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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)
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  • 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



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