Definition of the HOBBit metric (Highest Order Bifurcation Bit)
Also called the maxPOI metric (max Position Of Difference)
*******************************
Hobbit metric in general
(on a continuous attributes??)
If Ai in R(A1..An) is a non-negative fixed point attribute.
Then we represent each x in Ai as a fixed-point binary number.
x = x(m)x(m-1)---x(1)x(0).x(-1)---x(-n) ( 0=.00...x(-inf) )
where . is the binary point position
Let m(x,y) = max{i|xi XOR yi=1}
= left-most position at which they differ.
d(x,y) = 2^m(x,y)
d(x,x) = 2^(-inf) = 0
1. Positive Definite: clear
2. Symmetry: clear
3. Triangle Inequality:
m(x,z) <= M=MAX{ m(x,y),m(y,z) } since x,y,z all the same beyond M
thus, either
a. 2^m(x,z) <= 2^m(x,y) or
b. 2^m(x,z) <= 2^m(y,z) (or both)
so, either
a. d(x,z) <= d(x,y) <= d(x,y) + d(y,z) or
b. d(x,z) <= d(y,z) <= d(x,y) + d(y,z) QED
**Illustration**********************************
v
x(m)x(m-1)---x(1)x(0).x(-1)---x(-n)
y(m)y(m-1)---y(1)y(0).y(-1)---y(-n)
v
y(m)y(m-1)---y(1)y(0).y(-1)---y(-n)
z(m)z(m-1)---z(1)z(0).z(-1)---z(-n)
M
v
x(m)x(m-1)---x(1)x(0).x(-1)---x(-n)
z(m)z(m-1)---z(1)z(0).z(-1)---z(-n)
************************************************
For signed fixed point, x = sgn(x)(m)x(m-1)---x(1)x(0).x(-1)---x(-n)
If x and y are same sign, d(x,y) = as above.
If x and y are opposite sign, d(x,y) = d(x,0) + d(0,y)
For floating point numbers, assume
x = x(m)x(m-1)---x(1)x(0).x(-1)---x(-n) * 2^exp
- shift both x and y until exp = 0 then apply HOBBit.
What are the r-discs of x? {y|d(x,y)<=1}
r=1: (all 1-discs are disjoint)
x__ x__ x__ x__
000 {000,001} 010 {010,011} 100 {100,101} 110 {110,111}
001 011 101 111
r=2: (all 2-discs are disjoint)
x__ x__
000 {000,001,010,011} 100 {100,101,110,111}
001 101
010 110
011 111
r=4: (1 4-disc)
x__
000 {000,001,010,011,100,101,110,111}
What are the r-circles? {y|d(x,y)=r}
r=1:
x__
000 {001} 010 {011} 100 {101} 110 {111}
001 {000} 011 {010} 101 {100} 111 {110}
r=2:
000 {010,011} 010 {000,001} 100 {110,111} 110 {100,101}
001 011 101 111
r=4:
000 {100,101,110,111}
001
010
011
What are the Hobbit Decision boundaries?
dim-2
11|- -:- -|- -:
| : | :
10|---+---|---+
| : | :
01|- -:- -A- -:
| : | :
00`---B---+---+
00 01 10 11 dim-1
a's are points closest to A
b's are points closest to B
blanks or s's are points same distance from A and B
dim-2
11 - - - - - -
| : | :
10 --- --- ---
| : | :
01a- -a- -B- -b
| : | :
00a---A---b---b
00 01 10 11 dim-1
11s- -s- -b- -B
| : | :
10s---s---b---b
| : | :
01a- -A- -s- -s
| : | :
00a---a---s---s
00 01 10 11
111|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
110|---.---:---.---|---.---:---.
| . : . | . : .
101|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
100|===.===:===.===|===.===:===.
| . : . | . : .
011|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
010|---.---:---.---|---.---:---.
| . : . | . : .
001|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
000`===.===:===.===|===.===:===.
000 001 010 011 100 101 110 111
111s- -s- -s- -s- -s- -s- -s- -s
| . : . | . : .
110s---s---s---s---s---s---s---s
| . : . | . : .
101s- -s- -s- -s- -s- -s- -s- -s
| . : . . . : .
100s===s===s===s===s===s===s===s
| . : . | . : .
011a- -a- -a- -a- -b- -b- -b- -b
| . : . | . : .
010a---a---a---a---B---b---b---b
| . : . | . : .
001a- -a- -a- -A- -b- -b- -b- -b
| . : . | . : .
000a===a===a===a===b===b===b===b
000 001 010 011 100 101 110 111
111a- -a- -a- -a- -|- -.- -:- -.
| . : . | . : .
110a---a---a---a---|---.---:---.
| . : . | . : .
101a- -a- -a- -a- -|- -.- -:- -.
| . : . | . : .
100a===A===a===a===|===.===:===.
| . : . | . : .
011|- -.- -:- -.- -b- -b- -b- -b
| . : . | . : .
010|---.---:---.---b---b---b---b
| . : . | . : .
001|- -.- -:- -.- -B- -b- -b- -b
| . : . | . : .
000`===.===:===.===b===b===b===b
000 001 010 011 100 101 110 111
111|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
110|---.---:---.---|---.---:---.
| . : . | . : .
101|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
100|===.===:===.===|===.===:===.
| . : . | . : .
011a- -A- -b- -b- -|- -.- -:- -.
| . : . | . : .
010a---a---B---b---|---.---:---.
| . : . | . : .
001|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
000`===.===:===.===|===.===:===.
000 001 010 011 100 101 110 111
111|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
110|---.---:---.---|---.---:---.
| . : . | . : .
101|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
100|===.===:===.===|===.===:===.
| . : . | . : .
011a- -A- -:- -.- -|- -.- -:- -.
| . : . | . : .
010a---a---:---.---|---.---:---.
| . : . | . : .
001|- -.- -b- -B- -|- -.- -:- -.
| . : . | . : .
000`===.===b===b===|===.===:===.
000 001 010 011 100 101 110 111
111|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
110|---.---:---.---|---.---:---.
| . : . | . : .
101|- -.- -:- -.- -|- -.- -:- -.
| . : . | . : .
100|===.===:===.===|===.===:===.
| . : . | . : .
011a- -a- -a- -a- -B- -b- -b- -b
| . : . | . : .
010a---a---a---a---b---b---b---b
| . : . | . : .
001a- -a- -a- -a- -b- -b- -b- -b
| . : . | . : .
000a===A===a===a===b===b===b===b
000 001 010 011 100 101 110 111
Notes from the above:
1. It seem correct that the dimension in which the
max occurs (for d(x,y)) a straight line perpendicular
to that access does separate the two hobbit ngbrhds,
Nx = {t|d(x,t) < d(y,t)} and
Ny = {t|d(y,t) < d(x,t)}.
However they can be very small sets (that's bad).
But they are very small precisely when
A & B are close (that's good).
We may get great advantage from breaking ties
in some way, e.g., d(x,y) defined as follows:
Sort the dimensional Hobbit distances,
( d(x1,y1),d(x2,y2),..,d(xn,yn) ) in decending order
( d(xi1,yi1),d(xi2,yi2),..,d(xin,yin) )
Define d(x,y) = SUM(j=1..n)[wj*d(xij,yij)] for some
rapidly decreasing sequence of fractions, wj.
For instance, using all 1 weights: Manhattan HOBBit distance
or L1-HOBBit (Note the original is then L-infinity-HOBBit)
d(x,y) = SUM(i=1..n)d(xi,yi)
111a- -a- -b- -b- - - - - - - -
| . : . | . : .
110a---a---b---b--- --- --- ---
| . : . | . : .
101a- -a- -b- -b- - - - - - - -
| . : . | . : .
100a===a=== ===b=== === === ===
| . : . | . : .
011a- -A- -a- - - -a- -a- -a- -a
| . : . | . : .
010a---a--- ---b--- ---a---a---a
| . : . | . : .
001b- - - -b- -B- -b- -b- -b- -b
| . : . | . : .
000 ===a===b===b===b===b===b===b
000 001 010 011 100 101 110 111
with lines removed:
111a a b b
110a a b b
101a a b b
100a a b
011a A a a a a a
010a a b a a a
001b b B b b b b
000 a b b b b b b
000 001 010 011 100 101 110 111
*******************************************************
Discussion:
I. Hobbit nbhds are
- eccentric (lop-sided) isn't bad.
- If the class-fctn (feature --> class_distribution)
IS continuous or smooth, then having more voters on
1 side does not necessarily skew the vote!
- It only skews the vote if there is some directional
discontinuity (where the fctn is only continuous in
certain feature directions)
- then Kriging works (Kriging finds veins of gold
which are directionally continuous and have
abrupt changes in other directions (sides of vein).
