Bootstrap technics is a resampling of a limited length sample without any new data. This method allows us to get new informations on statistical uncertaintes due to limited length sample.
Edit and compile if you like:
% Bootstrap resampling
% Germain Salvato-Vallverdu - mai 2009
% http://germain.salvato-vallverdu.perso.sfr.fr
\documentclass[11pt,a4paper,landscape]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[francais]{babel}
\usepackage[top=3cm,left=2cm,right=2cm,bottom=3cm]{geometry}
\usepackage{tikz}
\usepackage[active,tightpage]{preview}
\setlength\PreviewBorder{5pt}%
\usetikzlibrary{shadows}
\usepackage{hyperref}
\hypersetup{%
pdfauthor={Germain Salvato-Vallverdu},%
pdftitle={bootstrap method},%
pdfkeywords={Tikz,latex,bootstrap,uncertaintes},%
pdfcreator={PDFLaTeX},%
pdfproducer={PDFLaTeX},%
}
\title{Bootstrap method}
\author{Germain Salvato-Vallverdu}
\pagestyle{empty}
\begin{document}
\sffamily
\begin{preview}
\begin{center}
\begin{tikzpicture}[scale=1.4]
% style
\tikzstyle{rboule} = [circle,scale=0.7,ball color=red]
\tikzstyle{gboule} = [circle,scale=0.7,ball color=green]
\tikzstyle{bboule} = [circle,scale=0.7,ball color=blue]
\tikzstyle{nboule} = [circle,scale=0.7,ball color=black]
\tikzstyle{sample} = [->,thin]
% complete element space
\path[draw,<-,thick] (0,5.6) -- (4,5.6);
\path (6,5.6) node {Complete element space};
\path[draw,->,thick] (8,5.6) -- (12,5.6);
\path ( 0.00,5.25) node[bboule] (i01) {};
\path ( 0.25,5.25) node[rboule] (i02) {};
\path ( 0.50,5.25) node[gboule] (i03) {};
\path ( 0.75,5.25) node[bboule] (i04) {};
\path ( 1.00,5.25) node[rboule] (i05) {};
\path ( 1.25,5.25) node[bboule] (i06) {};
\path ( 1.50,5.25) node[bboule] (i07) {};
\path ( 1.75,5.25) node[rboule] (i08) {};
\path ( 2.00,5.25) node[bboule] (i09) {};
\path ( 2.25,5.25) node[rboule] (i10) {};
\path ( 2.50,5.25) node[bboule] (i11) {};
\path ( 2.75,5.25) node[bboule] (i12) {};
\path ( 3.00,5.25) node[nboule] (i13) {};
\path ( 3.25,5.25) node[rboule] (i14) {};
\path ( 3.50,5.25) node[bboule] (i15) {};
\path ( 3.75,5.25) node[bboule] (i16) {};
\path ( 4.00,5.25) node[rboule] (i17) {};
\path ( 4.25,5.25) node[rboule] (i18) {};
\path ( 4.50,5.25) node[bboule] (i19) {};
\path ( 4.75,5.25) node[bboule] (i20) {};
\path ( 5.00,5.25) node[rboule] (i21) {};
\path ( 5.25,5.25) node[bboule] (i22) {};
\path ( 5.50,5.25) node[rboule] (i23) {};
\path ( 5.75,5.25) node[bboule] (i24) {};
\path ( 6.00,5.25) node[bboule] (i25) {};
\path ( 6.25,5.25) node[rboule] (i26) {};
\path ( 6.50,5.25) node[rboule] (i27) {};
\path ( 6.75,5.25) node[gboule] (i28) {};
\path ( 7.00,5.25) node[rboule] (i29) {};
\path ( 7.25,5.25) node[bboule] (i30) {};
\path ( 7.50,5.25) node[bboule] (i31) {};
\path ( 7.75,5.25) node[nboule] (i32) {};
