☰

如何使用 TikZ/PGF 制作异方差图？

tag-icon • tag-icon tikz-pgf
如何使用 TikZ/PGF 制作异方差图？

该图是线性回归模型中异方差问题的典型表示。如果有人能指导我如何使用 tikz ，我将不胜感激。

问候

答案1

以下是使用 PGFPlots 的一种方法，基于在以下位置找到的 2D 版本使用 pgfplot 绘制人口回归函数

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}

\begin{document}

\begin{tikzpicture}[ % Define Normal Probability Function
declare function={
            normal(\x,\m,\s) = 1/(2*\s*sqrt(pi))*exp(-(\x-\m)^2/(2*\s^2));
        }
       ]
\begin{axis}[
    no markers,
    domain=0:12,
    zmin=0, zmax=1,
    xmin=0, xmax=3,
    samples=200,
   samples y=0,
    axis lines=middle,
    xtick={0.5,1.5,2.5},
    xmajorgrids,
    xticklabels={},
    ytick=\empty,
    xticklabels={$x_1$, $x_2$, $x_3$},
    ztick=\empty,
    xlabel=$x$, xlabel style={at={(rel axis cs:1,0,0)}, anchor=west},
    ylabel=$y$, ylabel style={at={(rel axis cs:0,1,0)}, anchor=south west},
    zlabel=Probability density, zlabel style={at={(rel axis cs:0,0,0.5)}, rotate=90, anchor=south},
    set layers
  ]


\addplot3 [samples=2, samples y=0, domain=0:3] (x, {1.5*(x-0.5)+3}, 0);
\addplot3 [cyan!50!black, thick] (0.5, x, {normal(x, 3, 0.5)});
\addplot3 [cyan!50!black, thick] (1.5, x, {normal(x, 4.5, 1)});
\addplot3 [cyan!50!black, thick] (2.5, x, {normal(x, 6, 1.5)});

\pgfplotsextra{
\begin{pgfonlayer}{axis background}
\draw [on layer=axis background] (0.5, 3, 0) -- (0.5, 3, {normal(0,0,0.5)}) (0.5,0,0) -- (0.5,12,0);
\draw (1.5, 4.5, 0) -- (1.5, 4.5, {normal(0,0,1)}) (1.5,0,0) -- (1.5,12,0);
\draw (2.5, 6, 0) -- (2.5, 6, {normal(0,0,1.5)}) (2.5,0,0) -- (2.5,12,0);
\end{pgfonlayer}
}
\end{axis}


\end{tikzpicture}

\end{document}

这是一个具有不同观点的版本，其中的点使用具有不同标准差的正态分布随机分布（使用绘制二维高斯样本)：

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}

\makeatletter
        \pgfdeclareplotmark{dot}
        {%
            \fill circle [x radius=0.02, y radius=0.08];
        }%
\makeatother

\begin{document}

\begin{tikzpicture}[ % Define Normal Probability Function
declare function={
            normal(\x,\m,\s) = 1/(2*\s*sqrt(pi))*exp(-(\x-\m)^2/(2*\s^2));
        },
    declare function={invgauss(\a,\b) = sqrt(-2*ln(\a))*cos(deg(2*pi*\b));}
       ]
\begin{axis}[
    %no markers,
    domain=0:12,
    zmin=0, zmax=1,
    xmin=0, xmax=3,
    samples=200,
   samples y=0,
    view={40}{30},
    axis lines=middle,
    enlarge y limits=false,
    xtick={0.5,1.5,2.5},
    xmajorgrids,
    xticklabels={},
    ytick=\empty,
    xticklabels={$x_1$, $x_2$, $x_3$},
    ztick=\empty,
    xlabel=$x$, xlabel style={at={(rel axis cs:1,0,0)}, anchor=west},
    ylabel=$y$, ylabel style={at={(rel axis cs:0,1,0)}, anchor=south west},
    zlabel=Probability density, zlabel style={at={(rel axis cs:0,0,0.5)}, rotate=90, anchor=south},
    set layers, mark=cube
  ]

\addplot3 [gray!50, only marks, mark=dot, mark layer=like plot, samples=200, domain=0.1:2.9, on layer=axis background] (x, {1.5*(x-0.5)+3+invgauss(rnd,rnd)*x}, 0);
\addplot3 [samples=2, samples y=0, domain=0:3] (x, {1.5*(x-0.5)+3}, 0);
\addplot3 [cyan!50!black, thick] (0.5, x, {normal(x, 3, 0.5)});
\addplot3 [cyan!50!black, thick] (1.5, x, {normal(x, 4.5, 1)});
\addplot3 [cyan!50!black, thick] (2.5, x, {normal(x, 6, 1.5)});

\pgfplotsextra{
\begin{pgfonlayer}{axis background}
\draw [gray, on layer=axis background] (0.5, 3, 0) -- (0.5, 3, {normal(0,0,0.5)}) (0.5,0,0) -- (0.5,12,0)
    (1.5, 4.5, 0) -- (1.5, 4.5, {normal(0,0,1)}) (1.5,0,0) -- (1.5,12,0)
    (2.5, 6, 0) -- (2.5, 6, {normal(0,0,1.5)}) (2.5,0,0) -- (2.5,12,0);

