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如何绘制 F 分布

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如何绘制 F 分布

我希望通过声明其密度函数来绘制 F 分布。但是，使用以下代码时，我收到一条错误消息，提示fdst尚未声明：

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

\begin{document}

\begin{tikzpicture}[
    declare function={gamma(\z)=
    2.506628274631*sqrt(1/\z)+ 0.20888568*(1/\z)^(1.5)+ 0.00870357*(1/\z)^(2.5)- (174.2106599*(1/\z)^(3.5))/25920- (715.6423511*(1/\z)^(4.5))/1244160)*exp((-ln(1/\z)-1)*\z;},
    declare function={fdst(\x,\n,\m))= (gamma((\n+\m)/2)/(gamma(\n/2) *gamma(\m/2))) *(\n/\m)^(\n/2) *((\x *((\n-2)/2))/((1+(\n *\x)/\m)^((\n+\m)/2)));}
]

\begin{axis}[
    axis lines=left,
    enlargelimits=upper,
    samples=50
]
 \addplot [very thick,yellow!80!black] {fdst(x,3,3)};

\end{axis}
\end{tikzpicture}
\end{document}

任何帮助都感激不尽。

答案1

您可以使用伽马函数的近似值绘制伽马分布的概率密度函数，用它来定义beta 函数然后根据以下定义 f 分布：

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

\begin{document}

\begin{tikzpicture}[
    declare function={
            gamma(\z)=2.506628274631*sqrt(1/\z)+ 0.20888568*(1/\z)^(1.5)+ 0.00870357*(1/\z)^(2.5)- (174.2106599*(1/\z)^(3.5))/25920- (715.6423511*(1/\z)^(4.5))/1244160)*exp((-ln(1/\z)-1)*\z;
        },
        declare function={
            beta(\x,\y)=gamma(\x)*gamma(\y)/gamma(\x+\y);
        },
    declare function={
        fdst(\x,\a,\b) = 1 / beta(\a/2, \b/2) * (\a/\b)^(\a/2) * \x^(\a/2-1) * (1 + \a/\b*\x)^(-(\a + \b)/2);
    }
]

\begin{axis}[
    axis lines=left,
    enlargelimits=upper,
    samples=100,
    xmin=0, ymin=0,
    domain=0.01:4,
    legend cell align=left
]

\addplot [very thick,blue] {fdst(x,1,1)}; \addlegendentry{$d_1=1,\hphantom{00} d_2=1$}
\addplot [very thick,orange] {fdst(x,100,100)}; \addlegendentry{$d_1=100, d_2=100$}
\addplot [very thick,purple] {fdst(x,5,2)}; \addlegendentry{$d_1=5,\hphantom{00} d_2=2$}

\end{axis}
\end{tikzpicture}
\end{document}

答案2

运行xelatex：

\documentclass{article}
\usepackage{pst-func}
\begin{document}

\psset{xunit=2cm,yunit=10cm,plotpoints=100}
\begin{pspicture*}(-0.5,-0.07)(5.5,0.8)
 \psline[linestyle=dashed](0.5,0)(0.5,0.75)
 \psline[linestyle=dashed](! 2 7 div 0)(! 2 7 div 0.75)
 \psset{linewidth=1pt}
 \psFDist{0.1}{5}
 \psFDist[linecolor=red,nue=3,mue=12]{0.01}{5}
 \psFDist[linecolor=blue,nue=12,mue=3]{0.01}{5}
 \psaxes[Dy=0.1,ticksize=-4pt 0]{->}(0,0)(5,0.75)
\end{pspicture*}

\end{document}

在此处输入图片描述

答案3

只是通过实现 中的函数来扩展 Jake 的答案lua。在转换为数字表示和从数字表示转换时有些麻烦pgfplots（这可能不太正确）。此外，还使用了不同的 beta 函数近似值：

\documentclass[tikz,border=5]{standalone}
\usepackage{pgfplots}
{\catcode`\~=11 \gdef\tilde{~}}
{\catcode`\%=11 \gdef\percent{%}}

\directlua{%

function lua2pgfplots(v)
  if v \tilde= v then
    return "3Y0e0]"
  end
  if v == math.huge then
    return "4Y0e0]"
  end
  if v == -math.huge then
    return "5Y0e0]"
  end
  if v == 0 then
    return "0Y0e0]"
  end
  if v > 0 then
    return string.format("1Y\percent e]", v)
  else
    return string.format("2Y\percent e]", v)
  end
end

function pgfplots2lua(v)
  local f, x
  f, x = string.match(v, "(\percent d)Y(.*).")
  f = tonumber(f)
  x = tonumber(x)
  if f == 1 then
    return x
  end
  if f == 2 then
    return -x
  end
  if f == 3 then
    return 0/0
  end
  if f == 4 then
    return math.huge
  end
  if f == 5 then
    return -math.huge
  end
  return x
end
}

\directlua{
function beta(x,y)
  return math.sqrt(2*math.pi)*(math.pow(x,x-.5)*math.pow(y,y-.5)) / 
         math.pow(x+y, x+y-.5)
end

function fdist(x, d1, d2)
  return 1/beta(d1/2,d2/2)*math.pow(d1/d2, d1/2) *
         math.pow(x, d1/2-1) *
         math.pow(1+d1/d2*x, -(d1+d2)/2)
end
}

\pgfmathdeclarefunction{fdist}{3}{%
  \edef\pgfmathresult{%
    \directlua{%
      local f
      f = fdist(pgfplots2lua("#1"),pgfplots2lua("#2"),pgfplots2lua("#3"))
      f = lua2pgfplots(f)
      tex.print(f)
    }%
  }%  
}
\begin{document}

\begin{tikzpicture}
\begin{axis}[
    axis lines=left,
    enlargelimits=upper,
    samples=100,
    xmin=0, ymin=0,
    domain=0.01:4,
    legend cell align=left
]

\addplot [very thick, red]   {fdist(x,1,1)};  \addlegendentry{$d_1=1,\hphantom{00} d_2=1$}
\addplot [very thick, black] {fdist(x,2,1)};  \addlegendentry{$d_1=2,\hphantom{00} d_2=1$}
\addplot [very thick, blue]  {fdist(x,5,2)};  \addlegendentry{$d_1=5,\hphantom{00} d_2=2$}
\addplot [very thick, green] {fdist(x,10,1)}; \addlegendentry{$d_1=10,\hphantom{0} d_2=1$}
\addplot [very thick, gray]  {fdist(x,100,100)}; \addlegendentry{$d_1=100, d_2=100$}

\end{axis}
\end{tikzpicture}
\end{document}

在此处输入图片描述

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