Re: Randomness of ten digits

On Sat, 22 Nov 2008 03:09:06 GMT, "David L. Wilson"
<dwilson314@xxxxxxxxxxx> wrote:


"Stig Holmquist" <stigfjorden@xxxxxxxxxxx> wrote in message
news:1t4ci4h324s7gi399fune92557f9qhcfb6@xxxxxxxxxx
There are 3 628 800 permutations of the ten digits 0 to 9.
Some of these sequences are ordered such as 0-1-2-3-4-5-6-7-8-9
while others seem totally patternless such as 3-6-1-8-0-9-2-7-4-5.

Is there a simple test to determine the degree of disoder in each of
the many sequences?

I've been playing around with a gap test in which I square the gap
between each digit and sum them up. The first sequence yields 9 while
the second yields 312. Is this a reasonable test of randomness, if a
sequence is considered random if no pattern can be found?

It would be interesting to find out the distribution curve for the sum
of all squared gaps of the 3+ million sequnces. Has this problem been
solved?

I have looked at similar before but just summed the absolute values of the
difference of sucessive numbers. There is no reason to restrict the problem
to 10 digits and I was looking at it from the view of permuations of n
sucessive nuimbers. (For some applications that makes sense-I see no
application related reason in this case to squaring before summing as this
is not a case of orthogonal dimensions.) This was about 25 years ago and I
was looking for permutations that yielded the largest sum of absolute values
of the difference of sucessive numbers. The formula I obtained was
different for even and odd length sequences of such numbers. The
application was in regard to transmitting data redundantly and trying to
prevent the effectiveness of certain sorts of jamming.

An example is for (1,2,3,4,5,6,7), the sequence (3,7,1,5,2,6,4) maximizes
the above sum (giving 4+6+4+3+4+2=23 compared to 6 for (1,2,3,4,5,6,7).

Exercise: Show that that for (1,2,3,...,n), there is a permutation that
maximizes the sum of absolute values of the difference of sucessive numbers
to be:
(n^2-3)/2, for n odd
and
(n^2-2)/2, for n even.

Your post is very interesting as I too have looked at the sums of
absolut differences but have discovered it can be misleading.
Using your set of seven digits one can make four sequences all
yielding a sum of 23 as follows with the sum of squares next and the
sum of cubes last:

3-5-2-7-1-6-4 103 509

4-6-2-7-1-5-3 101 485

3-6-2-7-1-5-4 103 497

3-7-1-5-2-6-4 97 443

Unless Ive made an error, which I often do , there is a substantial
difference between seemingly equally random sequences. I use the
Wikipedia definition of randomness : A numeric sequence is said to be
statistically random when it contains no recognizable patterns or
regularities.

Based on the above sums I would suggest the set with the highest sum
of cubes is the most random.

My primary interest is in trying to generate a 10x10 Latin square with
the highest degree of randomness, such that no equal cells share any
corners but also has no recycled sequnces two or three steps displaced
from the previous row.

I've tried to make a 5x5 Latin square without any shared coners but
been unable so far to do better than 3 shared corners.

When counting gaps I think of each sequence as a circle and thus must
add 1 to each value above..

Stig






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Relevant Pages

  • Re: Randomness of ten digits
    ... I've been playing around with a gap test in which I square the gap ... It would be interesting to find out the distribution curve for the sum ... I have looked at similar before but just summed the absolute values of the ... different for even and odd length sequences of such numbers. ...
    (sci.math.num-analysis)
  • Re: Randomness of ten digits
    ... I've been playing around with a gap test in which I square the gap ... It would be interesting to find out the distribution curve for the sum ... I have looked at similar before but just summed the absolute values of the ... different for even and odd length sequences of such numbers. ...
    (sci.math.num-analysis)
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