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# simulation::montecarlo(n) 0.1 tcllib "Tcl Simulation Tools"

## Name

simulation::montecarlo - Monte Carlo simulations

## Synopsis

- package require Tcl ?8.4?
- package require simulation::montecarlo 0.1
- package require simulation::random
- package require math::statistics

- ::simulation::montecarlo::getOption keyword
- ::simulation::montecarlo::hasOption keyword
- ::simulation::montecarlo::setOption keyword value
- ::simulation::montecarlo::setTrialResult values
- ::simulation::montecarlo::setExpResult values
- ::simulation::montecarlo::getTrialResults
- ::simulation::montecarlo::getExpResult
- ::simulation::montecarlo::transposeData values
- ::simulation::montecarlo::integral2D ...
- ::simulation::montecarlo::singleExperiment args

## Description

The technique of Monte Carlo simulations is basically simple:

- generate random values for one or more parameters.

- evaluate the model of some system you are interested in and record the interesting results for each realisation of these parameters.

- after a suitable number of such trials, deduce an overall characteristic of the model.

You can think of a model of a network of computers, an ecosystem of some kind or in fact anything that can be quantitatively described and has some stochastic element in it.

The package simulation::montecarlo offers a basic framework for such a modelling technique:

    #
    # MC experiments:
    # Determine the mean and median of a set of points and compare them
    #
    ::simulation::montecarlo::singleExperiment -init {
        package require math::statistics
        set prng [::simulation::random::prng_Normal 0.0 1.0]
    } -loop {
        set numbers {}
        for { set i 0 } { $i < [getOption samples] } { incr i } {
            lappend numbers [$prng]
        }
        set mean   [::math::statistics::mean $numbers]
        set median [::math::statistics::median $numbers] ;# ? Exists?
        setTrialResult [list $mean $median]
    } -final {
        set result [getTrialResults]
        set means   {}
        set medians {}
        foreach r $result {
            foreach {m M} $r break
            lappend means   $m
            lappend medians $M
        }
        puts [getOption reportfile] "Correlation: [::math::statistics::corr $means $medians]"
    } -trials 100 -samples 10 -verbose 1 -columns {Mean Median}

This example attemps to find out how well the median value and the mean value of a random set of numbers correlate. Sometimes a median value is a more robust characteristic than a mean value - especially if you have a statistical distribution with "fat" tails.

## PROCEDURES

The package defines the following auxiliary procedures:

::simulation::montecarlo::getOption keyword

  Get the value of an option given as part of the singeExperiment command.

  string keyword

    Given keyword (without leading minus)

::simulation::montecarlo::hasOption keyword

  Returns 1 if the option is available, 0 if not.

  string keyword

    Given keyword (without leading minus)

::simulation::montecarlo::setOption keyword value

  Set the value of the given option.

  string keyword

    Given keyword (without leading minus)

  string value

    (New) value for the option

::simulation::montecarlo::setTrialResult values

  Store the results of the trial for later analysis

  list values

    List of values to be stored

::simulation::montecarlo::setExpResult values

  Set the results of the entire experiment (typically used in the final phase).

  list values

    List of values to be stored

::simulation::montecarlo::getTrialResults

  Get the results of all individual trials for analysis (typically used in the final phase or after completion of the command).

::simulation::montecarlo::getExpResult

  Get the results of the entire experiment (typically used in the final phase or even after completion of the singleExperiment command).

::simulation::montecarlo::transposeData values

  Interchange columns and rows of a list of lists and return the result.

  list values

    List of lists of values

There are two main procedures: integral2D and singleExperiment.

::simulation::montecarlo::integral2D ...

  Integrate a function over a two-dimensional region using a Monte Carlo approach.

  Arguments PM

::simulation::montecarlo::singleExperiment args

  Iterate code over a number of trials and store the results. The iteration is gouverned by parameters given via a list of keyword-value pairs.

  int n

    List of keyword-value pairs, all of which are available during the execution via the getOption command.

The singleExperiment command predefines the following options:

- -init code: code to be run at start up

- -loop body: body of code that defines the computation to be run time and again. The code should use setTrialResult to store the results of each trial (typically a list of numbers, but the interpretation is up to the implementation). Note: Required keyword.

- -final code: code to be run at the end

- -trials n: number of trials in the experiment (required)

- -reportfile file: opened file to send the output to (default: stdout)

- -verbose: write the intermediate results (1) or not (0) (default: 0)

- -analysis proc: either "none" (no automatic analysis), standard (basic statistics of the trial results and a correlation matrix) or the name of a procedure that will take care of the analysis.

- -columns list: list of column names, useful for verbose output and the analysis

Any other options can be used via the getOption procedure in the body.

## TIPS

The procedure singleExperiment works by constructing a temporary procedure that does the actual work. It loops for the given number of trials.

As it constructs a temporary procedure, local variables defined at the start continue to exist in the loop.

## Keywords

math, montecarlo simulation, stochastic modelling

## Category

Mathematics

## Copyright

Copyright © 2008 Arjen Markus <arjenmarkus@users.sourceforge.net>