reorganized statistics exercises
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34
bootstrap/exercises/Makefile
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34
bootstrap/exercises/Makefile
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TEXFILES=$(wildcard exercises??.tex)
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EXERCISES=$(TEXFILES:.tex=.pdf)
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SOLUTIONS=$(EXERCISES:exercises%=solutions%)
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.PHONY: pdf exercises solutions watch watchexercises watchsolutions clean
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pdf : $(SOLUTIONS) $(EXERCISES)
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exercises : $(EXERCISES)
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solutions : $(SOLUTIONS)
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$(SOLUTIONS) : solutions%.pdf : exercises%.tex instructions.tex
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{ echo "\\documentclass[answers,12pt,a4paper,pdftex]{exam}"; sed -e '1d' $<; } > $(patsubst %.pdf,%.tex,$@)
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pdflatex -interaction=scrollmode $(patsubst %.pdf,%.tex,$@) | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $(patsubst %.pdf,%.tex,$@) || true
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rm $(patsubst %.pdf,%,$@).[!p]*
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$(EXERCISES) : %.pdf : %.tex instructions.tex
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pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
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watch :
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while true; do ! make -q pdf && make pdf; sleep 0.5; done
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watchexercises :
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while true; do ! make -q exercises && make exercises; sleep 0.5; done
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watchsolutions :
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while true; do ! make -q solutions && make solutions; sleep 0.5; done
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clean :
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rm -f *~ *.aux *.log *.out
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cleanup : clean
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rm -f $(SOLUTIONS) $(EXERCISES)
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BIN
bootstrap/exercises/UT_WBMW_Black_RGB.pdf
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BIN
bootstrap/exercises/UT_WBMW_Black_RGB.pdf
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17
bootstrap/exercises/bootstrapmean.m
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17
bootstrap/exercises/bootstrapmean.m
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function [bootsem, mu] = bootstrapmean( x, resample )
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% computes standard error by bootstrapping the data
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% x: vector with data
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% resample: number of resamplings
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% returns:
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% bootsem: the standard error of the mean
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% mu: the bootstrapped means as a vector
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mu = zeros( resample, 1 );
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nsamples = length(x);
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for i = 1:resample
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% resample:
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xr = x(randi(nsamples, nsamples, 1));
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% compute statistics on sample:
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mu(i) = mean(xr);
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end
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bootsem = std( mu );
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end
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92
bootstrap/exercises/bootstraptymus-datahist.pdf
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92
bootstrap/exercises/bootstraptymus-datahist.pdf
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BIN
bootstrap/exercises/bootstraptymus-meanhist.pdf
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BIN
bootstrap/exercises/bootstraptymus-meanhist.pdf
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BIN
bootstrap/exercises/bootstraptymus-samples.pdf
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bootstrap/exercises/bootstraptymus-samples.pdf
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47
bootstrap/exercises/bootstraptymus.m
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47
bootstrap/exercises/bootstraptymus.m
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%% (b) load the data:
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load( 'thymusglandweights.dat' );
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nsamples = 80;
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x = thymusglandweights(1:nsamples);
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%% (c) mean, sem and hist:
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sem = std(x)/sqrt(nsamples);
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fprintf( 'Mean of the data set = %.2fmg\n', mean(x) );
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fprintf( 'SEM of the data set = %.2fmg\n', sem );
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hist(x,20)
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xlabel('x')
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ylabel('count')
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savefigpdf( gcf, 'bootstraptymus-datahist.pdf', 6, 5 );
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pause( 2.0 )
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%% (d) bootstrap the mean:
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resample = 500;
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[bootsem, mu] = bootstrapmean( x, resample );
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hist( mu, 20 );
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xlabel('mean(x)')
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ylabel('count')
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savefigpdf( gcf, 'bootstraptymus-meanhist.pdf', 6, 5 );
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fprintf( ' bootstrap standard error: %.3f\n', bootsem );
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fprintf( 'theoretical standard error: %.3f\n', sem );
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%% (e) confidence interval:
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q = quantile(mu, [0.025, 0.975]);
