Re: random numbers according to user defined distribution ??
sturlamolden wrote: Alex wrote: I wonder if it is possible in python to produce random numbers according to a user defined distribution? Unfortunately the random module does not contain the distribution I need :-( There exist some very general algorithms to generate random numbers from arbitrary distributions. The most notable of these are Markov Chain Monte Carlo, e.g. the Metropolis-Hastings algorithm. It is very easy to implement in any programming language. The nature MCMC algorithms makes it inefficient when implemented in pure Python. But you can get tremendous speedup by simulating multiple Markov chains in parallel, by means of vectorizing with NumPy. A relative of Metropolis-Hastings which may also be applicable to your problem is pure rejection sampling. It is far less efficient, but produces no autocorrelation in the samples. I don't know. I think the certainty that rejection sampling actually gives you the desired distribution as opposed to MH's uncertainty is very much a worthwhile tradeoff for univariate distributions. Sure, you throw away fewer samples, but you know that MH isn't throwing away samples that it ought to. Personally, I view MH as a last resort when the dimensionality gets too large to do anything else. But then, that's just my opinion. -- Robert Kern I have come to believe that the whole world is an enigma, a harmless enigma that is made terrible by our own mad attempt to interpret it as though it had an underlying truth. -- Umberto Eco -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
Alex wrote: I wonder if it is possible in python to produce random numbers according to a user defined distribution? Unfortunately the random module does not contain the distribution I need :-( Have you looked at the numpy random number module? It seems to have quite a lot of distributions. See help(numpy.random). Jeremy -- Jeremy Sanders http://www.jeremysanders.net/ -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
On Aug 6, 3:02 pm, Alex [EMAIL PROTECTED] wrote: I wonder if it is possible in python to produce random numbers according to a user defined distribution? Unfortunately the random module does not contain the distribution I need :-( Sure there's a way but it won't be very efficient. Starting with an arbitrary probability density function over some range, you can run it through a quadrature routine to create a cumulative density function over that range. Use random.random() to create a uniform variate x. Then use a bisecting search to find x in the cumulative density function over the given range. from __future__ import division from random import random def integrate(f, lo, hi, steps=1000): dx = (hi - lo) / steps lo += dx / 2 return sum(f(i*dx + lo) * dx for i in range(steps)) def make_cdf(f, lo, hi, steps=1000): total_area = integrate(f, lo, hi, steps) def cdf(x): assert lo = x = hi return integrate(f, lo, x, steps) / total_area return cdf def bisect(target, f, lo, hi, n=20): 'Find x between lo and hi where f(x)=target' for i in range(n): mid = (hi + lo) / 2.0 if target f(mid): hi = mid else: lo = mid return (hi + lo) / 2.0 def make_user_distribution(f, lo, hi, steps=1000, n=20): cdf = make_cdf(f, lo, hi, steps) return lambda: bisect(random(), cdf, lo, hi, n) if __name__ == '__main__': def linear(x): return 3 * x - 6 lo, hi = 2, 10 r = make_user_distribution(linear, lo, hi) for i in range(20): print r() -- Raymond -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
Thanks for the many answers. So basically I have to get the inverse of the CDF and use this to transform my uniformly distributed random numbers. If my desired distribution is simple I can get an analytical solution for the inverse, otherwise I have to use numerical methods. Okay, things are now much clearer. Many thanks, Alex -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
Alex wrote: Thanks for the many answers. So basically I have to get the inverse of the CDF and use this to transform my uniformly distributed random numbers. If my desired distribution is simple I can get an analytical solution for the inverse, otherwise I have to use numerical methods. Okay, things are now much clearer. It's worth noting that unless if the PPF (the inverse of the CDF) is very straightforward, this method is not very good. The numerical errors involved cause very poor results in the tails of many distributions. Typically, rejection sampling, if done well, will work much better. There are techniques for doing this on nearly-arbitrary distributions: http://statmath.wu-wien.ac.at/projects/arvag/index.html If you implement any of these techniques in Python, I would love to see them. -- Robert Kern I have come to believe that the whole world is an enigma, a harmless enigma that is made terrible by our own mad attempt to interpret it as though it had an underlying truth. -- Umberto Eco -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
Alex wrote: I wonder if it is possible in python to produce random numbers according to a user defined distribution? Unfortunately the random module does not contain the distribution I need :-( There exist some very general algorithms to generate random numbers from arbitrary distributions. The most notable of these are Markov Chain Monte Carlo, e.g. the Metropolis-Hastings algorithm. It is very easy to implement in any programming language. The nature MCMC algorithms makes it inefficient when implemented in pure Python. But you can get tremendous speedup by simulating multiple Markov chains in parallel, by means of vectorizing with NumPy. A relative of Metropolis-Hastings which may also be applicable to your problem is pure rejection sampling. It is far less efficient, but produces no autocorrelation in the samples. If you can generate random deviates from the marginal distributions, but need to sample the joint distribution, look into using the Gibbs sampler. It is an efficient version of Metropolis-Hastings for this special case. http://en.wikipedia.org/wiki/Metropolis-Hastings_algorithm http://en.wikipedia.org/wiki/Rejection_sampling http://en.wikipedia.org/wiki/Gibbs_sampling -- http://mail.python.org/mailman/listinfo/python-list
random numbers according to user defined distribution ??
