       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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• Delivered-to: l-mathgroup@wolfram.com
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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[[i]]], {i, 1, 50}]]
>
> Out= {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

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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