我看到了这篇文章双变量它绘制了一个二元正态概率密度函数。我想要的是基于这个概率函数（统计地）绘制一些点，然后将其显示在与上述问题中的美丽图相同的图中。
以下是上述来源的源代码：
\documentclass{standalone}
\usepackage{pgfplots}
\begin{document}
\pgfplotsset{
colormap={whitered}{color(0cm)=(white); color(1cm)=(orange!75!red)}
}
\begin{tikzpicture}[
declare function={mu1=1;},
declare function={mu2=2;},
declare function={sigma1=0.5;},
declare function={sigma2=1;},
declare function={normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));},
declare function={bivar(\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2))/2;}]
\begin{axis}[
colormap name=whitered,
width=15cm,
view={45}{65},
enlargelimits=false,
grid=major,
domain=-1:4,
y domain=-1:4,
samples=26,
xlabel=$x_1$,
ylabel=$x_2$,
zlabel={$P$},
colorbar,
colorbar style={
at={(1,0)},
anchor=south west,
height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
title={$P(x_1,x_2)$}
}
]
\addplot3 [surf] {bivar(mu1,sigma1,mu2,sigma2)};
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth] (x,4,{normal(mu1,sigma1)});
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth] (-1,x,{normal(mu2,sigma2)});
\draw [black!50] (axis cs:-1,0,0) -- (axis cs:4,0,0);
\draw [black!50] (axis cs:0,-1,0) -- (axis cs:0,4,0);
\node at (axis cs:-1,1,0.18) [pin=165:$P(x_1)$] {};
\node at (axis cs:1.5,4,0.32) [pin=-15:$P(x_2)$] {};
\end{axis}
\end{tikzpicture}
\end{document}
我想要的是从概率分布中绘制一些点（比如 100 个），然后在同一个或另一个图上绘制这些点。
答案1
这是一个选项，我刚刚用过这其他帖子生成数字
\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
% https://tex.stackexchange.com/questions/148091/gaussian-random-numbers
\newcount\gaussF
\edef\gaussR{0}
\edef\gaussA{0}
\makeatletter
\pgfmathdeclarefunction{invgauss}{2}{%
\pgfmathln{#1}% <- might need parsing
\pgfmathmultiply@{\pgfmathresult}{-2}%
\pgfmathsqrt@{\pgfmathresult}%
\let\@radius=\pgfmathresult%
\pgfmathmultiply{6.28318531}{#2}% <- might need parsing
\pgfmathdeg@{\pgfmathresult}%
\pgfmathcos@{\pgfmathresult}%
\pgfmathmultiply@{\pgfmathresult}{\@radius}%
}
\pgfmathdeclarefunction{randnormal}{0}{%
\pgfmathrnd@
\ifdim\pgfmathresult pt=0.0pt\relax%
\def\pgfmathresult{0.00001}%
\fi%
\let\@tmp=\pgfmathresult%
\pgfmathrnd@%
\ifdim\pgfmathresult pt=0.0pt\relax%
\def\pgfmathresult{0.00001}%
\fi
\pgfmathinvgauss@{\pgfmathresult}{\@tmp}%
}
\makeatother
\pgfplotsset{width=7cm,compat=newest}
\pgfplotsset{%
colormap={whitered}{color(0cm)=(white);
color(1cm)=(orange!75!red)}
}
\begin{document}
\begin{tikzpicture}[
declare function = {mu1=1;},
declare function = {mu2=2;},
declare function = {sigma1=0.5;},
declare function = {sigma2=1;},
declare function = {normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));},
declare function = {bivar(\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2))/2;}]
\begin{axis}[
colormap name = whitered,
width = 15cm,
view = {45}{65},
enlargelimits = false,
grid = major,
domain = -1:4,
y domain = -1:4,
