313
4611
4989
3672
4423
3009
3986
3903
2732
3133
4017
4422
3409
4305
3340
4407
3789
8329
7370
8348
5323
7350
6227
7523
4166
7482
5150
4225
6370
5784
6967
5442
7582
6286
5874
6578
5403
7133
6274
7220
6858
6955 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
7366
7908
8178
6691
5250
5166
5070
5758
7186
6066
7037
7884
6603
5886
5419
5781
6933
4218
5798
6057
6023
6703
4247
5883
9283
7055
9863
8270
8973
11473
5340
10463
8802
10826
8250
7995
8990
6774
7325
8458
8474
12346
11895
6809
9563
6721
11602
7423
9268
10132
7256
7346
8457
9704
10342
12181
8500
7579
7729
11079
8837
8406
5148
9079
9162
8852
9658
9452
8622
7455
8229
8445
7306 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
148
149
150 Ld 51 49 47 46 50 54 44 48 43 58 57 52 55 45 53 70 64 69 65 63 66 59 60 61 56 67 62 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 68 71 72 73 74 75
d=1000 76 77 79 78 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
148
149
150
On IRIS150 d, precompute! XoX, Ld=Xod nk,L,d  Lmin(Ck&Ld)
xk,L,d  max(Ck&Ld)
p, (pre-ccompute?)
Ld,p(X-p)od=Ld-pod nk,L,d,pmin(Ck&Ld,p)=nk,L,d-pod xk,L,d.pmax(Ck&Ld,p)=xk,L,d-pod
Lp,d
=Ld-pod
d=e1
d=e2
d=e3
d=e4
p=AvgS p=AvgE p=AvgI
d=1000 p=0000
nk,L,d xk,L,d S E I
d=1000 p=AS=( )
-7 8
-1 20
d=1000 p=AE=( )
d=1000 p=AI=( ) S E I
d=0100 p=0000
nk,L,d xk,L,d
d=0100 p=AS=( )
d=0100 p=AE=( )
d=0100 p=AI=( )
S 10 19 E I
d=0010 p=0000
nk,L,d xk,L,d
d=0010 p=AS=( )
-5 4
15 36
d=0010 p=AE=( )
-13 8
d=0010 p=AI=( )
-25 -4
S 1 6 E I
d=0001 p=0000
nk,L,d xk,L,d
d=0001 p=AS=( )
-1 4
8 16
12 23
d=0001 p=AE=( )
-12 -7
-3 5
1 12
d=0001 p=AI=( )
We have introduce 36 linear bookends to the class hulls, 1 pair for each of 4 ds, 3 ps , 3 class. For fixed d, Ck, the pTree mask is the same over the 3 p's. However we need to differentiate anyway to calculate R correctly.
That is, for each d-line we get the same set of intervals for every p (just shifted by -pod). The only reason we need to have them all is to accurately compute R on each min-max interval. In fact, we computer R on all intervals (even those where a single class has been isolated) to eliminate False Positives (if FPs are possible - sometimes they are not, e.g., if we are to classify IRIS samples known to be Setosa, vErsicolor or vIriginica, then there is no "other").
Assuming Ld, nk,L,d and xk,L,d have been pre-computed and stored, the cut-pt pairs of (nk,L,d,p; xk,L,d,p) are computed without further pTree processing, by the scalar computations: nk,L,d,p = nk,L,d-pod xk,L,d.p = xk,L,d-pod.