- Hobbit deals best with "low gradient" continuity
or smoothness and the eccentricity isn't a problem
if the plateaus can be narrowed so that the vote
weights are better differentiated wrt the proper
distance (the 1 defining the continuity or smoothness).
- What is bad is the fact that hobbit plateau's
(between rings) are wide (too few different distances).
- What is important is that each selected neighbor votes
with the best weight and that there are sufficiently
many close ngbrs (even if they are all on 1 side) to
take advantage of the continuity.
- Need thin rings!
II. The feature --> class_distr fctn must be understood.
- Is it distance-based? Which distance?
- Note Hobbit distance is a computational convenience
- Which podium or radial basis function (RBF) is correct
for the ring plateau's
- Are the contours tight enough around the feature sample
to take advantage of the continuity?
III. Given a table, R(A1...An)
Its' BSQ bands are R[A1], ..., R[An]
Its' bSQ bands are B11 (=bit-1 of R[A1]),..,B1m, ... , Bnm
One can view the bSQ bands as categorical,
but with a cannonical weighting ( Bij has weight 2^(m-j) / 2^m )
and the bands can be weighted by 0 <= wi <=1 and SUM(wi) = 1
So there is a gobal weighting of Bij's with wj * (2^(m-j)/2^m)
****************************************************************
Position Of Inequality Metrics (POI metrics or Hawaiian Metrics)
==============================
Any time there are weighted categorical bands,
Rank order them in decreasing order of weight
Hobbit distance = Max Position Of Inequality (MaxPOI)
Manhattan dist. = Sums Weights of Pos Of Ineq (SumPOI or L1-POI)
Euclidean dist. = SQRT(Sum SQRs of Wts of Pos Of Ineq) (L2-POI)
Minkowski-q dis.= q-RT(Sum (Wts of Pos Of Ineq)^q) (Lq-POI)
Max-distance =??? Hobbit
Collectively call these the Hawaiian Categorical Distances.
==============================
When there are more than 2 categories (more than just 0 or 1),
1. Attributize the categories (ala MBR)?
2. Use Hawaiian Distance?
How might we relieve some of the eccentricity of Hobbit rings?
FIBBONACCI HAWAIIAN METRICS
===========================
Vasiliy's idea (I think I have it right?)
(Vasiliy, please correct me if I missunderstand ;-)
Recode numbers, using Fibbonacci sequence as placeholders
rather than 2^0, 2^1, 2^2... or
10^0, 10^1, 10^2... or
Fibbonacci numbers: 1 1 2 3 5 8 13 21 34 55 89 144 233 ...
( ni = n(i-1) + n(i-2) )
Assign them to positions the usual way (e.g., for byte data):
Recursviesly divide by highest FN that divides x,
where x is the remainder from the previous division.
(the initial remainder is x)
233 144 89 55 34 21 13 8 5 3 2 1
Positions: 11 10 9 8 7 6 5 4 3 2 1 0
So positive numbers, NUM, get expressed as:
Fibbonacci
starter
Fib:233 144 89 55 34 21 13 8 5 3 2 1
Pos: 11 10 9 8 7 6 5 4 3 2 1 0
NUM:
0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 1
2 0 0 0 0 0 0 0 0 0 0 1 0
3 0 0 0 0 0 0 0 0 0 1 0 0
4 0 0 0 0 0 0 0 0 0 1 0 1
5 0 0 0 0 0 0 0 0 1 0 0 0
6 0 0 0 0 0 0 0 0 1 0 0 1
7 0 0 0 0 0 0 0 0 1 0 1 0
8 0 0 0 0 0 0 0 1 0 0 0 0
9 0 0 0 0 0 0 0 1 0 0 0 1
10 0 0 0 0 0 0 0 1 0 0 1 0
11 0 0 0 0 0 0 0 1 0 1 0 0
12 0 0 0 0 0 0 0 1 0 1 0 1
13 0 0 0 0 0 0 1 0 0 0 0 0
14 0 0 0 0 0 0 1 0 0 0 0 1
15 0 0 0 0 0 0 1 0 0 0 1 0
16 0 0 0 0 0 0 1 0 0 1 0 0
17 0 0 0 0 0 0 1 0 0 1 0 1
18 0 0 0 0 0 0 1 0 1 0 0 0
19 0 0 0 0 0 0 1 0 1 0 0 1
20 0 0 0 0 0 0 1 0 1 0 1 0
21 0 0 0 0 0 1 0 0 0 0 0 0
22 0 0 0 0 0 1 0 0 0 0 0 1
23 0 0 0 0 0 1 0 0 0 0 1 0
24 0 0 0 0 0 1 0 0 0 1 0 0
25 0 0 0 0 0 1 0 0 0 1 0 1
26 0 0 0 0 0 1 0 0 1 0 0 0
27 0 0 0 0 0 1 0 0 1 0 0 1
28 0 0 0 0 0 1 0 0 1 0 1 0
29 0 0 0 0 0 1 0 1 0 0 0 0
30 0 0 0 0 0 1 0 1 0 0 0 1
31 0 0 0 0 0 1 0 1 0 0 1 0
32 0 0 0 0 0 1 0 1 0 1 0 0
33 0 0 0 0 0 1 0 1 0 1 0 1
34 0 0 0 0 1 0 0 0 0 0 0 0
35 0 0 0 0 1 0 0 0 0 0 0 1
36 0 0 0 0 1 0 0 0 0 0 1 0
37 0 0 0 0 1 0 0 0 0 1 0 0
38 0 0 0 0 1 0 0 0 0 1 0 1
39 0 0 0 0 1 0 0 0 1 0 0 0
40 0 0 0 0 1 0 0 0 1 0 0 1
41 0 0 0 0 1 0 0 0 1 0 1 0
42 0 0 0 0 1 0 0 1 0 0 0 0
43 0 0 0 0 1 0 0 1 0 0 0 1
44 0 0 0 0 1 0 0 1 0 0 1 0
45 0 0 0 0 1 0 0 1 0 1 0 0
46 0 0 0 0 1 0 0 1 0 1 0 1
47 0 0 0 0 1 0 1 0 0 0 0 0
48 0 0 0 0 1 0 1 0 0 0 0 1
49 0 0 0 0 1 0 1 0 0 0 1 0
50 0 0 0 0 1 0 1 0 0 1 0 0
51 0 0 0 0 1 0 1 0 0 1 0 1
52 0 0 0 0 1 0 1 0 1 0 0 0
53 0 0 0 0 1 0 1 0 1 0 0 1
54 0 0 0 0 1 0 1 0 1 0 1 0
55 0 0 0 1 0 0 0 0 0 0 0 0
56 0 0 0 1 0 0 0 0 0 0 0 1
57 0 0 0 1 0 0 0 0 0 0 1 0
58 0 0 0 1 0 0 0 0 0 1 0 0
59 0 0 0 1 0 0 0 0 0 1 0 1
60 0 0 0 1 0 0 0 0 1 0 0 0
61 0 0 0 1 0 0 0 0 1 0 0 1
62 0 0 0 1 0 0 0 0 1 0 1 0
63 0 0 0 1 0 0 0 1 0 0 0 0
64 0 0 0 1 0 0 0 1 0 0 0 1
65 0 0 0 1 0 0 0 1 0 0 1 0
66 0 0 0 1 0 0 0 1 0 1 0 0
67 0 0 0 1 0 0 0 1 0 1 0 1
68 0 0 0 1 0 0 1 0 0 0 0 0
69 0 0 0 1 0 0 1 0 0 0 0 1
70 0 0 0 1 0 0 1 0 0 0 1 0
71 0 0 0 1 0 0 1 0 0 1 0 0
72 0 0 0 1 0 0 1 0 0 1 0 1
73 0 0 0 1 0 0 1 0 1 0 0 0
74 0 0 0 1 0 0 1 0 1 0 0 1
75 0 0 0 1 0 0 1 0 1 0 1 0
76 0 0 0 1 0 1 0 0 0 0 0 0
77 0 0 0 1 0 1 0 0 0 0 0 1
78 0 0 0 1 0 1 0 0 0 0 1 0
79 0 0 0 1 0 1 0 0 0 1 0 0
80 0 0 0 1 0 1 0 0 0 1 0 1
81 0 0 0 1 0 1 0 0 1 0 0 0
82 0 0 0 1 0 1 0 0 1 0 0 1
83 0 0 0 1 0 1 0 0 1 0 1 0
84 0 0 0 1 0 1 0 1 0 0 0 0
85 0 0 0 1 0 1 0 1 0 0 0 1
86 0 0 0 1 0 1 0 1 0 0 1 0
87 0 0 0 1 0 1 0 1 0 1 0 0
88 0 0 0 1 0 1 0 1 0 1 0 1
89 0 0 1 0 0 0 0 0 0 0 0 0
90 0 0 1 0 0 0 0 0 0 0 0 1
91 0 0 1 0 0 0 0 0 0 0 1 0
92 0 0 1 0 0 0 0 0 0 1 0 0
93 0 0 1 0 0 0 0 0 0 1 0 1
94 0 0 1 0 0 0 0 0 1 0 0 0
95 0 0 1 0 0 0 0 0 1 0 0 1
96 0 0 1 0 0 0 0 0 1 0 1 0
97 0 0 1 0 0 0 0 1 0 0 0 0
98 0 0 1 0 0 0 0 1 0 0 0 1
99 0 0 1 0 0 0 0 1 0 0 1 0
100 0 0 1 0 0 0 0 1 0 1 0 0
101 0 0 1 0 0 0 0 1 0 1 0 1
102 0 0 1 0 0 0 1 0 0 0 0 0
103 0 0 1 0 0 0 1 0 0 0 0 1
104 0 0 1 0 0 0 1 0 0 0 1 0
105 0 0 1 0 0 0 1 0 0 1 0 0
106 0 0 1 0 0 0 1 0 0 1 0 1
107 0 0 1 0 0 0 1 0 1 0 0 0
108 0 0 1 0 0 0 1 0 1 0 0 1
109 0 0 1 0 0 0 1 0 1 0 1 0
110 0 0 1 0 0 1 0 0 0 0 0 0
111 0 0 1 0 0 1 0 0 0 0 0 1
112 0 0 1 0 0 1 0 0 0 0 1 0
113 0 0 1 0 0 1 0 0 0 1 0 0
114 0 0 1 0 0 1 0 0 0 1 0 1
115 0 0 1 0 0 1 0 0 1 0 0 0
116 0 0 1 0 0 1 0 0 1 0 0 1
117 0 0 1 0 0 1 0 0 1 0 1 0
118 0 0 1 0 0 1 0 1 0 0 0 0
119 0 0 1 0 0 1 0 1 0 0 0 1
120 0 0 1 0 0 1 0 1 0 0 1 0
121 0 0 1 0 0 1 0 1 0 1 0 0
122 0 0 1 0 0 1 0 1 0 1 0 1
123 0 0 1 0 1 0 0 0 0 0 0 0
124 0 0 1 0 1 0 0 0 0 0 0 1
125 0 0 1 0 1 0 0 0 0 0 1 0
126 0 0 1 0 1 0 0 0 0 1 0 0
127 0 0 1 0 1 0 0 0 0 1 0 1
128 0 0 1 0 1 0 0 0 1 0 0 0
129 0 0 1 0 1 0 0 0 1 0 0 1
130 0 0 1 0 1 0 0 0 1 0 1 0
131 0 0 1 0 1 0 0 1 0 0 0 0
132 0 0 1 0 1 0 0 1 0 0 0 1
133 0 0 1 0 1 0 0 1 0 0 1 0
134 0 0 1 0 1 0 0 1 0 1 0 0
135 0 0 1 0 1 0 0 1 0 1 0 1
136 0 0 1 0 1 0 1 0 0 0 0 0
137 0 0 1 0 1 0 1 0 0 0 0 1
138 0 0 1 0 1 0 1 0 0 0 1 0
139 0 0 1 0 1 0 1 0 0 1 0 0
140 0 0 1 0 1 0 1 0 0 1 0 1
141 0 0 1 0 1 0 1 0 1 0 0 0
142 0 0 1 0 1 0 1 0 1 0 0 1
143 0 0 1 0 1 0 1 0 1 0 1 0
144 0 1 0 0 0 0 0 0 0 0 0 0
145 0 1 0 0 0 0 0 0 0 0 0 1
146 0 1 0 0 0 0 0 0 0 0 1 0
147 0 1 0 0 0 0 0 0 0 1 0 0
148 0 1 0 0 0 0 0 0 0 1 0 1
149 0 1 0 0 0 0 0 0 1 0 0 0
150 0 1 0 0 0 0 0 0 1 0 0 1
151 0 1 0 0 0 0 0 0 1 0 1 0
152 0 1 0 0 0 0 0 1 0 0 0 0
153 0 1 0 0 0 0 0 1 0 0 0 1
154 0 1 0 0 0 0 0 1 0 0 1 0
155 0 1 0 0 0 0 0 1 0 1 0 0
156 0 1 0 0 0 0 0 1 0 1 0 1
157 0 1 0 0 0 0 1 0 0 0 0 0
158 0 1 0 0 0 0 1 0 0 0 0 1
159 0 1 0 0 0 0 1 0 0 0 1 0
160 0 1 0 0 0 0 1 0 0 1 0 0
161 0 1 0 0 0 0 1 0 0 1 0 1
162 0 1 0 0 0 0 1 0 1 0 0 0
163 0 1 0 0 0 0 1 0 1 0 0 1
164 0 1 0 0 0 0 1 0 1 0 1 0
165 0 1 0 0 0 1 0 0 0 0 0 0
166 0 1 0 0 0 1 0 0 0 0 0 1
167 0 1 0 0 0 1 0 0 0 0 1 0
168 0 1 0 0 0 1 0 0 0 1 0 0
169 0 1 0 0 0 1 0 0 0 1 0 1
170 0 1 0 0 0 1 0 0 1 0 0 0
171 0 1 0 0 0 1 0 0 1 0 0 1
172 0 1 0 0 0 1 0 0 1 0 1 0
173 0 1 0 0 0 1 0 1 0 0 0 0
174 0 1 0 0 0 1 0 1 0 0 0 1
175 0 1 0 0 0 1 0 1 0 0 1 0
176 0 1 0 0 0 1 0 1 0 1 0 0
177 0 1 0 0 0 1 0 1 0 1 0 1
178 0 1 0 0 1 0 0 0 0 0 0 0
179 0 1 0 0 1 0 0 0 0 0 0 1
180 0 1 0 0 1 0 0 0 0 0 1 0
181 0 1 0 0 1 0 0 0 0 1 0 0
182 0 1 0 0 1 0 0 0 0 1 0 1
183 0 1 0 0 1 0 0 0 1 0 0 0
184 0 1 0 0 1 0 0 0 1 0 0 1
185 0 1 0 0 1 0 0 0 1 0 1 0
186 0 1 0 0 1 0 0 1 0 0 0 0
187 0 1 0 0 1 0 0 1 0 0 0 1
188 0 1 0 0 1 0 0 1 0 0 1 0
189 0 1 0 0 1 0 0 1 0 1 0 0
190 0 1 0 0 1 0 0 1 0 1 0 1
191 0 1 0 0 1 0 1 0 0 0 0 0
192 0 1 0 0 1 0 1 0 0 0 0 1
193 0 1 0 0 1 0 1 0 0 0 1 0
194 0 1 0 0 1 0 1 0 0 1 0 0
195 0 1 0 0 1 0 1 0 0 1 0 1
196 0 1 0 0 1 0 1 0 1 0 0 0
197 0 1 0 0 1 0 1 0 1 0 0 1
198 0 1 0 0 1 0 1 0 1 0 1 0
199 0 1 0 1 0 0 0 0 0 0 0 0
200 0 1 0 1 0 0 0 0 0 0 0 1
201 0 1 0 1 0 0 0 0 0 0 1 0
202 0 1 0 1 0 0 0 0 0 1 0 0
203 0 1 0 1 0 0 0 0 0 1 0 1
204 0 1 0 1 0 0 0 0 1 0 0 0
205 0 1 0 1 0 0 0 0 1 0 0 1
206 0 1 0 1 0 0 0 0 1 0 1 0
207 0 1 0 1 0 0 0 1 0 0 0 0
208 0 1 0 1 0 0 0 1 0 0 0 1
209 0 1 0 1 0 0 0 1 0 0 1 0
210 0 1 0 1 0 0 0 1 0 1 0 0
211 0 1 0 1 0 0 0 1 0 1 0 1
212 0 1 0 1 0 0 1 0 0 0 0 0
213 0 1 0 1 0 0 1 0 0 0 0 1
214 0 1 0 1 0 0 1 0 0 0 1 0
215 0 1 0 1 0 0 1 0 0 1 0 0
216 0 1 0 1 0 0 1 0 0 1 0 1
217 0 1 0 1 0 0 1 0 1 0 0 0
218 0 1 0 1 0 0 1 0 1 0 0 1
219 0 1 0 1 0 0 1 0 1 0 1 0
220 0 1 0 1 0 1 0 0 0 0 0 0
221 0 1 0 1 0 1 0 0 0 0 0 1
222 0 1 0 1 0 1 0 0 0 0 1 0
223 0 1 0 1 0 1 0 0 0 1 0 0
224 0 1 0 1 0 1 0 0 0 1 0 1
225 0 1 0 1 0 1 0 0 1 0 0 0
226 0 1 0 1 0 1 0 0 1 0 0 1
227 0 1 0 1 0 1 0 0 1 0 1 0
228 0 1 0 1 0 1 0 1 0 0 0 0
229 0 1 0 1 0 1 0 1 0 0 0 1
230 0 1 0 1 0 1 0 1 0 0 1 0
231 0 1 0 1 0 1 0 1 0 1 0 0
232 0 1 0 1 0 1 0 1 0 1 0 1
233 1 0 0 0 0 0 0 0 0 0 0 0
234 1 0 0 0 0 0 0 0 0 0 0 1
235 1 0 0 0 0 0 0 0 0 0 1 0
236 1 0 0 0 0 0 0 0 0 1 0 0
237 1 0 0 0 0 0 0 0 0 1 0 1
238 1 0 0 0 0 0 0 0 1 0 0 0
239 1 0 0 0 0 0 0 0 1 0 0 1
240 1 0 0 0 0 0 0 0 1 0 1 0
241 1 0 0 0 0 0 0 1 0 0 0 0
242 1 0 0 0 0 0 0 1 0 0 0 1
243 1 0 0 0 0 0 0 1 0 0 1 0
244 1 0 0 0 0 0 0 1 0 1 0 0
245 1 0 0 0 0 0 0 1 0 1 0 1
246 1 0 0 0 0 0 1 0 0 0 0 0
247 1 0 0 0 0 0 1 0 0 0 0 1
248 1 0 0 0 0 0 1 0 0 0 1 0
249 1 0 0 0 0 0 1 0 0 1 0 0
250 1 0 0 0 0 0 1 0 0 1 0 1
251 1 0 0 0 0 0 1 0 1 0 0 0
252 1 0 0 0 0 0 1 0 1 0 0 1
253 1 0 0 0 0 0 1 0 1 0 1 0
254 1 0 0 0 0 1 0 0 0 0 0 0
255 1 0 0 0 0 1 0 0 0 0 0 1
It looks like we get more hobbit plateaus and that they
are thinner ;-))))
To push the idea a little further, consider a Fibbonacci starter
value of .1 rather than 1 (results in 16 bit representations and
results in more plateaus which should be even thinner ;-)))
Fib 159 98 61 37 23 14. 8.9 5.5 3.4 2.1 1.3 0.8 0.5 0.3 0.2 0.1
Pos: 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
num_
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
2 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1
3 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0
4 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
5 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0
6 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1
7 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1
8 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0
9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
10 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0
11 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0
12 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1
13 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1
14 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0
15 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
16 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0
17 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1
18 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1
19 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0
20 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0
21 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0
22 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0
23 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1
24 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1
25 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
26 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0
27 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0
28 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1
29 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1
30 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0
31 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0
32 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0
33 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0
34 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1
35 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1
36 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0
37 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0
38 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0
39 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1
40 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
41 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0
42 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0
43 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0
44 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0
45 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1
46 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1
47 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
48 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0
49 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0
50 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1
51 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1
52 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1
53 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0
54 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0
55 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0
56 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1
57 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1
