From 2787e55094949b099fec2cf7a967ef27d5317cac Mon Sep 17 00:00:00 2001 From: Jan Benda Date: Tue, 20 Oct 2015 01:43:18 +0200 Subject: [PATCH] New bootstrap exercises --- statistics/code/bootstrapsem.m | 23 + statistics/code/bootstraptymus.m | 24 + statistics/code/boxwhisker.m | 3 + statistics/code/correlations.m | 14 + statistics/code/tdistribution.m | 32 + statistics/exercises/bootstrap-01.tex | 132 + statistics/exercises/thymusglandweights.dat | 10000 ++++++++++++++++ statistics/lecture/boxwhisker.py | 47 + statistics/lecture/boxwhiskerdata.npy | Bin 0 -> 3280 bytes statistics/lecture/descriptivestatistics.tex | 85 + .../figures/2012-10-29_16-26-05_771.jpg | Bin 0 -> 740827 bytes .../figures/2012-10-29_16-29-35_312.jpg | Bin 0 -> 395085 bytes .../figures/2012-10-29_16-41-39_523.jpg | Bin 0 -> 472511 bytes statistics/lecture/figures/srs1.png | Bin 0 -> 60051 bytes statistics/lecture/figures/srs2.png | Bin 0 -> 56148 bytes statistics/lecture/figures/srs3.png | Bin 0 -> 74342 bytes statistics/material/Hesterberg2003.pdf | Bin 0 -> 589657 bytes 17 files changed, 10360 insertions(+) create mode 100644 statistics/code/bootstrapsem.m create mode 100644 statistics/code/bootstraptymus.m create mode 100644 statistics/code/boxwhisker.m create mode 100644 statistics/code/correlations.m create mode 100644 statistics/code/tdistribution.m create mode 100644 statistics/exercises/bootstrap-01.tex create mode 100644 statistics/exercises/thymusglandweights.dat create mode 100644 statistics/lecture/boxwhisker.py create mode 100644 statistics/lecture/boxwhiskerdata.npy create mode 100755 statistics/lecture/figures/2012-10-29_16-26-05_771.jpg create mode 100755 statistics/lecture/figures/2012-10-29_16-29-35_312.jpg create mode 100755 statistics/lecture/figures/2012-10-29_16-41-39_523.jpg create mode 100644 statistics/lecture/figures/srs1.png create mode 100644 statistics/lecture/figures/srs2.png create mode 100644 statistics/lecture/figures/srs3.png create mode 100644 statistics/material/Hesterberg2003.pdf diff --git a/statistics/code/bootstrapsem.m b/statistics/code/bootstrapsem.m new file mode 100644 index 0000000..b5a7cbe --- /dev/null +++ b/statistics/code/bootstrapsem.m @@ -0,0 +1,23 @@ +nsamples = 1000 +resample = 500 + + x = randn( nsamples, 1 ); +sem = std(x)/sqrt(nsamples); + +mu = zeros( resample, 1 ); +for i = 1:resample + % resample: + xr = x(randi(nsamples, nsamples, 1)); + % compute statistics on sample: + mu(i) = mean(xr); +end +bootsem = std( mu ); + +hold on +hist( x, 20 ); +hist( mu, 20 ); +hold off + +disp(['bootstrap standard error: ', num2str(bootsem)]); +disp(['standard error: ', num2str(sem)]); + diff --git a/statistics/code/bootstraptymus.m b/statistics/code/bootstraptymus.m new file mode 100644 index 0000000..e92a5b7 --- /dev/null +++ b/statistics/code/bootstraptymus.m @@ -0,0 +1,24 @@ +resample = 500 + +load( 'thymusglandweights.dat' ); +x = thymusglandweights; +nsamples = length( x ); +sem = std(x)/sqrt(nsamples); + +mu = zeros( resample, 1 ); +for i = 1:resample + % resample: + xr = x(randi(nsamples, nsamples, 1)); + % compute statistics on sample: + mu(i) = mean(xr); +end +bootsem = std( mu ); + +hold on +hist( x, 20 ); +hist( mu, 20 ); +hold off + +disp(['bootstrap standard error: ', num2str(bootsem)]); +disp(['standard error: ', num2str(sem)]); + diff --git a/statistics/code/boxwhisker.m b/statistics/code/boxwhisker.m new file mode 100644 index 0000000..65e8f24 --- /dev/null +++ b/statistics/code/boxwhisker.m @@ -0,0 +1,3 @@ +x = randn(40, 10); +boxplot(x, 'whisker', 100.0 ); + diff --git a/statistics/code/correlations.m b/statistics/code/correlations.m new file mode 100644 index 0000000..7816457 --- /dev/null +++ b/statistics/code/correlations.m @@ -0,0 +1,14 @@ +n = 1000 +x = randn( n, 1 ); +y = randn( n, 1 ) + 0.2*x; +r = corr(x,y) + + nsamples = 500; + rs = zeros( nsamples, 1 ); +for i = 1:nsamples + xs = x(randi(n,n,1)); +ys = x(randi(n,n,1)); + rs(i) = corr(xs,ys); +end + +hist( rs, 20 ) diff --git a/statistics/code/tdistribution.m b/statistics/code/tdistribution.m new file mode 100644 index 0000000..9d8920b --- /dev/null +++ b/statistics/code/tdistribution.m @@ -0,0 +1,32 @@ +n = 100000 +x=randn(n, 1); + +nsamples = 3; +nmeans = 10000; +means = zeros( nmeans, 1 ); +sdevs = zeros( nmeans, 1 ); +students = zeros( nmeans, 1 ); +for i=1:nmeans + sample = x(randi(n, nsamples, 1)); + means(i) = mean(sample); + sdevs(i) = std(sample); + students(i) = mean(sample)/std(sample)*sqrt(nsamples); +end +sdev = 1.0 +msdev = std(means) + +% scatter( means, sdevs ) + +hold on; +dxg=0.01; +xmax = 10.0 +xmin = -xmax +xg = [xmin:dxg:xmax]; +pg = exp(-0.5*(xg/sdev).^2)/sqrt(2.0*pi)/sdev; +hold on +plot(xg, pg, 'r', 'linewidth', 4) + +bins = xmin:0.1:xmax; +hist(means, bins, 1.0/(bins(2)-bins(1)) ); +hold off; + diff --git a/statistics/exercises/bootstrap-01.tex b/statistics/exercises/bootstrap-01.tex new file mode 100644 index 0000000..a5860e1 --- /dev/null +++ b/statistics/exercises/bootstrap-01.tex @@ -0,0 +1,132 @@ +\documentclass[12pt,a4paper,pdftex]{exam} + +\usepackage[german]{babel} +\usepackage{natbib} +\usepackage{graphicx} +\usepackage[small]{caption} +\usepackage{sidecap} +\usepackage{pslatex} +\usepackage{amsmath} +\usepackage{amssymb} +\setlength{\marginparwidth}{2cm} +\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref} + +%%%%% text size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry} +\pagestyle{headandfoot} +\header{{\bfseries\large \"Ubung 1}}{{\bfseries\large Bootstrap}}{{\bfseries\large 21. Oktober, 2015}} +\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email: +jan.benda@uni-tuebingen.de} +\runningfooter{}{\thepage}{} + +\setlength{\baselineskip}{15pt} +\setlength{\parindent}{0.0cm} +\setlength{\parskip}{0.3cm} +\renewcommand{\baselinestretch}{1.15} + +\newcommand{\qt}[1]{\textbf{#1}\\} +\newcommand{\pref}[1]{(\ref{#1})} +\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}} +\newcommand{\code}[1]{\texttt{#1}} + +\newcommand{\continue}{\ifprintanswers% +\else +\vfill\hspace*{\fill}$\rightarrow$\newpage% +\fi} +\newcommand{\continuepage}{\ifprintanswers% +\newpage +\else +\vfill\hspace*{\fill}$\rightarrow$\newpage% +\fi} +\newcommand{\newsolutionpage}{\ifprintanswers% +\newpage% +\else +\fi} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\vspace*{-6.5ex} +\begin{center} +\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[1ex] +{\large Jan Grewe, Jan Benda}\\[-3ex] +Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\ +\end{center} + +\begin{itemize} +\item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass sie +auch wirklich das macht, was sie machen soll! Teste dies mit kleinen +Beispielen direkt in der Kommandozeile. +\item Versuche die L\"osungen der folgenden Aufgaben m\"oglichst in +sinnvolle kleine Funktionen herunterzubrechen. +\item Sobald etwas \"ahnliches mehr als einmal berechnet werden soll, +lohnt es sich eine Funktion daraus zu schreiben! +\item Teste rechenintensive \code{for} Schleifen zuerst mit einer kleinen +Anzahl von Wiederholungen, und benutze erst am Ende, wenn alles +stimmt, eine gro{\ss}e Anzahl von Wiederholungen, um eine gute +Statistik zu bekommen. +\item Benutze die Hilfsfunktion von matlab und das Internet, um +herauszufinden wie bestimmte \code{matlab} Funktionen zu verwenden +sind und was f\"ur M\"oglichkeiten sie bieten. +\item Auch zu inhaltlichen Konzepten bietet das Internet oft viele Antworten! +\end{itemize} + + +\begin{questions} + +\question \qt{Bootstrap des Standardfehlers} +\begin{parts} + \part Lade von Ilias die Datei \code{thymusglandweights.dat} herunter. + Darin befindet sich ein Datensatz vom Gewicht der Thymus Dr\"use in 14-Tage alten + H\"uhnerembryos in mg. + \part Lade diese Daten in Matlab (\code{load} Funktion). + \part Bestimme Histogramm, Mittelwert und Standardfehler aus den ersten 80 Datenpunkten. + \part Bestimme den Standardfehler aus den ersten 80 Datenpunkten durch 500 Mal Bootstrappen. + \part Bestimme das 95\,\% Konfidenzintervall f\"ur den Mittelwert + aus der Bootstrap Verteilung (\code{quantile()} Funktion) --- also + das Interval innerhalb dessen mit 95\,\% Wahrscheinlichkeit der + wahre Mittelwert liegen wird. + \part Benutze den ganzen Datensatz und die Bootstrapping Technik, um die Abh\"angigkeit + des Standardfehlers von der Stichprobengr\"o{\ss}e zu bestimmen. + \part Vergleiche mit der bekannten Formel f\"ur den Standardfehler $\sigma/\sqrt{n}$. +\end{parts} + + +\continue +\question \qt{Student t-Verteilung} +\begin{parts} +\part Erzeuge 100000 normalverteilte Zufallszahlen. +\part Ziehe daraus 1000 Stichproben vom Umfang $m$ (3, 5, 10, 50). +\part Berechne den Mittelwert $\bar x$ der Stichproben und plotte die Wahrscheinlichkeitsdichte +dieser Mittelwerte. +\part Vergleiche diese Wahrscheinlichkeitsdichte mit der Gausskurve. +\part Berechne ausserdem die Gr\"o{\ss}e $t=\bar x/(\sigma_x/\sqrt{m}$ +(Standardabweichung $\sigma_x$) und vergleiche diese mit der Normalverteilung mit Standardabweichung Eins. Ist $t$ normalverteilt, bzw. unter welchen Bedingungen ist $t$ Normalverteilt? +\end{parts} + + +\question \qt{Korrelationen} +\begin{parts} +\part Erzeuge 1000 korrelierte Zufallszahlen $x$, $y$ durch +\begin{verbatim} +n = 1000 +a = 0.2; +x = randn(n, 1); +y = randn(n, 1) + a*x; +\end{verbatim} +\part Erstelle einen Scatterplot der beiden Variablen. +\part Warum ist $y$ mit $x$ korreliert? +\part Berechne den Korrelationskoeffizienten zwischen $x$ und $y$. +\part Was m\"usste man tun, um die Korrelationen zwischen den $x$-$y$ +Paaren zu zerst\"oren? +\part Mach genau dies 1000 mal und berechne jedes Mal den Korrelationskoeffizienten. +\part Bestimme die Wahrscheinlichkeitsdichte dieser Korrelationskoeffizienten. +\part Ist die Korrelation der urspr\"unglichen