1. No category

Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Dean S. Barron
President
twobluecats.com
[email protected]
A two sample test based
on rotationally
superimposable
permutations
Pawprints: A Cyclical Approach Based
On Kolmogoroff-Smirnoff
Conference on Nonparametric Statistics and Statistical Learning
The Blackwell and Pfahl Conference Center
The Ohio State University
May 19 -22, 2010
Pfahl 202 (Contributed) Nonparametric Tests
Thursday 20 May 2010
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 2.1. Sequences with four consecutive drawn from one population, n=8
location of
maximum
maximum
id
sequence (n/2)*ks consecutive
significant
consecutive
run
run
11112222 4
4
initial and final
yes
1
11122221 3
4
interior
no
5
11222211 2
4
interior
no
15
12222111 3
4
interior
no
35
21111222 3
4
interior
no
36
22111122 2
4
interior
no
56
22211112 3
4
interior
no
66
22221111 4
4
initial and final
yes
70
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
id=1 KS=1.00
griffe=11112222
1
id=5 KS=0.75
griffe=11122221
1
1
1
2
1
1
1
2
1
2
2
2
2
2
2
Figure 2.1. Circular representations. Green arrows are the sequence starts.
Definition 1. A unique permutation is called a griffe.
Definition 2. Operation ‫ ר‬is defined as the set of n rotations ‫ר‬k of griffes by 360k/n
degrees, for corresponding k=0, ... ,n-1.
This forms a cyclic abelian group by applying each of the n rotations to a griffe.
When present, the duplicate resultant transformed permutations are deleted to form
a reduced set.
Definition 3. Each such reduced set of griffes is called a patte.
This process is performed on every griffe, resulting in n!/[(n/2)!(n/2)!] pattes. When
present, the duplicate pattes are deleted to form a reduced set.
Definition 4. The reduced set of pattes is called a pawprint.
Each griffe has associated with it its original KS-value, called KSgriffe. Since each
set is comprised of equivalent data set sequences, the highest KS-value within a
patte is substituted for the original KS-value for each member griffe. This maximum
KS-value is called, KSpatte.
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 3.1. Operation, ‫ר‬, for n=8.
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
‫ר‬0
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬5
‫ר‬6
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬6
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 3.2. Duplicate generated pattes from two griffes from n=8
11112222 11122221
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
11112222
11122221
11122221
11222211
11222211
12222111
12222111
22221111
22221111
22211112
22211112
22111122
22111122
21111222
21111222
11112222
Table 3.3. Aligned duplicate generated pattes from two griffes from n=8
11112222 11122221
11112222
11122221
11222211
12222111
22221111
22211112
22111122
21111222
x
x
x
x
x
x
x
x
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
x
x
x
x
x
x
x
x
‫ר‬7
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
Two of these 10 pattes themselves contain duplicate griffes (Table 3.4). The
elimination of these elemental degenerative duplicates (light blue shading) results in
one patte of two griffes, and one patte of four griffes; the remaining eight pattes
consist of the full eight griffes. Thus, the remaining pattes are disjoint.
Table 3.4. The two pattes with degenerative duplicate griffes, n=8.
12121212 22112211
‫ר‬0
‫ר‬1
‫ר‬2
‫ר‬3
‫ר‬4
‫ר‬5
‫ר‬6
‫ר‬7
12121212
11221122
21212121
12211221
12121212
22112211
21212121
21122112
12121212
11221122
21212121
12211221
12121212
22112211
21212121
21122112
Note: Light blue shading indicates degenerative duplicates.
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 3.6 KS PP significance grid, n=8
PP significant
KS significant
0
0
1
total
1
68
2
70
total
0
0
0
68
2
70
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 3.7. KSgriffe KS patte grid, n=10
(n/2)*KS
(n/2)*PP
1
2
3
4
5
total
2
30
0
0
0
32
1
0
30
84
16
0
130
2
0
0
36
30
4
70
3
0
0
0
14
4
18
4
0
0
0
0
2
5
2
2
60
120
60
252
total
10
Note: Light blue areas represent sequences which are statistically significant.