58 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0
59 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0
60 0 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0
61 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1
62 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1
63 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1
64 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0
65 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0
66 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0
67 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1
68 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1
69 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0
70 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
71 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0
72 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0
73 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1
74 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1
75 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0
76 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0
77 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0
78 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1
79 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1
80 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0
81 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0
82 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0
83 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0
84 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1
85 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1
86 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0
87 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0
88 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0
89 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1
90 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1
91 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0
92 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0
93 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0
94 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0
95 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1
96 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1
97 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0
98 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0
99 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0
100 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1
101 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1
102 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0
103 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0
104 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0
105 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0
106 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1
107 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1
108 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0
109 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0
110 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0
111 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1
112 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1
113 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1
114 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0
115 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0
116 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0
117 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1
118 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1
119 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0
120 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0
121 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0
122 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1
123 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1
124 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1
125 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0
126 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0
127 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0
128 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 1
129 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1
130 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0
131 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0
132 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0
133 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0
134 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1
135 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1
136 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0
137 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0
138 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0
139 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1
140 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1
141 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0
142 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0
143 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0
144 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0
145 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1
146 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1
147 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0
148 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0
149 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0
150 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1
151 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1
152 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0
153 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0
154 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0
155 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0
156 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1
157 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1
158 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0
159 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0
160 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
161 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1
162 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1
163 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0
164 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0
165 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0
166 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0
167 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1
168 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1
169 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
170 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0
171 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0
172 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1
173 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1
174 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1
175 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
176 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0
177 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0
178 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1
179 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1
180 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
181 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0
182 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0
183 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1
184 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1
185 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1
186 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0
187 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
188 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0
189 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1
190 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1
191 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0
192 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0
193 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0
194 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0
195 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1
196 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1
197 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0
198 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0
199 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
200 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1
201 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1
202 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0
203 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
204 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0
205 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0
206 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1
207 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1
208 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0
209 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
210 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0
211 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1
212 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1
213 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0
214 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0
215 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0
216 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0
217 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1
218 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1
219 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0
220 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0
221 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0
222 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1
223 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
224 1 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0
225 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0
226 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0
227 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0
228 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1
229 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1
230 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0
231 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0
232 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0
233 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1
234 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1
235 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1
236 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0
237 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0
238 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0
239 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1
240 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1
241 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0
242 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0
243 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0
244 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1
245 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1
246 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1
247 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0
248 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0
249 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0
250 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1
251 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1
252 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0
253 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0
254 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0
255 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0
The question of "thin-ness of plateaus" needs to be studied before
it can be claimed.
NOTES on FIBBONACCI NUMBERS and SEQUENCES:
1. Taking the fraction to be 1/B where B is any of
the standard Fibbonacci numbers {1,2,3,5,8,13,21,34...}
gives a sequence of base numbers which will include 1.
******75 46 28. 17. 11 6.8 4.2 2.6 1.6 1 0.6 0.4 0.2 0.2 = 1/5
Pos:15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
num_
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1
2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1
3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1
4 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1
5 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1
6 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1
7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
8 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1
9 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1
10 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1
11 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1
12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1
13 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1
14 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1
15 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1
16 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1
17 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1
18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
19 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1
20 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1
21 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1
22 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1
23 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1
24 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1
25 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1
26 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1