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fprintf( '95%% confidence interval of the mean from %.2fmg to %.2fmg\n', q(1), q(2) );
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pause( 2.0 )
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%% (f): dependence on sample size:
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nsamplesrange = 10:10:1000;
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bootsems = zeros( length(nsamplesrange),1);
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for n=1:length(nsamplesrange)
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nsamples = nsamplesrange(n);
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% [bootsems(n), mu] = bootstrapmean(x, resample);
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bootsems(n) = bootstrapmean(thymusglandweights(1:nsamples), resample);
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end
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plot(nsamplesrange, bootsems, 'b', 'linewidth', 2);
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hold on
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plot(nsamplesrange, std(x)./sqrt(nsamplesrange), 'r', 'linewidth', 1)
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hold off
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xlabel('sample size')
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ylabel('SEM')
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legend('bootsrap', 'theory')
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savefigpdf( gcf, 'bootstraptymus-samples.pdf', 6, 5 );
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58
bootstrap/exercises/correlationsignificance.m
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58
bootstrap/exercises/correlationsignificance.m
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%% (a) generate correlated data
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n=1000;
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a=0.2;
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x = randn(n, 1);
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y = randn(n, 1) + a*x;
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%% (b) scatter plot:
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subplot(1, 2, 1);
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plot(x, a*x, 'r', 'linewidth', 3 );
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hold on
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%scatter(x, y ); % either scatter ...
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plot(x, y, 'o', 'markersize', 2 ); % ... or plot - same plot.
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xlim([-4 4])
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ylim([-4 4])
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xlabel('x')
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ylabel('y')
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hold off
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%% (d) correlation coefficient:
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%c = corrcoef(x, y); % returns correlation matrix
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%rd = c(1, 2);
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rd = corr(x, y);
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fprintf('correlation coefficient = %.2f\n', rd );
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%% (e) permutation:
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nperm = 1000;
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rs = zeros(nperm,1);
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for i=1:nperm
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xr=x(randperm(length(x))); % shuffle x
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yr=y(randperm(length(y))); % shuffle y
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rs(i) = corr(xr, yr);
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end
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%% (g) pdf of the correlation coefficients:
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[h,b] = hist(rs, 20 );
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h = h/sum(h)/(b(2)-b(1)); % normalization
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%% (h) significance:
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rq = quantile(rs, 0.95);
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fprintf('correlation coefficient at 5%% significance = %.2f\n', rq );
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if rd >= rq
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fprintf('--> correlation r=%.2f is significant\n', rd);
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else
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fprintf('--> r=%.2f is not a significant correlation\n', rd);
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end
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%% plot:
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subplot(1, 2, 2)
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hold on;
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bar(b, h, 'facecolor', 'b');
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bar(b(b>=rq), h(b>=rq), 'facecolor', 'r');
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plot( [rd rd], [0 4], 'r', 'linewidth', 2 );
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xlim([-0.2 0.2])
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xlabel('Correlation coefficient');
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ylabel('Probability density of H0');
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hold off;
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savefigpdf( gcf, 'correlationsignificance.pdf', 12, 6 );
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BIN
bootstrap/exercises/correlationsignificance.pdf
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bootstrap/exercises/correlationsignificance.pdf
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163
bootstrap/exercises/exercises01.tex
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bootstrap/exercises/exercises01.tex
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\documentclass[12pt,a4paper,pdftex]{exam}
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\usepackage[german]{babel}
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\usepackage{pslatex}
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\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
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\usepackage{xcolor}
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\usepackage{graphicx}
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\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
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%%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
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\pagestyle{headandfoot}
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\ifprintanswers
