Re: Re: correlation among random numbers
- To: mathgroup at smc.vnet.net
- Subject: [mg41144] Re: [mg41120] Re: [mg41103] correlation among random numbers
- From: Bobby Treat <drmajorbob+MathGroup3528 at mailblocks.com>
- Date: Mon, 5 May 2003 02:44:06 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
This uses Daniel Lichtblau's suggestion to get rid of the correlation
problem, but it may slow things down.
Unprotect@NormalDistribution;
NormalDistribution /: Random[NormalDistribution[mu_:0, sigma_:1]] := \
normal[mu, sigma, myRandom, myRandom]
Protect@NormalDistribution;
myRandom := ((Random[Integer, 2^30 - 1]/2^30.) + Random[Integer,
2^30 - 1])/2^30.
It's far too tedious to do it this way for every distribution in the
inventory, so fixing Random[] itself seems preferable.
But WRI should do that.
Bobby
-----Original Message-----
From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
To: mathgroup at smc.vnet.net
Subject: [mg41144] [mg41120] Re: [mg41103] correlation among random numbers
Most likely you have encountered the problem discussed (with a
solution) in:
http://forums.wolfram.com/mathgroup/archive/2000/May/msg00088.html
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/
On Saturday, May 3, 2003, at 04:27 pm, Andrea Romanino wrote:
>
> A Mathematica program I wrote that uses a Montecarlo method to
generate
> a probability distribution gives incorrect results.
>
> The problem turns out to be due to an unexpected correlation among
> independent random variables. In its simplest form, the problem is
> illustrated in the notebook you can find attached.
>
> I am using Mathematica 4.2.0 for Linux. The problem also arises with
> the
> version 4.1.
>
> While I can easily get rid of that correlation with "empirical"
> modifications to the program, I am afraid that other correlations, not
> leading to obviously incorrect results, may be hidden in the results.
I
> would therefore be extremely grateful if anyone could help me
> understanding the origin of the problem and how to avoid it.
>
> Thank you very much,
> Andrea Romanino
>
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> Cell[TextData[{
> "Correlation between independent random numbers (",
> StyleBox["Mathematica",
> FontSlant->"Italic"],
> " 4.2.0 for Linux)"
> }], "Section"],
>
> Cell["\<\
> Omitting one of the random number generations between r and x or \
> between x and y destroys the correlation.
> Same if Random[NormalDistribution[0,1]] is replaced by Random[]\
> \>", \
> "Subsection"],
>
> Cell["\<\
> To see that the correlation indeed arises evaluate the cells below
> The correlation arises between the variables r, x and y in the form
> Mod[x-y, \
> 1] - r = 0
> Omitting one of the random number generations between r and x (or
> between x \
> and y) eliminates the correlation
> Idem if one of the \"Random[NormalDistribution[0,1]]\" is replaced by
\
> \"Random[]\" or viceversa\
> \>", "Subsection"],
>
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> General::"\<spell1\>"]\), "\[IndentingNewLine]",
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>
> Cell[BoxData[{
> \(\(nrnd = 10;\)\), "\n",
> \(tab = {};
> Do[\[IndentingNewLine]r = Random[]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0,
> 1]]; \[IndentingNewLine]Random[];
> \[IndentingNewLine]Random[]; \
> \[IndentingNewLine]Random[]; \[IndentingNewLine]x =
> Random[]; \[IndentingNewLine]Random[];
> \[IndentingNewLine]Random[]; \
> \[IndentingNewLine]Random[]; \[IndentingNewLine]y =
> Random[]; \[IndentingNewLine]tab =
> Append[tab, {StringJoin["\<[x-y] = \>", ToString[Mod[x - y,
> 1]]],
> StringJoin["\<r = \>", ToString[r]],
> StringJoin["\<difference = \>",
> ToString[Chop[Mod[x - y, 1] - r]]]}], {w,
> nrnd}];\), "\[IndentingNewLine]",
> \(TableForm[tab]\)}], "Input"],
>
> Cell["A graphical representation of the correlation:", "Subsection"],
>
> Cell[BoxData[{
> \(\(nrnd = 1000;\)\), "\n",
> \(tab = {PointSize[0.02]};
> Do[\[IndentingNewLine]r = Random[]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0,
> 1]]; \[IndentingNewLine]Random[];
> \[IndentingNewLine]Random[]; \
> \[IndentingNewLine]Random[]; \[IndentingNewLine]x =
> Random[]; \[IndentingNewLine]Random[];
> \[IndentingNewLine]Random[]; \
> \[IndentingNewLine]Random[]; \[IndentingNewLine]y =
> Random[]; \[IndentingNewLine]tab =
> Append[tab, Point[{Mod[x - y, 1], r}]], {w,
> nrnd}];\), "\[IndentingNewLine]",
> \(\(Show[Graphics[tab], Frame -> True, AspectRatio -> 1];\)\)}],
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>
> Cell["\<\
> What one should obtain (I have just removed one line from the cell \
> above)\
> \>", "Subsection"],
>
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> \(tab = {PointSize[0.02]};
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> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
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> NormalDistribution[0, 1]]; \[IndentingNewLine]Random[
> NormalDistribution[0,
> 1]]; \[IndentingNewLine]Random[];
> \[IndentingNewLine]Random[]; \
> \[IndentingNewLine]x =
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> \[IndentingNewLine]Random[]; \
> \[IndentingNewLine]Random[]; \[IndentingNewLine]y =
> Random[]; \[IndentingNewLine]tab =
> Append[tab, Point[{Mod[x - y, 1], r}]], {w,
> nrnd}];\), "\[IndentingNewLine]",
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