19. LSR IRIS150 e1 only
Analyze R:RnR1 (and S:RnR1?) projections on each interval formed by consecutive L:RnR1 cut-pts.
Sp  (X-p)o(X-p) = XoX + L-2p + pop nk,S,p = min(Ck&Sp) xk,S,p  max(Ck&Sp)
Rp,d Sp-L2p,d = L-2p-(2pod)d + pop + pod2 + XoX - L2d nk,R,p,d = min(Ck&Rp,d) xk,R,p,d  max(Ck&Rp,d)
Ld d=1000 p=origin
Setosa
vErsicolor
vIrginica
d=1000 p=AS=( )
d=1000 p=AE=( )
d=1000 p=AI=( )
270
1558
2568 16
128 24 6 34 99
393
1096
1217
1826 12
2081
3445
26 32 1
517,4 79
633 16
723
1258
279 5
171
186
748
998 12
249
794 16
1641
2391 24
132
730
1622
2281 12 17
220
26 32
388
1369
Recursion works wonderfully on IRIS: The only hull overlaps after only d=1000 are
And the 4 i's common to both are {i24 i27 i28 i34}. We could call those "errors".
with AI 17
220
with AE 1
517,4 78
633
eliminates
FPs better?
What is the cost for these additional cuts (at new p-values in an L-interval)?
It looks like: make the one additional calculation: L-2p-(2pod)d
then AND the interval masks, then AND the class masks?
(Or if we already have all interval-class mask, only one mask AND step.)
If we have computed, S:RnR1, how can we utilize it?.
We can, of course simply put spherical hulls boundaries by centering on the class Avgs, e.g.,
Sp p=AvgS
Setosa E=50 I=11
vErsicolor
vIrginica
If on the L 1000,avgE interval, [-1, 11) we recurse using SavgI we get
7 4 36
540,4 72
170
Thus, for IRIS at least, with only d=e1=(1000), with only the 3 ps avgS, avgE, avgI, using full linear rounds, 1 R round on each resulting interval and 1 S, the hulls end up completely disjoint. That's pretty good news!
There is a lot of interesting and potentially productive (career building) engineering to do here.
What is precisely the best way to intermingle p, d, L, R, S? (minimizing time and False Positives)?

20. A pTree Pillar k-means clustering method
(The k is not specified - it reveals itself.) m1 m4
Choose m1 as a pt that maximizes Distance(X, avgX)
Choose m2 as a pt that maximizes Distance(X, m1)
Choose m3 as a pt that maximizes h=1..2Distance(X, mh)
Choose m4 as a pt that maximizes h=1..3Distance(X,mh)
Do until minimumh=1..kDistance(X,mh) < Threshold m3 m2
(or Do until mk < Threshold)
This gives k. Apply pk-means. (Note we already have all Dis(X,mh)s for the first round.
Note: D=m1m2 line. Treat PCCs like parentheses - ( corresponds to a PCI and ) corresponds to a PC.
Each matched pair should indicate a cluster somewhere in that slice. Where?
One could take the VoM as the best-guess centroid? Then proceed by restricting to that slice.
Or 1st apply R and do PCC parenthesizing on R values to identify radial slice where the cluster occurs.
VoM of that combo slice (linear and radial) as the centroid. Apply S to confirm.
Note: A possible clustering method for identifying density clusters (as opposed to round or convex clusters)
(Treating PCCs like parentheses)
PCI
PCD
PCI
PCD
d-line

21. Clustering: 1. For Anomaly Detection
2. To develop Classes against which we future unclassified objects are classified.
( Classification = moving up a concept hierarchy using a class assignment function, caf:X{Classes} )
When is it important not to over partition? Sometimes it is but sometimes it is not. In 2. it usually isn't.
With gap clustering we don't ever over partition, but with PCC based clustering we can.
If it is important that each cluster be whole, when using a k=means type clusterer, each round we can fuse Ci and Cj iff on Lmimj their projections touch or overlap.
NewClu (k is discovered, not specified. Assign each (object,class) a ClassWeight, CWReals (could be <0).
Classes "take next ticket" as they're discovered (tickets are 1,2,... Initially, all classes empty; All CWs=0.
Do for next d, compute Ld = Xod until after masking off new cluster, count is too high (doesn't drop enough)
For the next PCI in Ld (next-larger starting from smallest)
If followed by a PCD, declare next Classk and define it to be the set spherically gapped (or PCDed)
around centroid, Ck=Avg or VoMk over Ld-1[PCI, PCD]. Mask off this ticketed new Classk and contin
If followed by a PCI, declare next Classk and define it to be the set spherically gapped (or PCDed) around the centroid, Ck=Avg or VoMk over Ld-1[ (3PCI1+PCI2)/4, PCI2 ) Mask off this ticketed new Classk and continue.
For the next-smaller PCI (starting from largest) in Ld
If preceded by a PCD, declare next Classk and define it to be the set spherically gapped (or PCDed) around centroid Ck=Avg or VoMk over Ld-1[PCD, PCI]. Mask off this ticketed new Classk, contin.
If preceded by a PCI, declare next Classk and define it to be the set spherically gapped (or PCDed) around the centroid, Ck=Avg or VoMk over Ld-1( PCI2, (3PCI1+PCI2)/4] Mask off this ticketed new Classk and continue.