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\newcommand{\stitle}{: L\"osungen}
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\else
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\newcommand{\stitle}{}
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\fi
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\header{{\bfseries\large \"Ubung 3\stitle}}{{\bfseries\large Statistik}}{{\bfseries\large 21. Oktober, 2015}}
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\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
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jan.benda@uni-tuebingen.de}
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\runningfooter{}{\thepage}{}
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\setlength{\baselineskip}{15pt}
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\setlength{\parindent}{0.0cm}
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\setlength{\parskip}{0.3cm}
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\renewcommand{\baselinestretch}{1.15}
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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\lstset{
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language=Matlab,
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basicstyle=\ttfamily\footnotesize,
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numbers=left,
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numberstyle=\tiny,
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title=\lstname,
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showstringspaces=false,
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commentstyle=\itshape\color{darkgray},
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breaklines=true,
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breakautoindent=true,
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columns=flexible,
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frame=single,
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xleftmargin=1em,
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xrightmargin=1em,
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aboveskip=10pt
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}
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%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{bm}
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\usepackage{dsfont}
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\newcommand{\naZ}{\mathds{N}}
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\newcommand{\gaZ}{\mathds{Z}}
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\newcommand{\raZ}{\mathds{Q}}
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\newcommand{\reZ}{\mathds{R}}
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\newcommand{\reZp}{\mathds{R^+}}
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\newcommand{\reZpN}{\mathds{R^+_0}}
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\newcommand{\koZ}{\mathds{C}}
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%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\continue}{\ifprintanswers%
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\else
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\vfill\hspace*{\fill}$\rightarrow$\newpage%
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\fi}
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\newcommand{\continuepage}{\ifprintanswers%
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\newpage
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\else
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\vfill\hspace*{\fill}$\rightarrow$\newpage%
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\fi}
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\newcommand{\newsolutionpage}{\ifprintanswers%
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\newpage%
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\else
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\fi}
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%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\qt}[1]{\textbf{#1}\\}
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\newcommand{\pref}[1]{(\ref{#1})}
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\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
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\newcommand{\code}[1]{\texttt{#1}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\input{instructions}
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\begin{questions}
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\question \qt{Bootstrap des Standardfehlers}
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\begin{parts}
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\part Lade von Ilias die Datei \code{thymusglandweights.dat} herunter.
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Darin befindet sich ein Datensatz vom Gewicht der Thymus Dr\"use in 14-Tage alten
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H\"uhnerembryos in mg.
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\part Lade diese Daten in Matlab (\code{load} Funktion).
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\part Bestimme Histogramm, Mittelwert und Standardfehler aus den ersten 80 Datenpunkten.
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\part Bestimme den Standardfehler aus den ersten 80 Datenpunkten durch 500-mal Bootstrappen.
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\part Bestimme das 95\,\% Konfidenzintervall f\"ur den Mittelwert
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aus der Bootstrap Verteilung (\code{quantile()} Funktion) --- also
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||||
das Interval innerhalb dessen mit 95\,\% Wahrscheinlichkeit der
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wahre Mittelwert liegen wird.
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\part Benutze den ganzen Datensatz und die Bootstrapping Technik, um die Abh\"angigkeit
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des Standardfehlers von der Stichprobengr\"o{\ss}e zu bestimmen.
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\part Vergleiche mit der bekannten Formel f\"ur den Standardfehler $\sigma/\sqrt{n}$.
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||||
\end{parts}
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\begin{solution}
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\lstinputlisting{bootstrapmean.m}
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\lstinputlisting{bootstraptymus.m}
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\includegraphics[width=0.5\textwidth]{bootstraptymus-datahist}
|
||||
\includegraphics[width=0.5\textwidth]{bootstraptymus-meanhist}
|
||||
\includegraphics[width=0.5\textwidth]{bootstraptymus-samples}
|
||||
\end{solution}
|
||||
|
||||
|
||||
\continue
|
||||
\question \qt{Student t-Verteilung}
|
||||
\begin{parts}
|
||||
\part Erzeuge 100000 normalverteilte Zufallszahlen.
