posted February 03, 2007 05:09 PM            

quote:

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Assuming that RND is uniformly distributed then there is an x% probability that it will produce results which only occur x% of the time.

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quote:

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This leads to a remarkable rule of thumb.

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There is nothing remarkable about it at all.

What would have been remarkable would have been not the case as this would have cast doubt on RND being uniformly distributed.

If an event is expected to occur only 1% of the time assuming some null hypothesis is true and that event occurs on the first and only test then we can rightly question the truth of the null hypothesis.

However, if we did the test 100 times then we should not be surprised to see one of the tests in the same vein as our single test.

I looked at 1000 RNDs 10000 times.

I then saw "...there is an x% probability that it will produce results which only occur x% of the time." Well, it would wouldn't it?

Oh, dear.

posted February 03, 2007 05:51 PM            

This is the magic number I remember hearing most often, but don't ask
me where it came from. (Maybe because most statisticians are
pentadactyl??)

Some texts might advise dropping rather than pooling classes.

But in general statistics is best viewed as a supplement rather
than a substitute for common sense, so I would interpret Sokal & Rohlf
accordingly.

posted February 03, 2007 06:50 PM            

In this test the tails have dropped out themselves because
less than two were expected.

Each seed seems to have its own Chi-Squared profile. It may be better to pick your seeds
specifically to get the best profile, rather than choose seeds at random.

This is one example. Whereas seed=1 produces a large #12 value
this one peaks in #9.

Results for Randomize(2)

#9 deviates from the norm. (1% significance = 50 @ 30 degrees of freedom)

Chi Squared Variance Probability Distribution for p=.5 with 1000 repeats of 26 in 30 sessions
Comparing with binomial distribution for (.5+.5)^26
30 degrees of freedom for each of 26 categories.
Randomizer seed=2

| #0 ChiSq = 0.
| #1 ChiSq = 0.
| #2 ChiSq = 0.
| #3 ChiSq = 0.
| #4 ChiSq = 0.
| #5 ChiSq = 0.
|_______________________________ #6 ChiSq = 27.1
|________________________________________ #7 ChiSq = 34.3
|__________________________ #8 ChiSq = 22.5
|____________________________________________________________ #9 ChiSq = 51.9
|_______________________ #10 ChiSq = 20.1
|________________________________________ #11 ChiSq = 34.6
|_________________________________ #12 ChiSq = 28.2
|______________________________________ #13 ChiSq = 32.5
|______________________ #14 ChiSq = 19.1
|_____________________________ #15 ChiSq = 24.9
|_________________________________ #16 ChiSq = 28.6
|_________________________________ #17 ChiSq = 28.6
|_____________________________ #18 ChiSq = 24.8
|___________________________________ #19 ChiSq = 30.4
|____________________________________________ #20 ChiSq = 37.8
| #21 ChiSq = 0.
| #22 ChiSq = 0.
| #23 ChiSq = 0.
| #24 ChiSq = 0.
| #25 ChiSq = 0.
| #26 ChiSq = 0.

total chiSq=445.398224885069 at 780 degrees of freedom

posted February 03, 2007 07:10 PM            

The algorithm used to generate the values of RND is probably the linear congruence method ...
see http://www.cs.utsa.edu/~wagner/laws/rng.html

posted February 03, 2007 07:40 PM            

By the way, In my tests above, there is only one seeding
at the beginning, and the Chi profile remains consistent, even when
the number of experiment cycles or 'sessions' vary.

posted February 03, 2007 09:23 PM            

David:
Bob Zale made a brief statement about PB's random number generator
in the PB-DOS forum, in 2004 & said there the period is 2^32 http://www.powerbasic.com/support/forums/Forum3/HTML/001821.html

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posted February 04, 2007 02:58 AM            

What caught my eye was

quote:

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In fact, excessive use of RANDOMIZE (or any other changes) will cause deterioration.

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posted February 04, 2007 10:36 AM            

Very true of statistics in general. The statistical methods & tabled
values of probabilities that R.A. Fisher and his students put forth in the
1930's are archaic in the light of what can be done by way of computing
and Monte Carlo methods with a PC as compared with an abacus or a slide
rule.