27 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1
28 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1
29 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
30 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1
31 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1
32 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1
33 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1
34 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1
35 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1
36 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1
37 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1
38 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1
39 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1
40 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1
41 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1
42 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1
43 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1
44 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1
45 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1
46 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1
47 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1
48 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1
49 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1
50 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1
51 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1
52 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1
53 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1
54 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1
55 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1
56 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1
57 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1
58 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1
59 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1
60 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1
61 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1
62 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1
63 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1
64 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1
65 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1
66 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1
67 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1
68 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 1
69 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1
70 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1
71 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1
72 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1
73 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1
74 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1
75 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1
76 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1
77 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1
78 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1
79 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1
80 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1
81 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1
82 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1
83 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1
84 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1
85 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1
86 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1
87 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1
88 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1
89 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1
90 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1
91 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1
92 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1
93 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1
94 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1
95 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1
96 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1
97 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1
98 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1
99 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1
100 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 1
101 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1
102 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1
103 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1
104 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1
105 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1
106 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1
107 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1
108 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1
109 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1
110 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1
111 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1
112 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1
113 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1
114 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 1
115 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 1
116 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1
117 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1
118 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1
119 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1
120 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1
121 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1
122 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1
123 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1
124 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1
125 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1
126 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1
127 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1
128 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1
129 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1
130 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1
131 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1
132 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1
133 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1
134 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1
135 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1
136 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1
137 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1
138 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1
139 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1
140 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1
141 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1
142 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1
143 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1
144 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1
145 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1
146 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1
147 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1
148 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 1
149 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1
150 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1
151 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1
152 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1
153 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1
154 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1
155 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1
156 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1
157 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1
158 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1
159 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1
160 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1
161 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1
162 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1
163 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1
164 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1
165 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1
166 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1
167 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1
168 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1
169 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1
170 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1
171 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1
172 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1
173 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1
174 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1
175 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 1
176 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1
177 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1
178 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1
179 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
180 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 1
181 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 1
182 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1
183 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1
184 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1
185 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1
186 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1
187 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 1
188 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1
189 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 1
190 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1
191 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1
192 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1
193 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1
194 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1
195 0 1 0 1 0 1 0 1 0 0 1 0 0 0 0 1
196 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1
197 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1
198 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
199 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1
200 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1
201 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1
202 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1
203 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1
204 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1
205 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1
206 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1
207 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1
208 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1
209 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1
210 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1
211 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1
212 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1
213 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1
214 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1
215 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1
216 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1
217 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
218 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1
219 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1
220 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1
221 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1
222 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1
223 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1
224 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1
225 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1
226 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1
227 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1
228 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1
229 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1
230 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1
231 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1
232 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1
233 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1
234 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1
235 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1
236 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1
237 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 1
238 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1
239 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1
240 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1
241 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1
242 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1
243 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1
244 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 1
245 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1
246 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1
247 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1
248 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1
249 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1
250 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1
251 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1
252 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
253 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1
254 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1
255 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1
2. In the canonical Fibbonacci sequence,
Lim(n -> inf)Fn/Fn-1 = ~1.61803... =gm, the golden mean
(and the convergences is oscillatory above and below gm).
