Animated distributions

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The animate.sty package is a useful tool for creating Javascript driven animations in PDF files. In this example it is used to animate probability distributions with varying parameters. GNUPLOT is used to compute the distribution curves. Download the PDF to see the animation.

Update: Example recompiled using animate.sty version 2008-05-16. Animation should now render correctly in all versions of Acrobat Reader v >= 6.0.

Note

Acrobat Reader v >= 6.0 is required to view this example.

This example takes some time to compile. A large number of auxiliary files will be created.

Docutils System Messages

System Message: ERROR/3 (./animated-distributions.txt)

Anonymous hyperlink mismatch: 0 references but 1 targets. See "backrefs" attribute for IDs.


animated-distributions

Edit and compile if you like:

% Animated distributions
% Author: Brian S. Marks 
%
% Note that shell-escape has to be enabled to generate the plots.
% The animations can only be viewed in Acrobat (Reader) v >= 6.
\documentclass{beamer}


\usepackage{tikz}
\usepackage{animate}

% Define a the counter cnt. Used to identify files generated for use
% with Gnuplot.
\newcounter{cnt}
\setcounter{cnt}{0}

% Macro for drawing one frame of the F-distribution animation.
\newcommand{\fdst}[4]{%
    % shade the critical region tail
    \draw[fill,orange]  (#1,0) -- plot[id=5\thecnt,domain=#1:5.5,samples=50]
        function {#4*(x**(0.5*#2-1))*((1+#2*x/#3)**(-0.5*#2-0.5*#3))}
            -- (5.5,0) -- cycle;

    % draw the F distribution curve
    \draw[color=blue!50!black,thick]
        plot[id=f4\thecnt,smooth,domain=0:5.5,samples=100]
        function {#4*(x**(0.5*#2-1))*((1+#2*x/#3)**(-0.5*#2-0.5*#3))};

    % draw the F axis
    \draw[->] (0,0) -- (6,0) node[right] {$F$};
    % label the critical region boundary
    \draw (#1,0) -- (#1,-0.02) node[below] {$#1$};
    % label 0
    \draw (0,0) -- (0,-0.02) node[below] {$0$};

    % add some lables for degrees of freedom and alpha level
    \draw (2,0.5) node[right] {$df_1 = #2$};
    \draw (2,0.4) node[right] {$df_2 = #3$};
    \draw (2,0.3) node[right] {$\alpha = 0.10$};

    % draw the y axis
    \draw[very thin,->] (0,0) -- (0,0.8);
}

\newcommand{\distpic}[3]{
    % First draw the upper distribution.
    % Shade the critical region:
    \fill[red!30] (0.658,0)  -- plot[id=f3,domain=0.658:3,samples=50]
        function {exp(-x*x*0.5/0.16)} -- (3,0) -- cycle;

    % Draw the normal distribution curve
    \draw[blue!50!black,smooth,thick] plot[id=f1,domain=-2:3,samples=50]
    function {exp(-x*x*0.5/0.16)};
    % Draw the x-axis
    \draw[->,black] (-2.2,0) -- (3.2,0);
    % Put some ticks and tick labels in:
    \foreach \x in {-2,-1,0,1,2,3}
    \draw (\x,0) -- (\x,-0.1) node[below] {$\x$};
    % Put in a label for the critical region boundary:
    \draw[red!50!black,thick] (0.658,0) node[below,yshift=-0.5cm] {0.658}
    -- (0.658,0.85);

    % Put in labels for accepting or rejecting the null hypothesis with
    % the corresponding regions:
    \draw[red!50!black,thick,->] (0.688,0.7) -- (1.3,0.7)
        node[anchor=south] {Reject  $H_0$};
    \draw[red!50!black,thick,->] (0.628,0.7) -- (-1,0.7)
        node[anchor=south]{\parbox{1.5cm}{\raggedright Fail to reject $H_0$}};

    % Add a label to the upper picture, when the null is true
    \draw (-3,1) node[above,draw,fill=green!30] {$H_0$ is true:};

