ph@miro.Berkeley.EDU (Paul Heckbert) (10/30/90)
I'm looking for good references on techniques for generating random numbers with arbitrary density functions. For example, to generate a gaussian random number from uniform random numbers: r = sqrt(-2*log(U)); t = 2*pi*U; x = r*sin(t); y = r*cos(t); where U is a random number between 0 and 1, different every time x and y will then be gaussian-distributed between -inf and inf. Similarly, you can generate a point uniformly distributed inside a circle with: r = radius*sqrt(U); theta = 2*pi*U; I assume the method is well known in probability and statistics. What are good references for such techniques of distribution shaping? Paul Heckbert, Computer Science Dept. 570 Evans Hall, UC Berkeley INTERNET: ph@miro.berkeley.edu Berkeley, CA 94720 UUCP: ucbvax!miro.berkeley.edu!ph
cedman@lynx.ps.uci.edu (Carl Edman) (10/30/90)
In article <29358@pasteur.Berkeley.EDU> ph@miro.Berkeley.EDU (Paul Heckbert) writes:
I'm looking for good references on techniques for generating random numbers
with arbitrary density functions. For example, to generate a gaussian
random number from uniform random numbers:
r = sqrt(-2*log(U));
t = 2*pi*U;
x = r*sin(t);
y = r*cos(t);
where U is a random number between 0 and 1, different every time
x and y will then be gaussian-distributed between -inf and inf.
Similarly, you can generate a point uniformly distributed inside a circle with:
r = radius*sqrt(U);
theta = 2*pi*U;
I assume the method is well known in probability and statistics.
What are good references for such techniques of distribution shaping?
There is one general method of getting random variables given an arbitrary
distribution. It is known under the handy name of Metropolis-Rosenbluth-
-Rosenbluth-Teller-Teller-algorithm.
As such random numbers are very important for the numerical integration
of high-dimensional integrals (High meaning something of the order of 10^23
dimensions) using Monte-Carlo methods, this algorithm should be described
in any text-book on numerical methods. I am having "Computational Physics"
by Steven E.Koonin here at hand. The algorithm is quite simple, really,
so if there is demand I can describe it to the net.
Carl Edman
Theorectial Physicist,N.:A physicist whose | Send mail
existence is postulated, to make the numbers | to
balance but who is never actually observed | cedman@golem.ps.uci.edu
in the laboratory. | edmanc@uciph0.ps.uci.eduaipdc@castle.ed.ac.uk (Paul D. Crowley) (10/30/90)
cedman@lynx.ps.uci.edu (Carl Edman) writes: >There is one general method of getting random variables given an arbitrary >distribution. It is known under the handy name of Metropolis-Rosenbluth- >-Rosenbluth-Teller-Teller-algorithm. Is that for real? Excuse me while I crosspost to rec.humor... anyone know how it got such a wonderful name? >The algorithm is quite simple, really, With a name like that, at least it could have the common decency to be fiendishly complicated... -- \/ o\ Paul Crowley aipdc@uk.ac.ed.castle /\__/ "Trust me, I know what I'm doing" - Sledge Hammer
mroussel@alchemy.chem.utoronto.ca (Marc Roussel) (10/31/90)
In article <6936@castle.ed.ac.uk> aipdc@castle.ed.ac.uk (Paul D. Crowley) writes >cedman@lynx.ps.uci.edu (Carl Edman) writes: >>There is one general method of getting random variables given an arbitrary >>distribution. It is known under the handy name of Metropolis-Rosenbluth- >>-Rosenbluth-Teller-Teller-algorithm. > >Is that for real? Excuse me while I crosspost to rec.humor... anyone >know how it got such a wonderful name? At least in the Chemistry community, the technique is usually simply known as Metropolis sampling. I have heard it called rejection sampling in other places. The idiosyncratic MRRTT name which Carl gave it comes from the list of authors on the paper in which the method was first described. (I can post the reference if anyone is interested.) Apparently (so the story goes), Metropolis, the Rosenbluths and the Tellers developed the method on a paper napkin at a dinner party. Marc R. Roussel mroussel@alchemy.chem.utoronto.ca
brnstnd@kramden.acf.nyu.edu (Dan Bernstein) (10/31/90)