\path ( 8.00,5.25) node[bboule] (i33) {};
\path ( 8.25,5.25) node[rboule] (i34) {};
\path ( 8.50,5.25) node[gboule] (i35) {};
\path ( 8.75,5.25) node[bboule] (i36) {};
\path ( 9.00,5.25) node[bboule] (i37) {};
\path ( 9.25,5.25) node[rboule] (i38) {};
\path ( 9.50,5.25) node[bboule] (i39) {};
\path ( 9.75,5.25) node[rboule] (i40) {};
\path (10.00,5.25) node[bboule] (i41) {};
\path (10.25,5.25) node[rboule] (i42) {};
\path (10.50,5.25) node[bboule] (i43) {};
\path (10.75,5.25) node[bboule] (i44) {};
\path (11.00,5.25) node[rboule] (i45) {};
\path (11.25,5.25) node[nboule] (i46) {};
\path (11.50,5.25) node[rboule] (i47) {};
\path (11.75,5.25) node[bboule] (i48) {};
\path (12.00,5.25) node[bboule] (i49) {};
% title initial sample
\path (3.5,3.75) node[anchor=east]
{\parbox{0.15\textwidth}{\centering initial sample with N elements}};
% labels
\path (3.5,3) node[anchor=east] {1};
\path (3.5,2.5) node[anchor=east] {2};
\path (3.5,2.0) node[anchor=east] {3};
\path (3.5,1.5) node[anchor=east] {4};
\path (3.5,0.5) node[anchor=east] {$N_b$};
\path (3.5,1) node[anchor=east] {$\vdots$};
\path[draw,dashed] (4,1) -- (8.25,1);
\path[draw,<->,thick] (2.9,0.5) -- (2.9,3) node[anchor=east,pos=0.5]
{\parbox{0.1\textwidth}{\centering $N_b$ \textit{bootstrap} samples}};
% initial sample
\path ( 3.75,3.75) node[bboule] (j01) {};
\path ( 4.00,3.75) node[bboule] (j02) {};
\path ( 4.25,3.75) node[rboule] (j03) {};
\path ( 4.50,3.75) node[rboule] (j04) {};
\path ( 4.75,3.75) node[rboule] (j05) {};
\path ( 5.00,3.75) node[nboule] (j06) {};
\path ( 5.25,3.75) node[bboule] (j07) {};
\path ( 5.50,3.75) node[rboule] (j08) {};
\path ( 5.75,3.75) node[bboule] (j09) {};
\path ( 6.00,3.75) node[bboule] (j10) {};
\path ( 6.25,3.75) node[bboule] (j11) {};
\path ( 6.50,3.75) node[rboule] (j12) {};
\path ( 6.75,3.75) node[gboule] (j13) {};
\path ( 7.00,3.75) node[rboule] (j14) {};
\path ( 7.25,3.75) node[gboule] (j15) {};
\path ( 7.50,3.75) node[bboule] (j16) {};
\path ( 7.75,3.75) node[rboule] (j17) {};
\path ( 8.00,3.75) node[rboule] (j18) {};
\path ( 8.25,3.75) node[bboule] (j19) {};
\path ( 8.50,3.75) node[rboule] (j20) {};
% arrows from the complete space to the initial sample
\path[sample] (i01.south) edge[out=-60,in=120] (j01.north west);
\path[sample] (i04.south) edge[out=-60,in=120] (j02.north west);
\path[sample] (i05.south) edge[out=-60,in=120] (j03.north west);
\path[sample] (i08.south) edge[out=-60,in=120] (j04.north west);
\path[sample] (i10.south) edge[out=-60,in=120] (j05.north west);
\path[sample] (i13.south) edge[out=-60,in=120] (j06.north west);
\path[sample] (i16.south) edge[out=-90,in=90] (j07.north);
\path[sample] (i18.south) edge[out=-90,in=90] (j08.north);
\path[sample] (i20.south) edge[out=-90,in=90] (j09.north);