    % Label the critical region with an alpha level:
    \draw[<-,thick] (0.75,0.05) -- (1.6,0.2) node[right,yshift=0.3cm]
    {\begin{tabular}{l} $\alpha=0.05$ \\ (Type I error rate) \end{tabular}};


    % Add a label showing the effect size between the two plots:
    \draw[very thin] (0,-1) -- (0,-0.5);
    \draw[<->,thick] (0,-1) node[left] {Effect size:  #1} -- (#1,-1);
    \draw[thick] (0,-.9) -- (0,-1.1);

    \draw[very thin] (#1,-1) -- (#1,-1.7);
    \draw[thick] (#1,-.9) -- (#1,-1.1);

    % Now draw the lower distribution showing the effect size:
    \begin{scope}[yshift=-3cm]
    % Shade the "reject H0" region red
    \fill[red!30] (0.658,0)  -- plot[id=f3\thecnt,domain=0.658:3,samples=50]
        function {exp(-(x-#1)*(x-#1)*0.5/0.16)} --
        (3,0) -- cycle;
        % Shade the "accept H0" region blue
    \fill[blue!30] (-2,0) -- plot[id=f4\thecnt,domain=-2:0.658,samples=50]
        function {exp(-(x-#1)*(x-#1)*0.5/0.16)} --
        (0.658,0) -- cycle;

        % Draw the shifted normal distribution:
    \draw[blue!50!black,smooth,thick] plot[id=f1\thecnt,domain=-2:3,samples=50]
            function {exp(-(x-#1)*(x-#1)*0.5/0.16)};

        % Draw the x-axis and put in some ticks and tick labels
    \draw[->,black] (-2.2,0) -- (3.2,0);
    \foreach \x in {-2,-1,0,1,2,3}
            \draw (\x,0) -- (\x,-0.1) node[below] {$\x$};

        % Draw and label the critical region boundary
    \draw[red!50!black,very thick] (0.658,0) node[below,yshift=-0.5cm] {0.658}
        -- (0.658,1.0);
    \draw[red!50!black,very thick,->] (0.688,0.7) -- (2.7,0.7)
        node[anchor=south west] {Reject  $H_0$};
    \draw[red!50!black,very thick,->] (0.628,0.7) -- (-0.5,0.7)
        node[anchor=south]{\parbox{1.5cm}{\raggedright Fail to reject $H_0$}};

    % Add a label to the lower picture, when the alternative hypothesis is true:
    \draw (-3,1) node[above,draw,fill=green!30] {$H_a$ is true:};

        % Add labels showing the statistical power and the Type II error rate:
    \draw[<-,thick] (1.5,0.1) -- (3,0.2) node[anchor=south west]
        {Power = \large #2};
    \draw[<-,thick] (0.4,0.1) -- (-1,0.2) node[left,yshift=0.3cm]
        {\begin{tabular}{l}
        $\beta$ = {\large #3} \\ (Type II error rate) \end{tabular}};
    \end{scope}
}


\begin{document}

\begin{frame}
  General form for the $F$ distribution with $\nu_1$ and $\nu_2$ degrees of
  freedom:
  \[
    f(F; \nu_1, \nu_2) = \frac{\Gamma\left(\frac{\nu_1+\nu_2}{2}\right)}
    {\Gamma\left(\frac{\nu_1}{2}\right)\Gamma\left(\frac{\nu_2}{2}\right)}
    \left(\frac{\nu_1}{\nu_2}\right)^{\frac{\nu_1}{2}}
    \frac{F^{\frac{\nu_1-2}{2}}}{\left[1 +
    \left(\frac{\nu_1}{\nu_2}\right)F\right]^{\frac{\nu_1+\nu_2}{2}}}
  \]