3. There is a closed form formula for Fn
(nth element of the canoical Fibbonacci sequence):
Fn = ( (1+SQRT(5)/2)^n - (1-SQRT(5)/2)^n )
-------------------------------------
SQRT(5)
4. A rectangle with aspect ratio = gm has the nice recursive
(fractal?) property: Removing a maximal square leaves a
recatangle with aspect ratio = gm
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5. Given any Fibbonacci base sequence (FBS), {bn, bn-1, ..., b1, b0}
where b0 is the seed, b1=b0+0, bn=bn-1+bn-2 for n>1
the canonical FBS(b0) representation of a positive integer, x, is
QnQn-1...Q1Q0, where the Qi's are generated recursvely by:
Ri=x initially, for i = n, n-1, ... , 1, 0
IF bi <= Ri THEN Qi=1 and Ri-1 = Ri - bi ELSE Qi=0.
____Qi___
bi | Ri
Qi*bi
-----
Ri-1
This is taking out the maximum each time (left to right)
=======
6. What about taking out the minimum (right to left?
Ri=x initially, for i = 0, 1, 2, ... , n
IF bi <= Ri THEN Qi=1 and Ri+1 = Ri - bi ELSE Qi=0.
____Qi___
bi | Ri
Qi*bi
-----
Ri+1
Problems: Only an estimate of x is produce
and multiple x's can have the same representative.
That about taking a geometric sequence with base (~1.61803)^2 = gm^2 ?
************76. 46. 29. 17. 11. 6.8 4.2 2.6 1.6 1 0.6 0.3 .
Pos:15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1/(gm)^2
num_
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1
3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
4 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1
5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0
6 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0
7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
8 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
9 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1
10 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1
11 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1
12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
13 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
14 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0
15 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0
16 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0
17 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0
18 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
19 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0
20 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1
21 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1
22 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1
23 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0
24 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0
25 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0
26 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0
27 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1
28 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1
29 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1
30 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
31 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0
32 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0
33 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
34 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0
35 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0
36 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0
37 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0
38 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1
39 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1
40 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1
41 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0
42 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0
43 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0
44 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0
45 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0
46 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0
47 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
48 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0
49 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1
50 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1
51 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1
52 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0
53 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0
54 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
55 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0
56 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1
57 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1
58 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1
59 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
60 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0
61 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0
62 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0
63 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0
64 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0
65 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0
66 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
67 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1
68 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1
69 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1
70 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0
71 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0
72 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0
73 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0
74 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1
75 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1
76 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1
77 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
78 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0
79 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0
80 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0
81 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0
82 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0
83 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0
84 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0
85 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1
86 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1
87 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1
88 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0
89 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
90 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0
91 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0
92 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0
93 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0
94 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0
95 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0
96 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1
97 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1
98 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1
99 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0
100 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0
101 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0
102 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0
103 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1
104 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1
105 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1
106 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0
107 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0
108 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0
109 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0
110 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0
111 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0
112 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0
113 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0
114 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1
115 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1
116 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1
117 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0
118 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0
119 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0
120 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0
121 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0
122 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0
123 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
124 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0
125 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1
126 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1
127 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1
128 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
129 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0
130 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0
131 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0
132 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1
133 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1
134 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1
135 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0
136 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0
137 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0
138 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0
139 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0
140 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0
141 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
142 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0
143 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1
144 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
145 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1
146 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0
147 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0
148 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0
149 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0
150 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1
151 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1
152 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1
153 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0
154 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0
155 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0
156 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0
157 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0
158 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0
159 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0
160 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0
161 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1
162 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1
163 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1
164 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0
165 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0
166 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0
167 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0
168 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0
169 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0
170 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0
171 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0
172 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1
173 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1
174 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1
175 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0
176 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0
177 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0
178 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0
179 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1
180 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1
181 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1
182 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0
183 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0
184 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0
185 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0
186 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0
187 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0
188 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0
189 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0
190 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1
191 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1
192 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1
193 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0
194 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0
195 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0
196 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0
197 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1
198 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1
199 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
200 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
201 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1
202 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1
203 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1
204 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0
205 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0
206 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
207 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0
208 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1
209 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1
210 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 1
211 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
212 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0
213 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0
214 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0
215 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0
216 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0
217 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0
218 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0
219 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1
220 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1
221 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1
222 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0
223 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0
224 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0
225 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0
226 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1
227 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1
228 0 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1
229 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0
230 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0
231 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0
232 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0
233 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0
234 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0
235 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0
236 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0
237 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1
238 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1
239 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1
240 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0
241 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0
242 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0
243 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0
244 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0
245 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0
246 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0
247 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0
248 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1
249 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1
250 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1
251 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0
252 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0
253 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0
254 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0
255 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1
Appendix:
========
TOPOLOGIES, Definition:
For a finite set, we will define the Point-Topology
of a point with respect to a metric to be the set of
all intersections and unions that can be formed from
the generating set of "all open nontrivial spheres"
of positive radius containing the point. Non-trivial
means that the singleton point itself is not included.
With Hobbit (High order basic bits) distance we don't get
a superset until the radius is > 1 and then the supersets
are eccentric (as shown below).
The Hobbit "point-topology is different from all Lp
point-topologies (p=1..infinity) (which are centered).
HOBBIT (positive radius neighborhoods in 1-D)
------
1-bit:
0 {0,1}
1
2-bit:
00 0 {00,01} = {0,1} {0,1,2,3}
01
10 1 {10,11} = {2,3}
11
3-bit:
000 00 {000,001} = {0,1} {0,1,2,3} {0,1,4,5} {0,1,6,7} {0,1,2,3,4,5} {0,1,2,3,6,7} {0,1,4,5,6,7} {0,1,2,3,4,5,6,7}
001
010 01 {010,011} = {2,3} {2,3,4,5} {2,3,6,7} {2,3,4,5,6,7}
011
100 10 {100,101} = {4,5} {4,5,6,7}
101
110 11 {110,111} = {6,7}
111
4-bit:
0000 000 {0000,0001} = {0,1} plus all unions.
0001
0010 001 {0010,0011} = {2,3}
0011
0100 010 {0100,0101} = {4,5}
0101
0110 011 {0110,0111} = {6,7}
0111
1000 100 {1000,1001} = {8,9}
1001
1010 101 {1010,1011}={10,11}
1011
1100 110 {1100,1101}={12,13}
1101
1110 111 {1110,1111}={14,15}
1111
HOBIT podium steps around, e.g., 1000 (i.e., 1-D):
(there are no others using the hobbit topology)
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| |
. . . . . . . . * . . . . . . .
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
1 1 1 1 1 1
In this 1-D case, taking the step centers will be correct.
(and, e.g., using Gaussian-derived heights based on those)
v
|_v
| |_v_
| |___v___
_______v________| |
| |
. . . . . . . . * . . . . . . .
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
1 1 1 1 1 1
However, the same does not work in higher dimensions.
HOBIT podium steps around, e.g., (1000,0010) (2-D):
_______________________________
-15| |
-14| |
-13| |
-12| |
-11| |
-10| |
-9 | |
-8 | ______________|
-7 | | |
-6 | | |
-5 | | |
-4 | |______ |
-3 | | | | |
-2 | ,__| | |
-1 | | | |
-0 |________________|______|_______|
. . . . . . . . . . . . . . . .
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
1 1 1 1 1 1
What are the proper weights of the steps?
What we have been doing is to take the vector of center
values as the "representative point for the step).
_______________________________
-15| |
-14| |
-13| |
-12| . |
-11| |
-10| |
-9 | |
-8 | ______________|
-7 | | |
-6 | | . |
-5 | | |
-4 | |______ |
-3 | |. | | |
-2 | ,__| | |
-1 | | . | |
-0 |________________|______|_______|
. . . . . . . . . . . . . . . .
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
1 1 1 1 1 1
Not correct!
Notice:
1. that these are not the centers of mass of the steps.
2. Each successive step will always be the rest of a square,
cornered at the previous step (one of 4 possible ways,
depending on the next bits)
3. The center of mass of a step is at the (2/3,2/3) point (right?):
-15
-14
-13
-12
-11
-10
-9
-8 __|____|__|___
-7 | |
-6 | . . |_
-5 | |
-4 |______ * |- * = (12 1/6, 4 1/6) correct!
