Re: improving speed of simulation using random numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg128677] Re: improving speed of simulation using random numbers*From*: Roland Franzius <roland.franzius at uos.de>*Date*: Fri, 16 Nov 2012 01:51:35 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <k82b4l$7kt$1@smc.vnet.net>

Am 15.11.2012 10:03, schrieb felipe.benguria at gmail.com: > Dear all, > > I am trying to compute an expected value using simulation. > I have a random number x with density function d[x]. I want to compute the expected value of function f[x], which is equal to the integral of f[x] times > d[x] over x. > In my case, it is difficult to compute the integral so I simulate N values for x and compute the average of f[x] over all N simulated values. > > My problem is that my code takes to long for my purposes: this is a part of a larger program and is making it unfeasible in terms of time. > > The following code provides an example of the situation, and my question is how could I reduce the time this takes. THanks a lot for your help > > g[x_]:= x^2 > > > mydensity[myparameter_]:= ProbabilityDistribution[myparameter*(t)^(-myparameter - 1), {t, 1, Infinity}] > > randomnum[myparameter_] := RandomVariate[draw[myparameter], 50] > > > Timing[Sum[g[randomnum[5][[i]]], {i, 1, 50}]] > > Out[1353]= {0.64, 81.7808} > > This takes 0.6 seconds in my computer and that is way too long for my full program ( I do this many times). As always for distributions with lengthy algebraic descriptions: Get a numerical exact expression in the real variable x for the distribution function of the random variable named "X" In: CDF["X"][x_] (* = Prob[ "X" < x ] *) := expression Make a table with coordinates exchanged In: TableCDF["X"]=Table[ {CDF["X"][x], x}, {x,xmin,xmax,dx}] If equally distributed points don't fit properly, distribute the interpolations points on the x-axis properly until by using a suitable monotone function of x instead of x itself, until you see a sufficiently accurate fitting by Plotting both graphs. Interpolate the table data as an approximation of the inverse of the distribution function CDF["X"]^-1 : {0,1} -> domainOfValues["X"] In: InverseCDF["X"] = Interpolation[TableCDF["X"]] Now you can use the standard definition of a random generator for CDF[X], namely the map given by CDF[X]^-1 of equally distributed reals in {0,1} draw[]:=lastdraw=InverseCDF["X"][RandomReal[{0,1}]] The variable lastdraw holds the last random choice executed in case you need the value of the last pick again immediately. The random generator for samples of n now has the form draw[n_]:=InverseCDF["X"]/@ RandomReal[{0,1},n] In my experience these kind of random generators work at no time nearly. I have not checked if part or all of this trivial procedure is present in Mathematica v. 8 already. Have a look at http://reference.wolfram.com/mathematica/tutorial/ContinuousDistributions.html http://reference.wolfram.com/mathematica/ref/RandomVariate.html?q=RandomVariate&lang=en Most commonly used random number generators for named distributions are incorporated as RandomVariates. -- Roland Franzius