|
||||
\part Ziehe daraus 1000 Stichproben vom Umfang $m=3$, 5, 10, oder 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}
|
||||
\begin{solution}
|
||||
\lstinputlisting{tdistribution.m}
|
||||
\includegraphics[width=1\textwidth]{tdistribution-n03}\\
|
||||
\includegraphics[width=1\textwidth]{tdistribution-n05}\\
|
||||
\includegraphics[width=1\textwidth]{tdistribution-n10}\\
|
||||
\includegraphics[width=1\textwidth]{tdistribution-n50}
|
||||
\end{solution}
|
||||
|
||||
|
||||
\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 die Stichprobengr\"o{\ss}e \code{n} und \"uberpr\"ufe
|
||||
auf gleiche Weise die Signifikanz.
|
||||
\end{parts}
|
||||
\begin{solution}
|
||||
\lstinputlisting{correlationsignificance.m}
|
||||
\includegraphics[width=1\textwidth]{correlationsignificance}
|
||||
\end{solution}
|
||||
|
||||
|
||||
\end{questions}
|
||||
|
||||
\end{document}
|
||||
41
bootstrap/exercises/instructions.tex
Normal file
41
bootstrap/exercises/instructions.tex
Normal file
@@ -0,0 +1,41 @@
|
||||
\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}
|
||||
|
||||
\ifprintanswers%
|
||||
\else
|
||||
|
||||
% Die folgenden Aufgaben dienen der Wiederholung, \"Ubung und
|
||||
% Selbstkontrolle und sollten eigenst\"andig bearbeitet und gel\"ost
|
||||
% werden. Die L\"osung soll in Form eines einzelnen Skriptes (m-files)
|
||||
% im ILIAS hochgeladen werden. Jede Aufgabe sollte in einer eigenen
|
||||
% ``Zelle'' gel\"ost sein. Die Zellen \textbf{m\"ussen} unabh\"angig
|
||||
% voneinander ausf\"uhrbar sein. Das Skript sollte nach dem Muster:
|
||||
% ``variablen\_datentypen\_\{nachname\}.m'' benannt werden
|
||||
% (z.B. variablen\_datentypen\_mueller.m).
|
||||
|
||||
\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 Aufgaben m\"oglichst in
|
||||
sinnvolle kleine Funktionen herunterzubrechen.
|
||||
Sobald etwas \"ahnliches mehr als einmal berechnet werden soll,
|
||||
lohnt es sich eine Funktion daraus zu schreiben!
|
||||
\item Teste rechenintensive \code{for} Schleifen, Vektoren, Matrizen
|
||||
zuerst mit einer kleinen Anzahl von Wiederholungen oder kleiner
|
||||
Gr\"o{\ss}e, und benutze erst am Ende, wenn alles \"uberpr\"uft
|
||||
ist, eine gro{\ss}e Anzahl von Wiederholungen oder Elementen, um eine gute
|
||||
Statistik zu bekommen.
|
||||
\item Benutze die Hilfsfunktion von \code{matlab} (\code{help
|
||||
commando} oder \code{doc commando}) und das Internet, um
|
||||
herauszufinden, wie bestimmte \code{matlab} Funktionen zu verwenden
|
||||
sind und was f\"ur M\"oglichkeiten sie bieten.
|
||||
Auch zu inhaltlichen Konzepten bietet das Internet oft viele
|
||||
Antworten!
|
||||
\end{itemize}
|
||||
|
||||
\fi
|
||||
28
bootstrap/exercises/savefigpdf.m
Normal file
28
bootstrap/exercises/savefigpdf.m
Normal file
@@ -0,0 +1,28 @@
|
||||
function savefigpdf( fig, name, width, height )
|
||||
% Saves figure fig in pdf file name.pdf with appropriately set page size
|
||||
% and fonts
|
||||
|
||||
% default width:
|
||||
if nargin < 3
|
||||
width = 11.7;
|
||||
end
|
||||
% default height:
|
||||
if nargin < 4
|
||||
height = 9.0;
|
||||
end
|
||||
|
||||
% paper:
|
||||
set( fig, 'PaperUnits', 'centimeters' );
|
||||
set( fig, 'PaperSize', [width height] );
|
||||
set( fig, 'PaperPosition', [0.0 0.0 width height] );
|
||||
set( fig, 'Color', 'white')
|
||||
|
||||
% font:
|
||||
set( findall( fig, 'type', 'axes' ), 'FontSize', 12 )
|
||||
set( findall( fig, 'type', 'text' ), 'FontSize', 12 )
|
||||
|
||||
% save:
|
||||
saveas( fig, name, 'pdf' )
|
||||
|
||||
end
|
||||
|
||||
BIN
bootstrap/exercises/tdistribution-n03.pdf
Normal file
BIN
bootstrap/exercises/tdistribution-n03.pdf
Normal file
Binary file not shown.
