p Statistics: Chain Distance
by JVSchmidt


General
When testing the difference of sums we knew that this test is not very sensitive due to the combinatorical proberty of the sum, generally: n + m = (n-x) + (m+x).
But we can treat any digit position as an coordinate value and ask about the distance of to chains.
Let us define the DISTANCE OF TWO CHAINS (DOC) as:

DOC = SUM i=1,L ABS(Xi - Yi)
where
L = length of chains X and Y
Xi,Yi = i-th digit of chain X respectively Y

Here is an example for p with L=5.
First sequence X=14159
First sequence Y=26535
DOC = abs(1-2) + abs(4-6) + abs(1-5) + abs(5-3) + abs(9-5) =
= 1 + 2 + 4 + 2 + 4 = 13

The recursive law for the expected distribution can easily be found.
Let w(L,d) be the probability that two chains of length L have a DOC = d.
Then w(L+1,d) = w(L,d) / 10 + sum w(L,d-i) x 2*(10-i)/100
where sums taken for i=1 to 9.

Starting condition is:
w(1,d) = 2*(10-d)/100 for d=1-9
w(1,0) = 1/10

We understand that the DOC-value can be an indicator for possible correlations between digits of neighbouring sequences.

Result's Overview
Digits analyzed: 4.2 * 10 9
Analysis started at digit: 1
Ellapsed computer time for one class: 3 min 40 sec


Chi2-values for the distributions of chain distances for different length of chains L

Length of
chains
L
Number of examined pairs of
chains
K = N/(2*L)
Chi2 Number of statistical
relevant classes
2 1.050.000.000 13,0688 19
5 420.000.000 41,7334 46
10 210.000.000 76,3766 72
20 105.000.000 108,3159 105
40 52.500.0000 161,6699 148
70 30.000.0000 214,09242 191
80 26.250.000 212,11474 203


Detailed results for this test you will find here: Details for Chain Distance Test (EXCEL file)

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