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 3.10. KS PP significance grid, n=10
PP significant
KS significant
0
0
1
total
1
242
0
242
8
2
10
total
250
2
252
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 3.13. KS PP grid, linearized, 2≤n≤30.
KS
KS
KS
not sign
not sign
sign
n
PP
PP
PP
not sign
sign
not sign
2
0
0
2
6
0
0
4
20
0
0
6
68
0
2
8
242
8
0
10
894
6
18
12
3278
126
0
14
12512
118
150
16
45946
1042
820
18
180818
1658
1258
20
678218
12584
6732
22
2537728
81420
32844
24
9846592
93548
336908
26
38476962
886158
280098
28
1095330
2919720
30 149950590
KS
sign
PP
sign
0
0
0
0
2
6
28
90
812
1022
7898
52164
123552
473382
1151880
KS sign
0
0
0
2
2
24
28
240
1632
2280
14630
85008
460460
753480
4071600
PP sign
0
0
0
0
10
12
154
208
1854
2680
20482
133584
217100
1359540
2247210
n griffes
2
6
20
70
252
924
3432
12870
48620
184756
705432
2704156
10400600
40116600
155117520
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Daily Average Temperature
Februrary 2000
90
80
70
60
o
temp/ F
50
40
30
20
10
0
01
05
09
anchorage
Figure 4.1. Graph of eurostate data (REF210)
13
17
day
honolulu
paris
21
25
brussels
29
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 4.1. Comparison of power, β, for eurostate data at α=0.05
n
n1=n2
n/N
KScrit
KSmin
KSmax
βKS
PP
βPP
4
0.0690
4
2
4 0.1120
8
4
0
5
0.0862
5
3
5 0.0518
5
1.0000
10
6
0.1034
5
3
6 0.1936
6
1.0000
12
7
0.1207
6
4
7 0.1020
7
1.0000
14
8
0.1379
6
4
8 0.2529
8
1.0000
16
9
0.1552
6
5
9 0.4703
9
1.0000
18
10
0.1724
7
5
10 0.2973
10
1.0000
20
11
0.1897
7
6
11 0.5045
11
1.0000
22
12
0.2069
7
6
12 0.7470
12
1.0000
24
13
0.2241
7
7
13 1.0000
13
1.0000
26
14
0.2414
8
7
14 0.7598
14
1.0000
28
15
0.2586
8
8
15 1.0000
15
1.0000
30
29
0.5000
11
15
29 1.0000
29
1.0000
58
58
1.0000
15
29
29 1.0000
29
1.0000
116
Note: N=58. Blue area represents region where KSmin≥KScrit, universally. Pink area represents
region where PPcrit does not exist. KScrit (REF209).
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Relative Efficiency
Pawprints to Kolmogoroff-Smirnoff
eurostate data
4.50
Relative Efficiency, e
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.0000
0.2500
0.5000
0.7500
beta
alpha=0.01
alpha=0.05
alpha=0.10
Figure 4.2. Graph of relative efficiency for eurostate data.
lim alpha --> 0
1.0000
Dean S. Barron
twobluecats.com
Presented at Conference on Nonparametric Statistics and Statistical Learning,
The Ohio State University, May 19 -22, 2010
Table 4.2. Asymptotic Relative Efficiency, e, for β=1 for eurostate data
z
L
α
kscritn116
ks n
pp n
e
1.23
0.902972
0.1
14
22
10
2.20
1.36
0.950512
0.05
15
30
10
3.00
1.63
0.990154
0.01
18
42
14
3.00
1.95
0.999004
0.001
22
62
18
3.44
2.23
0.999904
0.0001
25
78
22
3.55
2.47
0.99999
0.00001
27
98
24
4.08
2.70
0.999999
0.000001
30
118
28
4.21
2.90
0.9999999 0.0000001
32
n/a
32
n/a
Note: Pink areas represent level at which relative efficiency is not defined. z and L are from Smirnoff
(REF202).