BIN
bootstrap/exercises/tdistribution-n05.pdf
Normal file
BIN
bootstrap/exercises/tdistribution-n05.pdf
Normal file
Binary file not shown.
BIN
bootstrap/exercises/tdistribution-n10.pdf
Normal file
BIN
bootstrap/exercises/tdistribution-n10.pdf
Normal file
Binary file not shown.
BIN
bootstrap/exercises/tdistribution-n50.pdf
Normal file
BIN
bootstrap/exercises/tdistribution-n50.pdf
Normal file
Binary file not shown.
58
bootstrap/exercises/tdistribution.m
Normal file
58
bootstrap/exercises/tdistribution.m
Normal file
@@ -0,0 +1,58 @@
|
||||
%% (a) generate random numbers:
|
||||
n = 100000;
|
||||
x=randn(n, 1);
|
||||
|
||||
for nsamples=[3 5 10 50]
|
||||
nsamples
|
||||
%% compute mean, standard deviation and t:
|
||||
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
|
||||
|
||||
% Gaussian pdfs:
|
||||
msdev = std(means);
|
||||
tsdev = 1.0;
|
||||
dxg=0.01;
|
||||
xmax = 10.0;
|
||||
xmin = -xmax;
|
||||
xg = [xmin:dxg:xmax];
|
||||
pm = exp(-0.5*(xg/msdev).^2)/sqrt(2.0*pi)/msdev;
|
||||
pt = exp(-0.5*(xg/tsdev).^2)/sqrt(2.0*pi)/tsdev;
|
||||
|
||||
%% plots
|
||||
subplot(1, 2, 1)
|
||||
bins = xmin:0.2:xmax;
|
||||
[h,b] = hist(means, bins);
|
||||
h = h/sum(h)/(b(2)-b(1));
|
||||
bar(b, h, 'facecolor', 'b', 'edgecolor', 'b')
|
||||
hold on
|
||||
plot(xg, pm, 'r', 'linewidth', 2)
|
||||
title( sprintf('sample size = %d', nsamples) );
|
||||
xlim( [-3, 3] );
|
||||
xlabel('Mean');
|
||||
ylabel('pdf');
|
||||
hold off;
|
||||
|
||||
subplot(1, 2, 2)
|
||||
bins = xmin:0.5:xmax;
|
||||
[h,b] = hist(students, bins);
|
||||
h = h/sum(h)/(b(2)-b(1));
|
||||
bar(b, h, 'facecolor', 'b', 'edgecolor', 'b')
|
||||
hold on
|
||||
plot(xg, pt, 'r', 'linewidth', 2)
|
||||
title( sprintf('sample size = %d', nsamples) );
|
||||
xlim( [-8, 8] );
|
||||
xlabel('Student-t');
|
||||
ylabel('pdf');
|
||||
hold off;
|
||||
|
||||
savefigpdf( gcf, sprintf('tdistribution-n%02d.pdf', nsamples), 14, 5 );
|
||||
pause( 3.0 )
|
||||
end
|
||||
10000
bootstrap/exercises/thymusglandweights.dat
Normal file
10000
bootstrap/exercises/thymusglandweights.dat
Normal file
File diff suppressed because it is too large
Load Diff
Reference in New Issue
Block a user