\path[sample] (i24.south) edge[out=-90,in=90] (j10.north);
\path[sample] (i25.south) edge[out=-90,in=90] (j11.north);
\path[sample] (i27.south) edge[out=-90,in=90] (j12.north);
\path[sample] (i28.south) edge[out=-90,in=90] (j13.north);
\path[sample] (i34.south) edge[out=-90,in=90] (j14.north);
\path[sample] (i35.south) edge[out=-90,in=90] (j15.north);
\path[sample] (i37.south) edge[out=-90,in=60] (j16.north east);
\path[sample] (i40.south) edge[out=-90,in=60] (j17.north east);
\path[sample] (i42.south) edge[out=-90,in=60] (j18.north east);
\path[sample] (i44.south) edge[out=-90,in=60] (j19.north east);
\path[sample] (i47.south) edge[out=-90,in=60] (j20.north east);
% bootstrap 1
\path ( 3.75, 3.0) node[bboule] {};
\path ( 4.00, 3.0) node[bboule] {};
\path ( 4.25, 3.0) node[bboule] {};
\path ( 4.50, 3.0) node[gboule] {};
\path ( 4.75, 3.0) node[rboule] {};
\path ( 5.00, 3.0) node[bboule] {};
\path ( 5.25, 3.0) node[gboule] {};
\path ( 5.50, 3.0) node[bboule] {};
\path ( 5.75, 3.0) node[rboule] {};
\path ( 6.00, 3.0) node[rboule] {};
\path ( 6.25, 3.0) node[bboule] {};
\path ( 6.50, 3.0) node[rboule] {};
\path ( 6.75, 3.0) node[rboule] {};
\path ( 7.00, 3.0) node[rboule] {};
\path ( 7.25, 3.0) node[bboule] {};
\path ( 7.50, 3.0) node[nboule] {};
\path ( 7.75, 3.0) node[rboule] {};
\path ( 8.00, 3.0) node[rboule] {};
\path ( 8.25, 3.0) node[gboule] {};
\path ( 8.50, 3.0) node[gboule] (b1) {};
% bootstrap 2
\path ( 3.75, 2.5) node[bboule] {};
\path ( 4.00, 2.5) node[bboule] {};
\path ( 4.25, 2.5) node[rboule] {};
\path ( 4.50, 2.5) node[nboule] {};
\path ( 4.75, 2.5) node[rboule] {};
\path ( 5.00, 2.5) node[bboule] {};
\path ( 5.25, 2.5) node[bboule] {};
\path ( 5.50, 2.5) node[rboule] {};
\path ( 5.75, 2.5) node[rboule] {};
\path ( 6.00, 2.5) node[rboule] {};
\path ( 6.25, 2.5) node[bboule] {};
\path ( 6.50, 2.5) node[rboule] {};
\path ( 6.75, 2.5) node[rboule] {};
\path ( 7.00, 2.5) node[rboule] {};
\path ( 7.25, 2.5) node[rboule] {};
\path ( 7.50, 2.5) node[rboule] {};
\path ( 7.75, 2.5) node[bboule] {};
\path ( 8.00, 2.5) node[bboule] {};
\path ( 8.25, 2.5) node[bboule] {};
\path ( 8.50, 2.5) node[bboule] (b2) {};
% bootstrap 3
\path ( 3.75, 2.0) node[rboule] {};
\path ( 4.00, 2.0) node[bboule] {};
\path ( 4.25, 2.0) node[rboule] {};
\path ( 4.50, 2.0) node[rboule] {};
\path ( 4.75, 2.0) node[bboule] {};
\path ( 5.00, 2.0) node[bboule] {};
\path ( 5.25, 2.0) node[bboule] {};
\path ( 5.50, 2.0) node[bboule] {};
\path ( 5.75, 2.0) node[rboule] {};
\path ( 6.00, 2.0) node[rboule] {};
\path ( 6.25, 2.0) node[gboule] {};
\path ( 6.50, 2.0) node[gboule] {};
\path ( 6.75, 2.0) node[rboule] {};
\path ( 7.00, 2.0) node[rboule] {};
\path ( 7.25, 2.0) node[nboule] {};
\path ( 7.50, 2.0) node[bboule] {};
\path ( 7.75, 2.0) node[rboule] {};