Hi everybody, I wonder if it is possible in python to produce random numbers according to a user defined distribution? Unfortunately the random module does not contain the distribution I need :-( Many thanks axel -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
On Thursday 07 August 2008 00:02, Alex [EMAIL PROTECTED] wrote: Hi everybody, I wonder if it is possible in python to produce random numbers according to a user defined distribution? Unfortunately the random module does not contain the distribution I need :-( Many thanks axel I'm not aware of any module with that specific function, but it's algorithmically not too complex I'd think. If you're writing an application that does this I'll assume that you have a basic gist of how to implement it ;). It's been a while since I messed with the subject, so I'm not getting any further than graphs in my head right now. Good luck! -- Dominic van Berkel Bi-la Kaifa -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
On Wed, 06 Aug 2008 15:02:37 -0700, Alex wrote: Hi everybody, I wonder if it is possible in python to produce random numbers according to a user defined distribution? Unfortunately the random module does not contain the distribution I need :-( This is a strange question. Of course you can -- just write a function to do so! Here's some easy ones to get you started: from __future__ import division import random, maths def unbounded_rand(p=0.5): Return a random integer between 0 and infinity. if not (0 p = 1): raise ValueError n = 0 while random.random() p: n += 1 return n def pseudonorm(): Return a random float with a pseudo-normal distribution. The probability distribution is centered at 0 and bounded by -1 and +1. return (sum([random.random() for i in range(6)])-3)/3 def triangular(min=0, max=1, mode=0.5): Return a random float in the range (min, max) inclusive with a triangular histogram, and the peak at mode. u = random.random() if u = (mode-min)/(max-min): return min + math.sqrt(u*(max-min)*(mode-min)) else: return max - math.sqrt((1-u)*(max-min)*(max-mode)) def linear(): Return a random float with probability density function pdf(x)=2x. return math.sqrt(random.random()) There's no general way to create a random function for an arbitrary distribution. I don't think there's a general way to *describe* an arbitrary random distribution. However, there are some mathematical techniques you can use to generate many different distributions. Google on transformation method and rejection method. If you have a specific distribution you are interested in, and you need some help, please ask. -- Steven -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
On Aug 6, 8:26 pm, Steven D'Aprano [EMAIL PROTECTED] cybersource.com.au wrote: On Wed, 06 Aug 2008 15:02:37 -0700, Alex wrote: Hi everybody, I wonder if it is possible in python to produce random numbers according to a user defined distribution? Unfortunately the random module does not contain the distribution I need :-( This is a strange question. Of course you can -- just write a function to do so! Here's some easy ones to get you started: ... There's no general way to create a random function for an arbitrary distribution. I don't think there's a general way to *describe* an arbitrary random distribution. What about the quantile function? -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
Alex [EMAIL PROTECTED] writes: I wonder if it is possible in python to produce random numbers according to a user defined distribution? That can mean a lot of things. The book Numerical Recipes (there are editions for various languages, unfortunately not including Python last time I looked) has some discussion about how to do it. -- http://mail.python.org/mailman/listinfo/python-list
Re: random numbers according to user defined distribution ??
On Wed, 06 Aug 2008 21:09:30 -0700, Dan Bishop wrote: There's no general way to create a random function for an arbitrary distribution. I don't think there's a general way to *describe* an arbitrary random distribution. What about the quantile function? Well, sure, if you can write down the quantile function, c.d.f or p.d.f. of a distribution, I suppose that counts as describing it, in some sense. But even if we limit ourselves to distributions which are actually useful, as opposed to arbitrary distributions that can't be described in terms of any known mathematical function, there are serious practical difficulties. I quote from the Wikipedia article on quantile functions: The quantile functions of even the common distributions are relatively poorly understood beyond the use of simple lookup tables, which is at odds with their importance in Monte Carlo sampling, where a sample from a given distribution may be obtained in principle by applying its quantile function to a sample from a uniform distribution. The exponential case above is one of the very few distributions where there is a simple formula. http://en.wikipedia.org/wiki/Quantile_function -- Steven -- http://mail.python.org/mailman/listinfo/python-list
