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Topic: Gumbel standard vs. Normal by Fingerprints
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Luis A. Afonso

Posts: 4,518
From: LIsbon (Portugal)
Registered: 2/16/05
Gumbel standard vs. Normal by Fingerprints
Posted: Apr 28, 2013 1:33 PM
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The Gumbel PDF,
______F(x;a,b) = (1/b)*exp(-exp(z)) where z= (x-a)/b

The standard version, a=0, b=1 is such that
______F(x;0,1) = exp(-exp(x))

To attain distinctive Fingerprints against Normal Data we must have at least n= 350 because the frequencies are, from 400´000 samples
Standard Gumbel________Normal
[000]= 0.000____________0.908
[001]= 0.000____________0.042
[010]= 0.056____________0.041
[111]= 0.944____________0.009

The test H0: normal against Ha: Gumbel, n=350,
Output A: [000] or [001] ___0.000__0.950
Chose Normal__Power 100%,
Output B: [010] or [111]___ 0.950__0.050
Chose Gumbel__Power = 95% , 1- (0.041+ 0.009)

Luis A. Afonso

REM "GUV-GBEL"
CLS
COLOR 13: PRINT : PRINT " <GUV-GBEL> "
PRINT " (reduced) GUMBEL DISTRIBUTION "
PRINT " [ Geary , UALM , VALM ] "
DEFDBL A-Z
RANDOMIZE TIMER
COLOR 14
pi = 4 * ATN(1)
DATA 100,150,200,250,300,350
FOR k = 1 TO 6: READ n(k): NEXT k
DATA 1.64, 1.64, 1.64,1.64,1.64,1.64 : REM GEARY
FOR j = 1 TO 6: READ zwcrit(j): NEXT j
DATA 3.90,3.88,3.87,3.87,3.86,3.84 : REM U
FOR j = 1 TO 6: READ Ucrit(j): NEXT j
DATA 3.52,3.56,3.59,3.65,3.65,3.70 : REM V
FOR j = 1 TO 6: READ Vcrit(j): NEXT j
INPUT " How many "; many
PRINT
DIM x(350)
PRINT : PRINT : PRINT
LOCATE 10, 10
PRINT " [000] [001] [010] [011] "
PRINT : PRINT
REM
REM ! FOR k = 4 TO 6
k = 6
n = n(k)
REM
REM
REM
c1 = 6 * (n - 2) / ((n + 1) * (n + 3))
c2 = 3 * (n - 1) / (n + 1)
c = 24 * n * (n - 2) * (n - 3)
c3 = c / ((n + 1) ^ 2 * (n + 3) * (n + 5))
COLOR 7
REM
FOR j = 1 TO many
RANDOMIZE TIMER
LOCATE 8, 50: PRINT USING "#########"; many - j
REM
m = 0
FOR i = 1 TO n
7 g = RND
IF g < 1E-08 THEN GOTO 7
W = LOG(-LOG(g))
x(i) = W
m = m + x(i) / n
NEXT i
REM
m2 = 0: m3 = 0: m4 = 0
absx = 0
FOR i2 = 1 TO n: d = x(i2) - m
m2 = m2 + d * d / n
m3 = m3 + d * d * d / n
m4 = m4 + d * d * d * d / n
absx = absx + ABS(d)
NEXT i2
REM
REM
s = m3 / (m2 ^ 1.5): U = s * s / c1
ki = m4 / (m2 * m2) - c2: V = ki * ki / c3
sigma = SQR(m2 / n)
tau = absx / n
omega = 13.29 * LOG(sigma / tau)
zww = SQR(n + 2) * ((omega - 3) / 3.54)
COLOR 7
REM
Gcrit = zwcrit(k)
Ucrit = Ucrit(k)
Vcrit = Vcrit(k)
REM
i1 = 0
IF zww > Gcrit THEN i1 = 1
i2 = 0
IF U > Ucrit THEN i2 = 1
i3 = 0
IF V > Vcrit THEN i3 = 1
W(i1, i2, i3) = W(i1, i2, i3) + 1
NEXT j
REM
LOCATE 12 + k, 10
PRINT USING " #.### "; W(0, 0, 0) / many; : W(0, 0, 0) = 0
PRINT USING "#.### "; W(0, 0, 1) / many; : W(0, 0, 1) = 0
PRINT USING "#.### "; W(0, 1, 0) / many; : W(0, 1, 0) = 0
PRINT USING "#.### "; W(0, 1, 1) / many; : W(0, 1, 1) = 0
REM
W(0, 0, 0) = 0
W(0, 0, 1) = 0
W(0, 1, 0) = 0
W(0, 1, 1) = 0
PRINT USING " ### "; n;
PRINT USING "#.## #.## #.## "; zwcrit(k); Ucrit(k); Vcrit(k)
REM ! NEXT k
END



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