\end{pgfonlayer}
}
\end{axis}


\end{tikzpicture}

\end{document}

使用 x 中离散的现象：

\documentclass[border=5mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}

\makeatletter
        \pgfdeclareplotmark{dot}
        {%
            \fill circle [x radius=0.08, y radius=0.32];
        }%
\makeatother

\begin{document}

\begin{tikzpicture}[ % Define Normal Probability Function
declare function={
            normal(\x,\m,\s) = 1/(2*\s*sqrt(pi))*exp(-(\x-\m)^2/(2*\s^2));
        },
    declare function={invgauss(\a,\b) = sqrt(-2*ln(\a))*cos(deg(2*pi*\b));}
       ]
\begin{axis}[
    %no markers,
    domain=0:12,
    zmin=0, zmax=1,
    xmin=0, xmax=3,
    samples=200,
   samples y=0,
    view={40}{30},
    axis lines=middle,
    enlarge y limits=false,
    xtick={0.5,1.5,2.5},
    xmajorgrids,
    xticklabels={},
    ytick=\empty,
    xticklabels={$x_1$, $x_2$, $x_3$},
    ztick=\empty,
    xlabel=$x$, xlabel style={at={(rel axis cs:1,0,0)}, anchor=west},
    ylabel=$y$, ylabel style={at={(rel axis cs:0,1,0)}, anchor=south west},
    zlabel=Probability density, zlabel style={at={(rel axis cs:0,0,0.5)}, rotate=90, anchor=south},
    set layers, mark=cube
  ]

\pgfplotsinvokeforeach{0.5,1.5,2.5}{
\addplot3 [draw=none, fill=black, opacity=0.25, only marks, mark=dot, mark layer=like plot, samples=30, domain=0.1:2.9, on layer=axis background] (#1, {1.5*(#1-0.5)+3+invgauss(rnd,rnd)*#1}, 0);
}
\addplot3 [samples=2, samples y=0, domain=0:3] (x, {1.5*(x-0.5)+3}, 0);
\addplot3 [cyan!50!black, thick] (0.5, x, {normal(x, 3, 0.5)});
\addplot3 [cyan!50!black, thick] (1.5, x, {normal(x, 4.5, 1)});
\addplot3 [cyan!50!black, thick] (2.5, x, {normal(x, 6, 1.5)});

\pgfplotsextra{
\begin{pgfonlayer}{axis background}
\draw [gray, on layer=axis background] (0.5, 3, 0) -- (0.5, 3, {normal(0,0,0.5)}) (0.5,0,0) -- (0.5,12,0)
    (1.5, 4.5, 0) -- (1.5, 4.5, {normal(0,0,1)}) (1.5,0,0) -- (1.5,12,0)
    (2.5, 6, 0) -- (2.5, 6, {normal(0,0,1.5)}) (2.5,0,0) -- (2.5,12,0);

\end{pgfonlayer}
}
\end{axis}


\end{tikzpicture}

\end{document}

答案2

这里有一个纯粹的tikz解决方案，只是为了好玩。在当前状态下，如果您想更改常规函数的参数，代码需要进行一些手动计算才能正确获取辅助线的坐标。

\documentclass[tikz, border=6mm]{standalone}

\newcommand{\normal}[2]{{\x},{(1/sqrt(2*pi*#2^2))*exp(-(\x-#1)^2/(2*#2^2))}}

\begin{document}
  \begin{tikzpicture}[x={(.5,0,-1)}, % y-axis
                      y={(0,1,0)}, 
                      z={(2,-.25,2)}, % x-axis
                      thick]

  \begin{scope}[>=latex, ->, xshift=-3cm, yshift=-.5cm]
    \draw (0,0,0) -- ++(5,0,0) node [right] {$y$};
    \draw (0,0,0) -- ++(0,2,0) node [above] {$f(u)$};
    \draw (0,0,0) -- ++(0,0,8) node [right] {$x$};
  \end{scope}

  \begin{scope}[smooth]
    \draw [domain=-2:2] plot (\normal{0}{.2});
    \draw [domain=-2:2, xshift=2cm, yshift=-.5cm] plot (\normal{.5}{.5});
    \draw [domain=-2:2, xshift=4cm, yshift=-1cm] plot (\normal{1}{1});
  \end{scope}

  \draw [shorten <=-1.45cm] (0,0,0) -- (1,0,4) node [font=\scriptsize, below right] {$\beta_1+\beta2X_i$};
  \foreach \x\m\y\l in {0/0/2/x_1,2/.5/.8/x_2,4/1/.4/x_i} {
    \draw (-2,0,\x) -- ++(4,0,0) node [below, at start] {$\l$};
    \draw (\m,0,\x) --++(0,\y,0);
  }
  \end{tikzpicture}
\end{document}

渲染图像

相关内容