-3 | |
-2 , | . |_ (13 1/2, 5 1/2) not!
-1 | |
-0 |_______|
. . . . . . . . . . . . . . . .
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
1 1 1 1 1 1
What's more, we shouldn't be using the center of mass
of the step ("angle") anyway!
We want to take the 1st moment of the "angle" about the
original point , = (1000,0010)
Who will calculate the formula?
"Not I!" said the professor ;-)))
(reference to the "Lil Red Hen" story)
And the situation worsen with higher dimensions
(In 3-D, we have "caves" not "angles").
.------.------.------.------
/ / / / /|
/------/------/------/------/ |
/ / / / /| |
/------/------/------/------/ | |
/ / / / /| |/|
/------/------/------/------/ | | |
/ / / / /| |/| |
.------.------.------.------. | | |/|
| | | | | |/| | |
| | | | | | |/| |
|______|______|______|______|/| | | |
| | | | | |/| |/|
| | | | | | | | |
| | | | |/| |/|
`------+------+------+------| | | |/
| | | | | |/| /
| | | | | | |/
`------+--|______|______|/| /
| | | |/
| | | /
| | |/
`------+------'
When we use the product of the dimension-interval-centers
we are using this small cube (weighting votes as if all
points in this podium step are at the center of it).
.------.------
/ / /|
/------/------/ |
/ / /| |
.------.------. | |
| | | |/|
| | | | |
|______|______|/| '
| | | |/
| | | /
| | |/
`------+------'
and ignoring all of these other podium step points:
.------.------.
/ / /|
/------/------/ |
/ / /| |
/------/------/-------------
/ / / / /|
/------/------/------/------/ |
/ / / / /| |
.------.------.------.------. | |---
| | | | | |/| /|
| | | | | | |_/ |
|______|______|______|______|/| |/| |
| | | | | | | |/|
| | | | | |/| | |
| | | | | | |/| '
`------+------+------+------|/| | |/
| | | | | |/| /
| | | | | | |/
`------+--|______|______|/| /
| | | |/
| | | /
| | |/
`------+------'
Because the projections hide most of this step, we have been taking
the wrong "center":
.------.------.------.------
/ / / / /|
/------/------/------/------/ |
/ / / / /| |
/------/------/------/------/ | |
/ / / / /| |/|
/------/------/------/------/ | | |
/ / / / /| |/| |
. - - - .------.------.------.------. | | | |
| | | | | | |/| |/|
| | | | | | | | | |
center_| |______|______|______|______|/| |/| |
| / | | | | | | | |/|
| | | | | | |/| | |
| / | | | | | | |/| '
` - - - `------+------+------+------|/| | |/
/ | | | | | |/| /
. | | | | | | |/
| `------+--|______|______|/| /
| / | | | |/
| | | | /
/| / | | |/
r | `------+------'
e |/
t ` : :
n v v
e `------|------'
c c
e
n
t
e
r
Again, probably the 1st moment around the point is better.
And then we want to use Gaussian riser heights for the steps.
If we do that, I would guess that the accuracy of the Closed-NN
classification algorithm will be almost as good as Perfect Centering!!!
Who will develop an animation to show the Hobbit decision boundary
between 2 given points ;-))) (i.e., the locus of points equidistant
from the two given points)
Note, it will probably not be a line, but a solid region!
More precisely, I predict that it is always a set with positive measure
(and the points in those boundary sets should not vote for class-label
or centroid).
Hobbit Podium around (100,010) (riser heights shown as "linear", not Gaussian):
Match Step
.------.
/ * /|
________ _ _ .------. |-.------. _ _ _ ________
/ / | | |/1stBit/| / 7|
/------/- - - | | '------/ | - -/ |
/ / S t e p | |/ Step /| | 6| |
/------/ |------|------. | '------.------. / | '
/ / t | | | |3 / /| 5| |/
/------/ i /- | | | /- p ------/ | / | /
/ / B | | |2 e /| |4| |/
/------/ /- `--4---+--5---'- t ------/ | | | /
/ / d / / S 1| |/| |/
/------/ r /- /------ 2nd-bit ------/ | | | /
/ / 3 / / / 0| |/| |/
/------/- -/----- .------.------.--6---.--7---. | | | /
/ / / / | | | | | |/| |/
/------/------/------/ | | | | | | | /
/ / / / | | | | |/| |/
.--0---.--1---.--2---.--3---+------+------+------+------| | /
| | | | | | | | | |/
| | | | | | | | | /
| | | | | | | | |/
`------+------+------+------+------+------+------+------'
Hobbit Podium around (100,010) (riser heights ~Gaussian):
.------.------.---------------------------.------.------.
/ / / .-------------. / / 7|
/------/------/- .------. 1stBit/| -----/------/ '
/ / S t e p / * /| / | 6|/
/------/ - .------./ step/| /------.------. / /
/ / t / |------|------/ |3 / /| 5|/
/------/- i /- | | | /-- p -/------/ | / /
/ / B / | | |2 e / /| |4|/
/------/- /- `--4---+--5---'-- t -/------/ | | /
/ / d / / / / s / 1| |/|/
/------/- r /-- /------/- 2nd-bit -/------/ | | /
/ / 3 / / / / / 0| |/|/
/------/------/----- .------.------.--6---.--7---. | | /
/ / / | | | | | |/|/
/------/------/------/- | | | | | | /
/ / / / | | | | |/|/
.--0---.--1---.--2---.--3---+------+------+------+------| /
| | | | | | | | |/
`------+------+------+------+------+------+------+------'
Sometimes speed is not the over-riding issue (e.g., the KDD-Cup)
To optimize accuracy at the possible expense of speed we can use
perfectly centered steps using OR operations applied to individual
tuple Ptrees (what Maleq did). That, together with an optimal riser
height choice (Gaussian?) should be optimal. This will be a much
slower algorithm and does not use the Hobbit metric.
Within the context of the Hobbit metric (the best one to use wrt
speed when P-trees are involved), can we approach optimality even
closer than we have above?
Let's try to get better centering by tweaking the Hobbit method somewhat:
One can quickly convince oneself that it is impossible to
define a centered second step with only hobbit nrhds.
The reason is that the values y=001 and x=011
are not the boundary of any hobbit open set
___.------.------.------.------.------.------.------.___
/ / .------.-------------. 101/| /|
/- /- / _/_____ SL(4,2)/| -/ | / /
/ / /- / + /| / | 100/| |/|/111
/- /- / .------./ /| / -/ | | /
/ / /- `------'------/ |/011 /| |/|/110
/- /- / /| / -/ | | /
/ / .------.------.------. |/010 /| |/|/
/- /- | | | | / -/ | | /
/ / | | | |/001 /| |/|/
/- /- `-011--'-100--'-101--' -/ | | /
/ / 000| |/|/
/- .-001--.-010--.------.------.------.-110--.-111--/ | | /
/ | | | | | | | | |/|/
/- | | | | | | | | | /
/ | | | | | | | |/|/
.-000--+------+------+------+------+------+------+------| /
| | | | | | | | |/
`------+------+------+------+------+------+------+------'
One improvement possible is to move the center as we step down the
podium so that the steps never expand in the same directions twice:
#=SS-1,SS+1 0,1
%=SS,SS+1 1,1
000 001 __010____011____100____101_ 110 111
______________/ # / % /|_____________
/(SS-1,SS+1) _/______.-----.-------.___ // (SS,SS+1) /|
@=(01,10) S-1,S+1 /(0,1) _____/___@___/_____ (S,S) /|* /|// (1,1) / /
*=(10,10) S,S+1 / / 100,010/| 10,01/ | / |/ /|/111
/ / S-1,S) /------// /| |/ / / /
/ / (01,01) | |------. | |__/__________ /|/110
.---------/_____________|______|______| |/ /| / /
/ |______|_____/ /| / (SS,SS) / |/|/101
/ / / (S,S-1) / |/ (1,0) /| | /
/ / S-1,S-1) / (10,00) /| / / |/|/100
/ / (01,00) /_____________/ |/ /| | /
/ .------.------| | | / / |/|/011
/ (SS-1,SS) | | | | |/ /| | /
/ (0,0) | | | | |------.------. |/|/010
.------.------| | | | | | | | /
| | | | | | | | |/|/001
| | | | | | | | | /
| | | | | | | | |/000
`-000--+-011--+-010--+-011--+-100--+-101--+-110--+-111--'
Etc.
The relative riser heights needs to be studied for optimality.
Also note that the potential data points only occur at the following positions
.-------------------------------------------------------.