Daten signifikant? +\part Variiere den Parameter $a$ und \"uberpr\"ufe auf gleiche Weise die Signifikanz. +\end{parts} + + +\end{questions} + +\end{document} \ No newline at end of file diff --git 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+2.857215052708918890e+01 +3.231958561935496022e+01 +3.412899323633600090e+01 +4.082470280559310538e+01 +4.461123475557004525e+01 +4.558433869366629665e+01 +4.291763671642394939e+01 +4.591699280429043739e+01 +4.120022785530896670e+01 +4.540300034573863286e+01 +2.392026459841191866e+01 +4.761895006540471087e+01 +3.881330900350112501e+01 +2.273765697154177445e+01 diff --git a/statistics/lecture/boxwhisker.py b/statistics/lecture/boxwhisker.py new file mode 100644 index 0000000..1209f7e --- /dev/null +++ b/statistics/lecture/boxwhisker.py @@ -0,0 +1,47 @@ +import numpy as np +import matplotlib.pyplot as plt + +#x = np.random.randn( 40, 10 ) +#np.save('boxwhiskerdata', x ) +x = np.load('boxwhiskerdata.npy') + +plt.xkcd() +fig = plt.figure( figsize=(6,4) ) +ax = fig.add_subplot( 1, 1, 1 ) +ax.spines['right'].set_visible(False) +ax.spines['top'].set_visible(False) +ax.yaxis.set_ticks_position('left') +ax.xaxis.set_ticks_position('bottom') +ax.set_xlabel('Experiment') +ax.set_ylabel('x') +ax.set_ylim( -4.0, 4.0) +ax.annotate('Median', + xy=(3.9, 0.1), xycoords='data', + xytext=(3.5, -2.5), textcoords='data', ha='right', + arrowprops=dict(arrowstyle="->", relpos=(0.8,1.0), + connectionstyle="angle3,angleA=-110,angleB=60") ) +ax.annotate('1. quartile', + xy=(5.8, -0.7), xycoords='data', + xytext=(5.5, -3.5), textcoords='data', ha='right', + arrowprops=dict(arrowstyle="->", relpos=(0.5,1.0), + connectionstyle="angle3,angleA=30,angleB=70") ) +ax.annotate('3. quartile', + xy=(6.1, 0.6), xycoords='data', + xytext=(6.5, 3.0), textcoords='data', ha='left', + arrowprops=dict(arrowstyle="->", relpos=(0.0,0.0), + connectionstyle="angle3,angleA=30,angleB=70") ) +ax.annotate('minimum', + xy=(6.1, -2.3), xycoords='data', + xytext=(7.2, -3.3), textcoords='data', ha='left', + arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5), + connectionstyle="angle3,angleA=10,angleB=100") ) +ax.annotate('maximum', + xy=(5.9, 2.8), xycoords='data', + xytext=(4.9, 3.5), textcoords='data', ha='right', + arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5), + connectionstyle="angle3,angleA=0,angleB=120") ) +ax.boxplot( x, whis=100.0 ) +plt.tight_layout() +plt.savefig('boxwhisker.pdf') +plt.show() + diff --git a/statistics/lecture/boxwhiskerdata.npy b/statistics/lecture/boxwhiskerdata.npy new file mode 100644 index 0000000000000000000000000000000000000000..6751afa775ea24f0c9cf7a38661516374ed7f54a GIT binary patch literal 3280 zcmXX|c{mh`+Z{`VY;}b;q(#=7rIhVejnpkm_DU&BCCXBbgv$-3EH^5ZB?;Fqlsi=J zJCQBQGLxk##@I3jGt3xSe|^vMo&V1H@0{nn?{gklJ6hSE5F#8Rgs6FXUG)f5Gg47A zIlos;OGWLxU*NUCb64E_0zJL{Pd{|d_o~+~{p!VY{$9WMPJ>-qD!RMWwN%1X{vQ%f ze}sF_M$TX(>(4un^Il+;vT=SevOwH4Y;UM%8(d$@FU-Et1}EUr(4&jO6bX6l^LNY# zQSyqh*?tBSo_)By`?dxH{`1ysE+y2#bBDE7+1x>h@a~z8(q>>yN`Y#ePa8O2p5$kI z=)$C2@m68aP9T%ckldb+z@&8cRSac9?;f8xkmY0V@+k55<$lgdtd zMMx>SUiD0`2AU<+L_~g6V}wj<%wbI)UL2~(i2Zv2pUSUp z*f9lnYr1QRdw3{GO-=Yqbr@P5Mbj|x*h83N(ok7t!7lYdKiqUOs_T}ifgBOr>f#nM zM8*+)BXcHTp 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