samples = 26,
xlabel = $x_1$,
ylabel = $x_2$,
zlabel = {$P$},
xmin = -1,
xmax = 4,
ymin = -1,
ymax = 4,
colorbar,
colorbar style = {
at = {(1,0)},
anchor = south west,
height = 0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
title = {$P(x_1,x_2)$}
}
]
\addplot3 [surf] {bivar(mu1,sigma1,mu2,sigma2)};
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth]
(x,4,{normal(mu1,sigma1)});
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth]
(-1,x,{normal(mu2,sigma2)});
\pgfplotsinvokeforeach{1,...,500} {
\pgfmathsetmacro{\xx}{mu1 + sigma1 * randnormal}
\pgfmathsetmacro{\yy}{mu2 + sigma2 * randnormal}
\addplot3 [mark=*, mark size=1pt] coordinates {(\xx, \yy, 0)};
}
\draw [black!50] (axis cs:-1,0,0) -- (axis cs:4,0,0);
\draw [black!50] (axis cs:0,-1,0) -- (axis cs:0,4,0);
\node at (axis cs:-1,1,0.18) [pin=165:$P(x_1)$] {};
\node at (axis cs:1.5,4,0.32) [pin=-15:$P(x_2)$] {};
\end{axis}
\end{tikzpicture}
\end{document}
答案2
另一项提议基于逆高斯技巧。
\documentclass{standalone}
\usepackage{pgfplots}
\begin{document}
\pgfplotsset{
colormap={whitered}{color(0cm)=(white); color(1cm)=(orange!75!red)}
}
\begin{tikzpicture}[
declare function={mu1=1;
mu2=2;
sigma1=0.5;
sigma2=1;
normal(\m,\s)=1/(2*\s*sqrt(pi))*exp(-(x-\m)^2/(2*\s^2));
bivar(\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((x-\ma)^2/\sa^2 + (y-\mb)^2/\sb^2))/2;
mybivar(\x,\y,\ma,\sa,\mb,\sb)=
1/(2*pi*\sa*\sb) * exp(-((\x-\ma)^2/\sa^2 + (\y-\mb)^2/\sb^2))/2;
invgauss(\a,\b) = sqrt(-2*ln(\a))*cos(deg(2*pi*\b));}]
% invgauss based on https://tex.stackexchange.com/questions/254484/how-to-make-a-graph-of-heteroskedasticity-with-tikz-pgf/254494#254494
\foreach\X in {1,...,100}
{\pgfmathsetmacro{\myx}{mu1+0.5*sigma1*invgauss(rnd,rnd)}
\pgfmathsetmacro{\myy}{mu2+0.5*sigma2*invgauss(rnd,rnd)}
\pgfmathsetmacro{\myz}{mybivar(\myx,\myy,mu1,sigma1,mu2,sigma2)}
\ifnum\X=1
\xdef\lstOne{(\myx,\myy,\myz)}
\else
\xdef\lstOne{\lstOne (\myx,\myy,\myz)}
\fi}
\begin{axis}[
colormap name=whitered,
width=15cm,
view={45}{65},
enlargelimits=false,
grid=major,
domain=-1:4,
y domain=-1:4,
samples=26,
xlabel=$x_1$,
ylabel=$x_2$,
zlabel={$P$},
colorbar,
colorbar style={
at={(1,0)},
anchor=south west,
height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
title={$P(x_1,x_2)$}
}
]
\addplot3 [surf] {bivar(mu1,sigma1,mu2,sigma2)};
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth] (x,4,{normal(mu1,sigma1)});
\addplot3 [domain=-1:4,samples=31, samples y=0, thick, smooth] (-1,x,{normal(mu2,sigma2)});
\addplot3 [only marks, mark=*, mark layer=like plot,mark size=1pt] coordinates {\lstOne};
\draw [black!50] (axis cs:-1,0,0) -- (axis cs:4,0,0);
\draw [black!50] (axis cs:0,-1,0) -- (axis cs:0,4,0);
\node at (axis cs:-1,1,0.18) [pin=165:$P(x_1)$] {};
\node at (axis cs:1.5,4,0.32) [pin=-15:$P(x_2)$] {};
\end{axis}
\end{tikzpicture}
\end{document}