In article <29358@pasteur.Berkeley.EDU> ph@miro.Berkeley.EDU (Paul Heckbert) writes: > I'm looking for good references on techniques for generating random numbers > with arbitrary density functions. For example, to generate a gaussian > random number from uniform random numbers: [ etc. ] Knuth's Art of Computer Programming, in the chapter on random numbers. Looking everything up in Knuth is an excellent habit for programmers to acquire. Often you don't have to look any further. And each book in the series is probably one of the best mathematics texts and references ever published: it's amazing how graduate students spend entire courses on things that Knuth covers in several exercises. ---Dan
mccool@dgp.toronto.edu (Michael McCool) (11/01/90)
>There is one general method of getting random variables given an arbitrary >distribution. It is known under the handy name of Metropolis-Rosenbluth- >-Rosenbluth-Teller-Teller-algorithm. >As such random numbers are very important for the numerical integration >of high-dimensional integrals (High meaning something of the order of 10^23 >dimensions) using Monte-Carlo methods, this algorithm should be described >in any text-book on numerical methods. I am having "Computational Physics" >by Steven E.Koonin here at hand. The algorithm is quite simple, really, >so if there is demand I can describe it to the net. > Carl Edman And this is why the patent by Pixar for stochastic sampling is so bogus: it is patenting theory that is already well-known and in the public domain. You can't patent EVERY APPLICATION of the same method. Michael McCool@dgp.toronto.edu
jcburt@ipsun.larc.nasa.gov (John Burton) (11/01/90)
In article <1990Oct31.112427.694@jarvis.csri.toronto.edu> mccool@dgp.toronto.edu (Michael McCool) writes: >>There is one general method of getting random variables given an arbitrary ^^^ Just ONE method??? >>distribution. It is known under the handy name of Metropolis-Rosenbluth- >>-Rosenbluth-Teller-Teller-algorithm. >>As such random numbers are very important for the numerical integration >>of high-dimensional integrals (High meaning something of the order of 10^23 >>dimensions) using Monte-Carlo methods, this algorithm should be described >>in any text-book on numerical methods. I am having "Computational Physics" ^^^ Perhaps you should read the article by Park & Miller "Random Number Generators: Good Ones are Hard to Find", CACM 31/10 (1988) pp1192-1201. Many results obtained from "common" random number generators are questionable primarily due to the lack of randomness of the generator. >>by Steven E.Koonin here at hand. The algorithm is quite simple, really, >>so if there is demand I can describe it to the net. > >> Carl Edman > >And this is why the patent by Pixar for stochastic sampling is so bogus: >it is patenting theory that is already well-known and in the public domain. >You can't patent EVERY APPLICATION of the same method. > Agreed! >Michael McCool@dgp.toronto.edu If you are interested in learning something about generating random numbers, a good introduction is the Park & Miller article. Another good article on RNGs is "Random Numbers for Simulation", by Pierre L'Ecuyer, CACM 33/10, October 1990, pp 85-97 John Burton (jcburt@cs.wm.edu) (jcburt@ipsun.larc.nasa.gov)
cedman@lynx.ps.uci.edu (Carl Edman) (11/01/90)
In article <1990Oct31.170640.15585@abcfd20.larc.nasa.gov> jcburt@ipsun.larc.nasa.gov (John Burton) writes: In article <1990Oct31.112427.694@jarvis.csri.toronto.edu> mccool@dgp.toronto.edu (Michael McCool) writes: >>There is one general method of getting random variables given an arbitrary ^^^ Just ONE method??? >>distribution. It is known under the handy name of Metropolis-Rosenbluth- >>-Rosenbluth-Teller-Teller-algorithm. >>As such random numbers are very important for the numerical integration >>of high-dimensional integrals (High meaning something of the order of 10^23 >>dimensions) using Monte-Carlo methods, this algorithm should be described >>in any text-book on numerical methods. I am having "Computational Physics" ^^^ Perhaps you should read the article by Park & Miller "Random Number Generators: Good Ones are Hard