Graphics, Figures & Tables ⇒ Branching Brownian Motion

[dr_kelly]

I would like to plot a branching brownian motion but i couldn't find any function for brownian paths. Is it possible to plot it using TikZ?

[Stefan Kottwitz]

Welcome to the forum!

Yes, it would be easily possible with TikZ, or PSTricks, or calculations done with Lua. I would use pgfplots, since it gives us a coordinate system with axes and many options for customizing.

Jake gave an example here:

Code: Select all

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots, pgfplotstable}


% Create a function for generating inverse normally distributed numbers using the Box–Muller transform
\pgfmathdeclarefunction{invgauss}{2}{%
  \pgfmathparse{sqrt(-2*ln(#1))*cos(deg(2*pi*#2))}%
}
% Code for brownian motion
\makeatletter
\pgfplotsset{
    table/.cd,
    brownian motion/.style={
        create on use/brown/.style={
            create col/expr accum={
                (\coordindex>0)*(
                    max(
                        min(
                            invgauss(rnd,rnd)*0.1+\pgfmathaccuma,
                            \pgfplots@brownian@max
                        ),
                        \pgfplots@brownian@min
                    )
                ) + (\coordindex<1)*\pgfplots@brownian@start
            }{\pgfplots@brownian@start}
        },
        y=brown, x expr={\coordindex},
        brownian motion/.cd,
        #1,
        /.cd
    },
    brownian motion/.cd,
            min/.store in=\pgfplots@brownian@min,
        min=-inf,
            max/.store in=\pgfplots@brownian@max,
            max=inf,
            start/.store in=\pgfplots@brownian@start,
        start=0
}
\makeatother
%

% Initialise an empty table with a certain number of rows
\pgfplotstablenew{201}\loadedtable % How many steps?



\begin{document}
\pgfplotsset{
        no markers,
        xmin=0,
        enlarge x limits=false,
        scaled y ticks=false,
        ymin=-1, ymax=1
}
\tikzset{line join=bevel}
\pgfmathsetseed{3}
\begin{tikzpicture}
\begin{axis}
   \addplot table [brownian motion] {\loadedtable};
   \addplot table [brownian motion] {\loadedtable};
\end{axis}
\end{tikzpicture}

\pgfmathsetseed{3}
\begin{tikzpicture}
\begin{axis}
    \addplot table [
        brownian motion={%
            max=0.5,
            min=-0.75
        }
    ] {\loadedtable};
    \addplot table [
        brownian motion={%
            start=0.5,
            min=-0.5, max=0.75
        }
    ] {\loadedtable};
\end{axis}
\end{tikzpicture}
\end{document} 

brownian-motion.png (23.08 KiB) Viewed 2072 times

Jake explained it:

This approach uses pgfplotstable to calculate the Brownian motions as cumulative sums of random normally distributed values (thanks to horchler for pointing out the need for normality). You have to first initialise an empty table, using something like \pgfplotstablenew{200}\loadedtable, and then you can draw the brownian motions using \addplot table [brownian motion] {\loadedtable};.

You can set the initial value and the maximum and minimum values using

Code: Select all

\addplot table [brownian motion={start=0.5, min=-1, max=1}] {\loadedtable};

Stefan