/ _______/______ /|
/ .-----------_/___,_ , /| / /
/ / , , / , /| / | /|/111
/ / .------./ ' /| | / /
/ / , , | |------. | |_____________ /|/110
.---------/_____________|______|______| |/ /| / /
/ , , | | / /| / ' ' / |/|/101
/ |-----------/ ' ' / |/ /| | /
/ , , / ' ' / , , /| / ' ' / |/|/100
/ / , , /_____________/ |/ , , /| | /
/ , , .------.------| | | / / |/|/011
/ | | | | |/ , , /| | /
/ , , | | | | |------.------. |/|/010
.------.------| | | | | | | | /
| | | | | | | | |/|/001
| | | | | | | | | /
| | | | | | | | |/000
`-000--+-011--+-010--+-011--+-100--+-101--+-110--+-111--'
So my best guess would be one of the following two podium schemes:
1.
--------------------------------------------------------
/ --------------------------- /|
/ / , , / , , /| / |
/ 0 / | /| |
/ 1 ' ' / ' ' /| / / | |
/ / - - - - - _______ - - - - /------------- /| | /
/ / , , /100, /| , , , /| / | |/111
/ 1 /___010/ / / / |/| | /
/ 0 ' ' | |/ ' ' ' /| | | |/110
/ / - - - - - `------' - - / - - - - - - / | | | /
/ / , , , , , , /| | | |/101
/ 0 / / / | | | /
/ 0 ' ' ' ' ' ' /| | | |/100
/ /______01_____/_____10______/_____11______/ | | | /
/ | | | | | | |/011
/ | | | | | | | | | /
/_____00______| | | | | | | | |/010
| | | | | | | | | /
| | | | | | | | | |/001
| | | | | | | | | /
| | | | | | | | |/000
`-000--+-011--+-010--+-011--+-100--+-101--+-110--+-111--'
--------------------------------------------------------
/ ----------------------------------------- /|
1 / , , / , , / , , /| / |
1 0 / |/| |
/ - 1 ' ' / ' ' / ' ' /| | | |
/ / - - - - - _______ - - - - -------------/ | | | /
/ / , , /100, /| , , , /| | | |/111
/ 1 /___010/ / / / | | | /
/ - 0 ' ' | |/ ' ' ' /| | | |/110
/ / - - - - - `------' - - / - - - - - - / | | | /
/ / , , , , , , /| | | |/101
/ 0 / / / | | | /
/ - 0 ' ' ' ' ' ' /| | | |/100
/ /______01_____/_____10______/_____11______/ | | | /
/ | | | | | | |/011
/ | | | | | | | | | /
/_____00______| | | | | | | | |/010
| | | | | | | | | /
| | | | | | | | | |/001 ^
| | | | | | | | | / /
| | | | | | | | |/000 /
`-000--+-011--+-010--+-011--+-100--+-101--+-110--+-111--' y
x -->
I think I favor this last one. The algorithm is easy:
***************************
1. STEP1 = S1 = P(100,010) and assign 1st (peak) riser value.
2. S2 = P(1,0) OR P(01,0) OR P(1,10) OR P(01,10); assign 2nd value to S2 & S1'.
3. Assign 3rd value to (S2 & S1)'
***************************
In general (for any sample point, (x, y) = (x1x2x3, y1y2y3)
***************************
1. STEP1 = S1 = P(x,y) and assign 1st (peak) riser value.
2. S2 = P(x1,y1) OR P(x1'x1,y1) OR P(x1,y1'y1) OR P(x1'x1,y1'y1); 2nd val to S2 & S1'
3. Assign 3rd value to (S2 & S1)'
***************************
Question: As the initial sample point gets close to the edge of the space,
do we want to modify this scheme so as to be more fair (accurate)???
Now, let's step back and look at the topology of the Hobbit metric
for the whole space. (all non-singlton sets realizable as Hobbit spheres).
The generator sets come directly from the "right shift" concept hierarchy:
[]
0 1
00 01 10 11
000 001 010 011 100 101 110 111
________________________________________________________
/ , , / , , / , , / , , /|
11 / |
/ ' ' / ' ' / ' ' / ' ' / |
/ - - - - - - - - - - - - - - - - - - - - - - - - - - - / | |
/ , , / , , / , , / , , /| | |
10 / | | |
/ ' ' / ' ' / ' ' / ' ' /| | | |
/ - - - - - - - - - - - - - - - - - - - - - - - - - - - / | | | /
/ , , / , , / , , , , /| | | |/111
01 / / | | | /
/ ' ' / ' ' / ' ' ' ' /| | | |/110
/ - - - - - - - - - - - - - - - - - - - - / - - - - - - / | | | /
/ , , , , , , , , /| | | |/101
00 / / / / | | | /
/ ' ' ' ' ' ' ' ' /| | | |/100
/_____00______/______01_____/_____10______/_____11______/ | | | /
| | | | | | | |/011
| | | | | | | | | | | /
| | | | | | | | | | |/010
| | | | | | | | | | /
| | | | | | | | | |/001 ^
| | | | | | | | | / /
| | | | | | | | |/000 /
`-000--+-011--+-010--+-011--+-100--+-101--+-110--+-111--' y
x -->
________________________________________________________
/ , , , , , , , , /|
11 / / |
/ ' ' ' ' ' ' ' ' / |
/ / / | |
/ , , , , , , , , /| | |
10 / / | | |
/ ' ' ' ' ' ' ' ' /| | | |
/ - - - - - - - - - - - - - / - - - - - - - - - - - - - / | | | /
/ , , , , , , , , /| | | |/111
01 / / | | | /
/ ' ' ' ' ' ' ' ' /| | | |/110
/ / / | | | /
/ , , , , , , , , /| | | |/101
00 / / | | | /
/ ' ' ' ' ' ' ' ' /| | | |/100
/_____________0_____________/_____________1_____________/ | | | /
| | | | | | | |/011
| | | | | | | | | | | /
| | | | | | | | | | |/010
| | | | | | | | | | /
| | | | | | | | | |/001 ^
| | | | | | | | | / /
| | | | | | | | |/000 /
`-000--+-011--+-010--+-011--+-100--+-101--+-110--+-111--' y
All sorts of hierarchical and grid-based clustering methods spring to mind
1. basically be examining carefully the "right-shift-rectangles" or RSRs
one should be able to see where the data is clustering.
- Of course we have already considered the P-cube, which is the collection
of all unit RSRs. In general, is is too large to examine for density points
2. If we start at the other extreme (divisive approach) at
the collection of all "full-right-shift-rectanges", FRSRs, of R(A1..An)
(assuming each is a byte) we search the root counts of
P(i1,,,,,,,;i2,,,,,,,; . . .;in,,,,,,,) = Pi11 ^ ... ^ Pin1
thowing out quadrants with low counts.
Within remaining quadrants, we back off the shift 1-bit and exam
P(i1,j1,,,,,,;i2,j2,,,,,,; . . .;in,jn,,,,,,)
thowing out quadrants with low counts.
etc.
Notes:
1. The above approach assumes the right-shift hierarchy, there are others!
2. The main point is: There aren't many hobit sets in total!
Should be able to search them effectively for clusters.
3. One could take a binary approach:
each bit is either predominantly 0 or predominantly 1.
If Pij is below threshold, fix that bit a 0
If Pij' is below threshold, fix that bit a 1
For a mask what remains.
This gives us a smaller search space
(raising the threshold makes it even smaller!)
4. One could find clusters iteratively (best cluster first, etc.)
At various levels in the right-shift hierarchy, for each Pij,
If RC(Pij) < max/2, fix that bit a 0
If RC(Pij) >= max/2, fix that bit a 1
The result should be the best cluster of that spatial granularity.
Mask it out and repeat...