to Find", CACM 31/10 (1988) pp1192-1201. Many results obtained from "common" random number generators are questionable primarily due to the lack of randomness of the generator. I am sorry if I expressed myself ambigously (what I meant was clear from the original question , but this context has in usual usenet-manner been removed). Of course, I and everybody else is aware that there are dozens of different random number generators , each with different strengths and weaknesses, and many with interesting weaknesses (yes, we can share anecdotes on the topic). But that was not the originial question. The question in the article was (roughly): Given a uniform random number generator and a desired random number distribution, is there are general algorithm to create random numbers with this distribution ? I replied to this: yes, there is and gave the most famous one. OK ? Carl Edman PS: yes, i know that some lazy people refer to the Metropolis-Rosenbluth- -Rosenbluth-Teller-Teller algorithm , merely as Metropolis algorithm :-). Theorectical Physicist,N.:A physicist whose | Send mail existence is postulated, to make the numbers | to balance but who is never actually observed | cedman@golem.ps.uci.edu in the laboratory. | edmanc@uciph0.ps.uci.edu
dtate@unix.cis.pitt.edu (David M Tate) (11/02/90)
In article <29358@pasteur.Berkeley.EDU> ph@miro.Berkeley.EDU (Paul Heckbert) writes: >I'm looking for good references on techniques for generating random numbers >with arbitrary density functions. > >What are good references for such techniques of distribution shaping? The ultimate reference for this stuff is "Nonuniform Random Number Generation" by Luc Devroye, from Springer-Verlag. If it's been done, and isn't extremely recent, it's in there. Knuth has some excellent stuff on this topic, but most of the important parts are in Devroye anyway. -- David M. Tate | What the hell does "30% chance of rain" mean? dtate@unix.cis.pitt.edu | What do I do, carry a third of an umbrella? "A Man for all Seasonings" | --Lawrence Block
shirley@iuvax.cs.indiana.edu (peter shirley) (11/03/90)
ph@miro.Berkeley.EDU (Paul Heckbert) writes: >I'm looking for good references on techniques for generating random numbers >with arbitrary density functions. For example, to generate a gaussian >random number from uniform random numbers: > r = sqrt(-2*log(U)); > t = 2*pi*U; > x = r*sin(t); > y = r*cos(t); > where U is a random number between 0 and 1, different every time >x and y will then be gaussian-distributed between -inf and inf. >Similarly, you can generate a point uniformly distributed inside a circle with: > r = radius*sqrt(U); > theta = 2*pi*U; >I assume the method is well known in probability and statistics. >What are good references for such techniques of distribution shaping? >Paul Heckbert, Computer Science Dept. >570 Evans Hall, UC Berkeley INTERNET: ph@miro.berkeley.edu >Berkeley, CA 94720 UUCP: ucbvax!miro.berkeley.edu!ph Here is an appendix (in postscript) from my thesis that may or may not be useful. It is on Monte Carlo methods, And the last section addresses Paul's question. If you can't print it, I can send you a dvi file. pete shirley shirley@cs.indiana.edu Save file in file "whatever" and then, after leaving the news reader, type "uudecode whatever" and a file called "mc.ps" should appear. begin 644 mc.ps M)2%04RU!9&]B92TR+C`*)25#<F5A=&]R.B!D=FEP<R!B>2!2861I8V%L($5Y M92!3;V9T=V%R90HE)5!A9V5S.B`X(#$*)25";W5N9&EN9T)O>#H@,"`P(#8Q M,B`W.3(*)25%;F1#;VUM96YT<PHE)4)E9VEN1&]C=6UE;G0Z('1E>"YP<F\* M+U1E6$1I8W0@,C`P(&1I8W0@9&5F(%1E6$1I8W0@8F5G:6X@+V)D9GMB:6YD M(&1E9GUD968@+TEN8VA[4F5S;VQU=&EO;B!M=6Q]"F)D9B`O1&]T<WLW,B!D M:78@4F5S;VQU=&EO;B!M=6Q]8F1F("]D;W!A9V5[-S(@4F5S;VQU=&EO;B!D M:78@9'5P(&YE9R!S8V%L90IT<F%N<VQA=&5]8F1F("]`;&5T=&5R>U)E<V]L M=71I;VX@9'5P("TQ,"!M=6P@9&]P86=E?6)D9B`O0&YO=&5[0&QE='1E<GUB M9&8*+T!A-'M297-O;'5T:6]N(&1U<"`M,3`N-CDR.3$S,S@U."!M=6P@9&]P M86=E?6)D9B`O0'1R86YS;&%T97MT<F%N<VQA=&5]8F1F"B]`<V-A;&5[<V-A M;&5]8F1F("]`<F]T871E>W)O=&%T97UB9&8@+T!L86YD<V-A<&5[6S`@,2`M M,2`P(#`@,%UC;VYC870*4F5S;VQU=&EO;B!D=7`@9&]P86=E?6)D9B`O0&QE 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