\path ( 8.00, 2.0) node[bboule] {};
\path ( 8.25, 2.0) node[rboule] {};
\path ( 8.50, 2.0) node[nboule] (b3) {};
% bootstrap 4
\path ( 3.75, 1.5) node[nboule] {};
\path ( 4.00, 1.5) node[bboule] {};
\path ( 4.25, 1.5) node[rboule] {};
\path ( 4.50, 1.5) node[gboule] {};
\path ( 4.75, 1.5) node[bboule] {};
\path ( 5.00, 1.5) node[gboule] {};
\path ( 5.25, 1.5) node[bboule] {};
\path ( 5.50, 1.5) node[rboule] {};
\path ( 5.75, 1.5) node[bboule] {};
\path ( 6.00, 1.5) node[rboule] {};
\path ( 6.25, 1.5) node[gboule] {};
\path ( 6.50, 1.5) node[nboule] {};
\path ( 6.75, 1.5) node[bboule] {};
\path ( 7.00, 1.5) node[rboule] {};
\path ( 7.25, 1.5) node[rboule] {};
\path ( 7.50, 1.5) node[rboule] {};
\path ( 7.75, 1.5) node[rboule] {};
\path ( 8.00, 1.5) node[bboule] {};
\path ( 8.25, 1.5) node[bboule] {};
\path ( 8.50, 1.5) node[gboule] (b4) {};
% bootstrap N
\path ( 3.75, 0.5) node[nboule] {};
\path ( 4.00, 0.5) node[gboule] {};
\path ( 4.25, 0.5) node[rboule] {};
\path ( 4.50, 0.5) node[rboule] {};
\path ( 4.75, 0.5) node[bboule] {};
\path ( 5.00, 0.5) node[rboule] {};
\path ( 5.25, 0.5) node[bboule] {};
\path ( 5.50, 0.5) node[gboule] {};
\path ( 5.75, 0.5) node[bboule] {};
\path ( 6.00, 0.5) node[bboule] {};
\path ( 6.25, 0.5) node[gboule] {};
\path ( 6.50, 0.5) node[rboule] {};
\path ( 6.75, 0.5) node[rboule] {};
\path ( 7.00, 0.5) node[gboule] {};
\path ( 7.25, 0.5) node[bboule] {};
\path ( 7.50, 0.5) node[bboule] {};
\path ( 7.75, 0.5) node[bboule] {};
\path ( 8.00, 0.5) node[rboule] {};
\path ( 8.25, 0.5) node[bboule] {};
\path ( 8.50, 0.5) node[gboule] (bN) {};
% arrows
\path[sample] (j20.east) edge [out=0, in=60] (b1.east);
\path[sample] (j20.east) edge [out=0, in=60] (b2.east);
\path[sample] (j20.east) edge [out=0, in=60] (b3.east);
\path[sample] (j20.east) edge [out=0, in=60] (b4.east);
\path[sample] (j20.east) edge [out=0, in=60] (bN.east);
\end{tikzpicture}
\end{center}
\vspace{\baselineskip}
\begin{center}
\begin{tikzpicture}
\definecolor{monred}{rgb}{0.79 0.0 0.1}
\tikzstyle{rec} = [rectangle,rounded corners,ultra thick,draw=monred!90,fill=white]
\node[rec,text=black!80,inner sep=3mm,drop shadow] (th)
{\parbox{0.6\textwidth}{
When N tend to infinity, the distribution of average values computed from bootstrap samples
is equal to the distribution of average values obtained from ALL samples with N elements
which can be constructed from the complete space. Thus the width of the distribution gives
an evaluation of the sample quality.}};
\node[rec,text=monred!95,anchor=south west,xshift=3mm,yshift=-1mm]
at (th.north west) {\small Theorem \scriptsize (B. Efron, Ann. Statist. 1979)};
\end{tikzpicture}
\end{center}
\end{preview}
\end{document}
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