  Critical region for an $F$ test for various degrees of freedom:

  \begin{center}

  \begin{animateinline}[autoplay,palindrome,
    begin={\begin{tikzpicture}[yscale=4]},
    end={\stepcounter{cnt}\end{tikzpicture}}]{8}
    \fdst{5.3426}{4}{3}{6.6667};\newframe
    \fdst{4.1072}{4}{4}{6};\newframe
    \fdst{3.5202}{4}{5}{5.6};\newframe
    \fdst{3.1808}{4}{6}{5.3333};\newframe
    \fdst{2.9605}{4}{7}{5.1429};\newframe
    \fdst{2.8064}{4}{8}{5};\newframe
    \fdst{2.6927}{4}{9}{4.8889};\newframe
    \fdst{2.6053}{4}{10}{4.8};\newframe
    \fdst{2.5216}{5}{10}{10.3684};\newframe
    \fdst{2.6106}{5}{9}{10.7120};\newframe
    \fdst{2.7264}{5}{8}{11.1463};\newframe
    \fdst{2.8833}{5}{7}{11.7125};\newframe
    \fdst{3.1075}{5}{6}{12.4807};\newframe
    \fdst{3.4530}{5}{5}{13.5812};\newframe
    \fdst{4.0506}{5}{4}{15.2856};\newframe
    \fdst{5.3092}{5}{3}{18.2638};\newframe
    \fdst{5.2847}{6}{3}{52.5};\newframe
    \fdst{4.0097}{6}{4}{40.5};\newframe
    \fdst{3.4045}{6}{5}{34.02};\newframe
    \fdst{3.0546}{6}{6}{30};\newframe
    \fdst{2.8274}{6}{7}{27.2755};\newframe
    \fdst{2.6683}{6}{8}{25.3125};\newframe
    \fdst{2.5509}{6}{9}{23.8333};\newframe
    \fdst{2.4606}{6}{10}{22.6800};\newframe
    \fdst{2.4140}{7}{10}{50.4952};\newframe
    \fdst{2.5053}{7}{9}{54.1009};\newframe
    \fdst{2.6241}{7}{8}{58.8076};\newframe
    \fdst{2.7849}{7}{7}{65.1899};\newframe
    \fdst{3.0145}{7}{6}{74.2894};\newframe
    \fdst{3.3679}{7}{5}{88.1895};\newframe
    \fdst{3.9790}{7}{4}{111.6641};\newframe
    \fdst{5.2662}{7}{3}{158.1280};\newframe
    \fdst{5.2517}{8}{3}{497.7779};\newframe
    \fdst{3.9549}{8}{4}{320};\newframe
    \fdst{3.3393}{8}{5}{236.544};\newframe
    \fdst{2.9830}{8}{6}{189.6296};\newframe
    \fdst{2.7516}{8}{7}{160.0933};\newframe
    \fdst{2.5893}{8}{8}{140};\newframe
    \fdst{2.4694}{8}{9}{125.5418};\newframe
    \fdst{2.3772}{8}{10}{114.688};\newframe
    \fdst{2.3473}{9}{10}{266.0};\newframe
    \fdst{2.4403}{9}{9}{298.0};\newframe
    \fdst{2.5612}{9}{8}{341.7};\newframe
    \fdst{2.7247}{9}{7}{404.0};\newframe
    \fdst{2.9577}{9}{6}{498.7};\newframe
    \fdst{3.3163}{9}{5}{655.8};\newframe
    \fdst{3.9357}{9}{4}{941.5};\newframe
    \fdst{5.2400}{9}{3}{1633.2};
  \end{animateinline}
  \end{center}

\end{frame}

\begin{frame}
  Statistical power in hypothesis testing:

  \begin{animateinline}[autoplay,loop,
    begin={\begin{tikzpicture}[scale=1.3]},
    end={\stepcounter{cnt}\end{tikzpicture}}]{3}
    \distpic{0.5}{.346}{.654}
    \newframe
    \distpic{0.7}{.542}{.458}\newframe
    \distpic{0.9}{.727}{.273}\newframe
    \distpic{1.1}{.865}{.135}\newframe
    \distpic{1.3}{.946}{.054}\newframe
    \distpic{1.5}{.982}{.018}\newframe
    \distpic{1.7}{.995}{.005}\newframe
    \distpic{1.9}{.999}{.001}
  \end{animateinline}
\end{frame}

\end{document}

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