Documentation

# NAME

simulation::montecarlo - Monte Carlo simulations

# SYNOPSIS

package require Tcl ?8.5 9?
package require simulation::montecarlo 0.1
package require simulation::random
package require math::statistics

# 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 singleExperiment command.

• string keyword

• ::simulation::montecarlo::hasOption keyword

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

• string keyword

• ::simulation::montecarlo::setOption keyword value

Set the value of the given option.

• string keyword

• 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.

Mathematics