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Re: Re: bimodal ditribution form counting signs of Pi digits differences
*To*: mathgroup at smc.vnet.net
*Subject*: [mg51834] Re: [mg51793] Re: bimodal ditribution form counting signs of Pi digits differences
*From*: DrBob <drbob at bigfoot.com>
*Date*: Wed, 3 Nov 2004 01:24:28 -0500 (EST)
*References*: <clst68$3nf$1@smc.vnet.net> <cm4rh8$6oo$1@smc.vnet.net> <200411020705.CAA21655@smc.vnet.net>
*Reply-to*: drbob at bigfoot.com
*Sender*: owner-wri-mathgroup at wolfram.com
>> It appears that both this version and the built in ( rule 30 based)
>> random are Markov based ( depend on their own previous history )
>> to produce randomness. It appears that Pi by measure doesn't and is,
>> thus, more "ideally" random.
The 1000th digit of Pi is completely determined by (a) the number 1000 and (b) the number Pi -- of which the first 999 digits are a part. How can you say, then, that it doesn't depend on the first 999 digits?
> By my experiments the "traditional" pseudorandom
> seems more random than the rule 30 based version.
You're just looking at pictures of finite samples. That proves nothing, and anyway, they DON'T seem more random (to anybody but you).
The point is that
(a) If Pi digits ARE pseudo-random in all the ways we'd like them to be, there's still no proof of it -- whereas the properties of CA-30 are mathematically proven.
(b) Computing Pi digits is MUCH less efficient than CA-30 and other standard generators. If we need a billion random variates, this is a very bad way to get them -- especially if we're not looking for small integers.
(c) These questions have been studied to death by experts.
Bobby
On Tue, 2 Nov 2004 02:05:21 -0500 (EST), Roger Bagula <tftn at earthlink.net> wrote:
> It appears that both this version and the built in ( rule 30 based)
> random are Markov based ( depend on their own previous history )
> to produce randomness.
> It appears that Pi by measure doesn't and is, thus, more "ideally" random.
> Questions associated with all such dependent randomness ( not just rule 30)
> are well known.
> By my experiments the "traditional" pseudorandom
> seems more random than the rule 30 based version.
> But still less than the ideal for which Pi seems better suited?
> I've been told that my experimentation with this area of thought for my
> own personal
> gratification is "futile".
> It seems mostly that there is a "doctrinaire" tide in place and if it
> questions
> Mathematica's integrity it is "futile".
> That "doctrinaire" tide is not a scientific
> or mathematical attitude that stand up to any critical comment.
> Association with such thought patterns is personally repulsive for me as
> well.
> Many current professional level development systems
> give access to more than one way to produce pseudorandom
> numbers for simulations. It is well known that not all such
> randomness systems are "equal" in their measures of randomness.
> Suppression of personal research for doctrinaire reasons
> is one of the worst results of a commerial enterprise
> in a scientific sense.
> Roger Bagula wrote:
>
>> A second crack at a null hypothesis using an
>> independent pseudorandom generator.
>> Results from this generator are more variable than the Mathematica built in
>> as you can change both the seed start number and the irrational it is
>> based on.
>> It too gives a different result than the Pi digits.
>>
>> Mathematica code:
>> Clear[r,s,a,c1,d1]
>> s=5
>> (*Pseudorandom number algorithm from Forcasting on Your
>> Microcomuter,nickell, tab books, 1983*)
>> SeedRandom[123]
>> r[n_Integer]:=r[n]=Mod[(E+r[n-1])^s,1]
>> r[0]=Random[]
>> digits =50000
>> a=Table[Mod[Floor[10*r[n]],10],{n,1,digits}];
>> c1=Drop[FoldList[Plus,0,Sign[Drop[a,1]-Drop[a,-1]]],1];
>> ListPlot[c1,PlotJoined->True];
>> (* Rowe Count*)
>> d1=Flatten@{0,Length/@Split[Sort@c1], 0}
>> ListPlot[d1,PlotJoined->True];
>>
>> Roger Bagula wrote:
>>
>>
>>
>>> This program is real slow on my machine.
>>> Show a lean toward positive differences that is "slight" at 2000 digits.
>>>
>>> Digits=2000
>>> $MaxExtraPrecision = Digits
>>> (* Sum of the sign of the differences between the first 2000 digits of Pi*)
>>> f[m_]=Sum[Sign[Floor[Mod[10^(n+1)*Pi,10]]-Floor[Mod[10^n*Pi,10]]],{n,0,m}]
>>> a=Table[{n,f[n]},{n,0,Digits-1}];
>>> ListPlot[a,PlotJoined->True]
>>> b=Table[a[[n]][[2]],{n,1,Dimensions[a][[1]]}];
>>> (* distribution of the noise that results*)
>>> c=Table[Count[b,m],{m,-12,12}]
>>> ListPlot[c,PlotJoined->True]
>>>
>>> Respectfully, Roger L. Bagula
>>> tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
>>> alternative email: rlbtftn at netscape.net
>>> URL : http://home.earthlink.net/~tftn
>>>
>>>
>>>
>>>
>>>
>>
>>
>>
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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