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如何在 tikz/pgf 中绘制布朗运动

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如何在 tikz/pgf 中绘制布朗运动

这里我想画一些布朗运动tikz，像这样： 在此处输入图片描述 此外，我想截断布朗运动的轨迹，像这样： 在此处输入图片描述

我曾多次尝试使用 tikz 中的随机函数，但总是失败。

顺便说一句，上传的图表是 Peter Mörters 和 Yuval Peres 撰写的“布朗运动 - 2008 年 5 月 25 日草稿版”的截图。

答案1

这个怎么样？它是伪随机的，但你可以通过设置使其可重复\pgfmathsetseed{integer}：

\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{tikz}

\begin{document}

\newcommand{\Emmett}[5]{% points, advance, rand factor, options, end label
\draw[#4] (0,0)
\foreach \x in {1,...,#1}
{   -- ++(#2,rand*#3)
}
node[right] {#5};
}

\begin{tikzpicture}
\draw[help lines] (0,-5) grid (15,5);
\Emmett{750}{0.02}{0.2}{red}{first one}
\Emmett{750}{0.02}{0.2}{green}{second one}
\Emmett{750}{0.02}{0.2}{blue}{third one}
\end{tikzpicture}

%\pgfmathsetseed{1337}

\end{document}

在此处输入图片描述

编辑1：截断也是可行的：

\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{tikz}

\begin{document}

\newcommand{\Emmett}[5]{% points, advance, rand factor, options, end label
\draw[#4] (0,0)
\foreach \x in {1,...,#1}
{   -- ++(#2,rand*#3)
}
node[right] {#5};
}

\newcommand{\Lathrop}[6]{% points, advance, rand factor, options, end label, truncate from point
\draw[#4] (0,0)
\foreach \x in {1,...,#6}
{   -- ++(#2,rand*#3)
}
coordinate (tempcoord) {};
\pgfmathsetmacro{\remaininglength}{(#1-#6)*#2}
\draw[#4] (tempcoord) -- ++ (\remaininglength,0) node[right] {#5};
}

\begin{tikzpicture}
\draw[help lines] (0,-5) grid (15,5);
\Lathrop{750}{0.02}{0.23}{red!70!black}{first one}{300}
\Lathrop{750}{0.02}{0.18}{green!70!black,thick}{second one}{400}
\Lathrop{750}{0.02}{0.21}{blue!70!black}{third one}{500}
\end{tikzpicture}

\end{document}

在此处输入图片描述

编辑2：啊，现在我得到了截断请求：现在您可以指定上限和下限并为它们绘制直线：

\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{tikz}
\usepackage{xifthen}

\begin{document}

\newcommand{\Emmett}[5]{% points, advance, rand factor, options, end label
\draw[#4] (0,0)
\foreach \x in {1,...,#1}
{   -- ++(#2,rand*#3)
}
node[right] {#5};
}

\newcommand{\Lathrop}[9]{% points, advance, rand factor, options, end label, upper, lower trunc, draw trunc  lines, trunc draw options
\begin{scope}
\pgfmathsetmacro{\picwidth}{#1*#2}
\clip (0,#6*28.453+0.5\pgflinewidth) rectangle (\picwidth,#7*28.453-0.5\pgflinewidth);
\ifthenelse{\equal{#8}{y}}
    {\draw[#9] (0,#6) -- (\picwidth,#6) (0,#7) -- (\picwidth,#7);}
    {}
\draw[#4] (0,0)
\foreach \x in {1,...,#1}
{   -- ++(#2,rand*#3)
}
coordinate (#5) ;
\end{scope}
\node[right,#4] at (#5) {#5};
}

\begin{tikzpicture}
\draw[help lines] (0,-5) grid (15,5);
\Lathrop{750}{0.02}{0.2}{red!70!black}{first one}{1.5}{-2.3}{n}{}
\Lathrop{750}{0.02}{0.2}{green!70!black,thick}{second one}{1.1}{-1.7}{y}{green!70!black,densely dashed}
\Lathrop{750}{0.02}{0.3}{blue!70!black}{third one}{2.4}{-2.7}{y}{blue!70!black,thin,densely dotted}
\end{tikzpicture}

\end{document}

在此处输入图片描述

PS 标签的放置还存在一些问题。该命令现在有 9 个参数，应切换到pgfkeys方便的键值接口。

答案2

这里有一个方法，用来pgfplotstable计算布朗运动作为随机正态分布值的累积和（感谢 horchler指出正常化的必要性）。您必须先使用类似 之类的东西初始化一个空表\pgfplotstablenew{200}\loadedtable，然后才能使用 绘制布朗运动\addplot table [brownian motion] {\loadedtable};。

您可以使用设置初始值以及最大值和最小值

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


\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}

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