Template:
.------.------.------.------.------.------.------.------
/ / / / / / / / /|
/------/------/------/------/------/------/------/------/ |
/ / / / / / / / /| |
/------/------/------/------/------/------/------/------/ | |
/ / / / / / / / /| |/|
/------/------/------/------/------/------/------/------/ | | |
/ / / / / / / / /| |/| |
/------/------/------/------/------/------/------/------/ | | | |
/ / / / / / / / /| |/| |/|
/------/------/------/------/------/------/------/------/ | | | | |
/ / / / / / / / /| |/| |/| |
/------/------/------/------/------/------/------/------/ | | | | | |
/ / / / / / / / /| |/| |/| |/|
/------/------/------/------/------/------/------/------/ | | | | | | |
/ / / / / / / / /| |/| |/| |/| |
.------.------.------.------.------.------.------.------. | | | | | | | |
| | | | | | | | | |/| |/| |/| |/|
| | | | | | | | | | | | | | | | |
| | | | | | | | |/| |/| |/| |/| |
|------+------+------+------+------+------+------+------| | | | | | | | |
| | | | | | | | | |/| |/| |/| |/|
| | | | | | | | | | | | | | | | |
| | | | | | | | |/| |/| |/| |/| |
|------+------+------+------+------+------+------+------| | | | | | | | |
| | | | | | | | | |/| |/| |/| |/|
| | | | | | | | | | | | | | | | |
| | | | | | | | |/| |/| |/| |/| |
|------+------+------+------+------+------+------+------| | | | | | | | |
| | | | | | | | | |/| |/| |/| |/|
| | | | | | | | | | | | | | | | |
| | | | | | | | |/| |/| |/| |/| |
|------+------+------+------+------+------+------+------| | | | | | | | /
| | | | | | | | | |/| |/| |/| |/7
| | | | | | | | | | | | | | | /
| | | | | | | | |/| |/| |/| |/ 6
|------+------+------+------+------+------+------+------| | | | | | /
| | | | | | | | | |/| |/| |/ 5
| | | | | | | | | | | | | /
| | | | | | | | |/| |/| |/ 4
|------+------+------+------+------+------+------+------| | | | /
| | | | | | | | | |/| |/ 3
| | | | | | | | | | | /
| | | | | | | | |/| |/ 2
|------+------+------+------+------+------+------+------| | /
| | | | | | | | | |/ 1
| | | | | | | | | /
| | | | | | | | |/ 0
------+------+------+------+------+------+------+------
0 1 2 3 4 5 6 7
***********************************************************************
***********************************************************************
Assuming x is weighted twice y, the influence function of (4,2) could be
constructed with maximum precision by considering the following podium steps:
--------------------------------------------------------
/ ----------------------------------------- /|
1 / , , / , , / , , /| / |
1 0 / |/| |
/ - 1 ' ' / ' ' / ' ' /| | | |
/ / - - - - - _______ - - - - -------------/ | | | /
/ / , /100, /|___ /| | | |/111
/ 1 -/___010/ | /|_________ / | | | /
/ - 0 ' ____| |/|_/___ / / /| | | |/110
/ / - - - -/ |------| /| / / / | | | /
/ / , /_____| |_____/ |/____ /| | | |/101
/ 0 |__________________| / /| / | | | /
/ - 0 ' |_____/ / /| | | |/100
/ /______01_____/_____10______/ |/ _/ | | | /
/ | | | | | | |/011
/ | | | | | | | | | /
/_____00______| | | | | | | | |/010
| | | | | | | | | /
| | | | | | | | | |/001 ^
| | | | | | | | | / /
| | | | | | | | |/000 /
`-000--+-011--+-010--+-011--+-100--+-101--+-110--+-111--' y
x - - >
000 001 010 011 100 101 110 111
.------.------.------.------.------.------.------.------.
| | | | | | | | |
000 | 15 | 13 | 10 | 8 | 6 | 8 | 10 | 13 |
| | | | | | | | |
|------+------+------+------+------+------+------+------|
| | | | | | | | |
001 | 14 | 9 | 7 | 4 | 3 | 4 | 7 | 9 |
| | | | | | | | |
|------+------+------+------+------+------+------+------|
| | | | | | | | |
010 | 6 | 5 | 3 | 2 | 1 | 2 | 3 | 5 |
| | | | | | | | |
|------+------+------+------+------+------+------+------|
| | | | | | | | |
011 | 14 | 9 | 7 | 4 | 3 | 4 | 7 | 9 |
| | | | | | | | |
|------+------+------+------+------+------+------+------|
| | | | | | | | |
100 | 15 | 13 | 10 | 8 | 6 | 8 | 10 | 13 |
| | | | | | | | |
|------+------+------+------+------+------+------+------|
| | | | | | | | |
101 | 22 | 19 | 17 | 12 | 11 | 12 | 17 | 19 |
| | | | | | | | |
|------+------+------+------+------+------+------+------|
| | | | | | | | |
110 | 26 | 24 | 20 | 18 | 16 | 18 | 20 | 24 |
| | | | | | | | |
|------+------+------+------+------+------+------+------|
| | | | | | | | |
111 | 28 | 27 | 25 | 23 | 21 | 23 | 25 | 27 |
| | | | | | | | |
`------+------+------+------+------+------+------+------'
000 001 010 011 100 101 110 111
Ulitmate Lazy Classification Algorithm:
--------------------------------------
R(A0,A1...An) with A0 = class label attribute
Decide weights for the dimensions (w1,...,wn), wi >= 0.
Decide podium shape functions (riser heights), (h1,...,hn)
hi = function of weighted distance from sample component, si.
Given a sample, s, each training point, x,
contributes a vote in
favor of its class label = SUM(i=1..n){hi(wi*(|xi-si|))}
Plurality vote wins:
V(c) = SUM(x=training|x0=c)h(SUM(i=1..n){wi*(|xi-si|)})
o Require a certain margin, else declare it an outlier??
o What if the vote distribution is bimodal? trimodal?
o Motivates examining vote distribution closely?
o e.g., if max >> mean then declare winner
else declare outlier.
Pros and Cons:
+ seems like it allows necessary flexibility for tuning
in a straight forward way.
+ KDD-cup: Use Medline abstracts to tune wi's and hi's???
+ No need to generate "mask Ptree" which defines "decision
- bdry partition" (very hard, except for simple cases)
+ Requires just one database scan to classify a sample
- Will Ptrees give better speed?
- It seems to be Manhattan-distance-based?
Would Euclidean be more accurate? slower?
Would Hobbit be faster and approximately as accurate?
Or take a particular distance (e.g., Manhattan) and one podium fctn:
distance
\
\
V(c) = SUM(x=training|x0=c){ h(d(x,s)) }
/
/
podium function
e.g.,
weighted Manhattan distance, d(x,s)
\
\
V(c) = SUM(x=training|x0=c){ h( SUM(i=1..n)wi*|xi-si| ) }
/
/
podium function
Anne_Denton suggests: d = Manhattan_distance
h = exp(-)
V(c) = SUM(x=training|x0=c){exp (-SUM(i=1..n)wi*|xi-si|)}
= SUM(x=training|x0=c){PROD{exp(-wi * |xi-si|)}
= SUM(x=training|x0=c){PROD{exp(-wi)^(|xi-si|)}
= SUM(x=training|x0=c){PROD{vi^(|xi-si|)} where vi = exp(-wi)
= SUM(x=training|x0=c){ PROD{vi^(SUM{(2^j)*(xij XOR sij)} }
Example: x = ( ,10,4,10) = ( , 1010, 0100, 1010)
B1: Yield(CL) B2: Green B3: Red B4: Blue
3 3 7 7 7 3 3 2 8 8 4 5 11 15 11 11
3 3 7 7 7 3 3 2 8 8 4 5 11 11 11 11
2 2 10 15 11 11 10 10 8 8 4 4 15 15 11 11
2 10 15 15 11 11 10 10 8 8 4 4 15 15 11 11
(w2,w3,w4) = (1,1,0)
h2= h3 = G = G(si,xi)=e^-(|si-xi|^2) (unnormalized Gaussian)
V(c) = SUM(x=training|x0=c)SUM(i=1..n){ wi*h(|xi,si|) }
x=training such that x0=c , contributes vote for c=x1 of:
2 2
-|s2-x2| -|s3-x3|
= e + e =
2 2
-|10-x2| -| 4-x3|
= e + e =
= e^-|10-x2|^2 + e^-|4-x3|^2
c=3: x=(7,8), (7,8), (3,8), (3,8)
V(3) = e^-|10-7|^2 + e^-|4-8|^2 +
e^-|10-7|^2 + e^-|4-8|^2 +
e^-|10-3|^2 + e^-|4-8|^2 +
e^-|10-3|^2 + e^-|4-8|^2
V(3) = .07350
V(7) = 3*e^-11
V(2) = 1.15858
V(10)= 1.38619
V(15)= 3
Using Ptrees?
------------
V(c)=SUM(x=training|x0=c)SUM(i=1..n){ wi*h(|xi-si|) }
= SUM(x in TR|x0=c)SUM(i=1..n){wi*h([(2^0)XOR(xi1,si1)+..+(2^7)XOR(xi7,si7)])}
If h is piecewise linear,
V(c)=SUM(x in TR|x0=c)SUM(i=1..n){wi*(h((2^0)XOR(xi1,si1))+..+h((2^7)XOR(xi7,si7))}
=SUM(x in TR|x0=c)SUM(i=1..n){wi*a1*(2^0)XOR(xi1,si1)+..+a7*(2^7)XOR(xi7,si7) + b}
=SUM(x in TR|x0=c)SUM(i=1..n){vi1*XOR(xi1,si1)+..+vi7*XOR(xi7,si7) + b}
=SUM(x in TR|x0=c)SUM(i=1..n){vi1*XOR(xi1,si1)+..+vi7*XOR(xi7,si7) + b}
This can be calculated using Ptrees.
A piecewise linear approximation can be constructed for any podium function
(Gaussian or??) therefore it can be done with Ptrees.