But that reminds me: In Ann Arbor, Michigan in the early 1950's there
used to be a yearly contest between the campus main computer and
one or more Asian students on an abacus. The last time I heard the abacus
was still winning. But then I left the U of M in 1952.

posted February 04, 2007 01:58 PM            

1) The powerBasic prng produces 2^32 or 4GB of random "values."
The size of each value can be up to an EXT in size, but for best
statistical randomness (Diehard, etc.) it's usually best to use
the BYTE size.

2) There is one sequence only of 4GB of random values. Various seed
numbers place you at different starting points in this sequence, and the
exact same starting point can be duplicated by many different seeds.

3) Sequences from a given starting point can overlap other sequences
from a different starting point significantly, and even nearly in
their entirety, even tho their seeds may not seem to be related,
eg. 1,2,3,4,5,6,7,8,9,10 overlaps 3,4,5,6,7,8,9,10,11,12,13

4) From any seed, after 4GB of values, you will repeat those values
exactly again. Since there is one sequence only of random values,
whatever seed is given will produce the same 4GB of values, just shifted
by a maximum of 2GB. In effect, all seeds produce sequences that
will overlap each other after a maximum of 2GB of values.

5) It follows that if you exceed 4GB of random values, simply
re-seeding the generator will not give you new random values.
Rather, it will repeat a portion of the sequence you have already
generated. If you re-seed before you reach 4GB of values, you risk
repeating part of your sequence even sooner.

posted February 04, 2007 06:59 PM            

A feasible procedure for avoiding potential problems (assuming that
one really needs more than 4 billion random numbers) would, I presume,
be to switch to different RNGs periodically. Or are there problems
with that too?

Charles:
I like the strategy James used (a few pages back) to circumvent
problems about PB's random number generator and get back to the more
basic math problem. I had asked what is the best estimate of
Pi one could get with PB, and my method was based on the ratio of two
long integers. Obviously, this population of numbers is not
infinitely large and no number was infinitely accurate in
precision. All possible outcomes could readily be enumerated --
which (at least for my problem) rendered data sampling
irrelevant & unnecessary.

By the same token, all possible values of TIMER could also be enumerated.
For the problem of the mean and for the problem of the shape of the
frequency distribution, the precise sequence in which the RNDs occur is
of no significance. Therefore both of these problems can be simplified considerably.

That reminds me of a simple program I wrote a couple of years ago
to demonstrate the influence of the precision of one's random numbers
on the precision of one's estimates of means, sd's and pi. The user is
given a choice of integer data or precision to 1,2,...18 decimals.
All possible numbers (and pairs of numbers) that one can get are then
dealt with, exhaustively. Of course if one wants even 6-place numbers one
one is going to have a long wait for the answer, but the point
of the exercise is usually clear long before that time. I suspect
that probability theorists might have a straightforward formula or principle
that will predict and explain such data, but I have not yet run across
it.

P.S. I hope that you are not advocating picking the right seed numbers
for hypothetically random numbers, to get the answers that one expects
on the basis of theory. My teachers would have turned pale at the
thought.

posted February 04, 2007 07:52 PM            

Blum Blum Shub is worth checking out. I found it a bit more
noisy than PB's RND function, but maybe I didnt pick the best
primes.

posted February 05, 2007 06:17 AM            

I tested about 3000 seeds with it last night and found only 47 which passed
the test.(Chi2<35)

One of the good things about the Chi-Squared is that it will tell you if your
results are 'too good', which in this case would be a value below about 15 in any
category. I found this did not happen very often.

The test of course applies specifically to a fixed range of numbers in the
pseudo-random sequence. In this case, 26*1000*30=780,000, with each number
being used as a 'coin flip'.

Here is a typical result:

Matching with expected binomial distribution:
New seed 2036
....
|______________________________________________ #6 ChiSq = 25.9
|________________________________________________ #7 ChiSq = 27.3
|__________________________________________________________ #8 ChiSq = 32.8
|________________________________________________ #9 ChiSq = 27.4
|_____________________________________________________ #10 ChiSq = 30.1
|______________________________________ #11 ChiSq = 21.7
|______________________________________ #12 ChiSq = 21.6
|___________________________________________________ #13 ChiSq = 28.7
|____________________________________________________________ #14 ChiSq = 34.
|______________________________________________ #15 ChiSq = 26.1
|___________________________________ #16 ChiSq = 19.7
|_____________________________________ #17 ChiSq = 21.2
|___________________________________________________ #18 ChiSq = 29.
|______________________________________ #19 ChiSq = 21.5
|_____________________________________________________ #20 ChiSq = 30.3
....