Merge branch 'master' of raven.am28.uni-tuebingen.de:scientificComputing

statistics/exercises/bootstrapmean.m

@@ -0,0 +1,17 @@
+function [bootsem, mu] = bootstrapmean( x, resample )
+% computes standard error by bootstrapping the data
+% x: vector with data
+% resample: number of resamplings
+% returns:
+% bootsem: the standard error of the mean
+% mu: the bootstrapped means as a vector
+    mu = zeros( resample, 1 );
+    nsamples = length(x);
+    for i = 1:resample
+        % resample:
+        xr = x(randi(nsamples, nsamples, 1));
+        % compute statistics on sample:
+        mu(i) = mean(xr);
+    end
+    bootsem = std( mu );
+end

statistics/exercises/bootstraptymus-datahist.pdf

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statistics/exercises/bootstraptymus.m

@@ -1,24 +1,47 @@
-resample = 500
+%% (b) load the data:
-
 load( 'thymusglandweights.dat' );
-x = thymusglandweights;
+nsamples = 80;
-nsamples = length( x );
+x = thymusglandweights(1:nsamples);
+
+%% (c) mean, sem and hist:
 sem = std(x)/sqrt(nsamples);
+fprintf( 'Mean of the data set = %.2fmg\n', mean(x) );
+fprintf( 'SEM of the data set = %.2fmg\n', sem );
+hist(x,20)
+xlabel('x')
+ylabel('count')
+savefigpdf( gcf, 'bootstraptymus-datahist.pdf', 6, 5 );
+pause( 2.0 )
 
-mu = zeros( resample, 1 );
+%% (d) bootstrap the mean:
-for i = 1:resample
+resample = 500;
-  % resample:
+[bootsem, mu] = bootstrapmean( x, resample );
-  xr = x(randi(nsamples, nsamples, 1));
+hist( mu, 20 );
-  % compute statistics on sample:
+xlabel('mean(x)')
-  mu(i) = mean(xr);
+ylabel('count')
-end
+savefigpdf( gcf, 'bootstraptymus-meanhist.pdf', 6, 5 );
-bootsem = std( mu );
+fprintf( '  bootstrap standard error: %.3f\n', bootsem );
+fprintf( 'theoretical standard error: %.3f\n', sem );
 
+%% (e) confidence interval:
+q = quantile(mu, [0.025, 0.975]);
+fprintf( '95%% confidence interval of the mean from %.2fmg to %.2fmg\n', q(1), q(2) );
+pause( 2.0 )
+
+%% (f): dependence on sample size:
+nsamplesrange = 10:10:1000;
+bootsems = zeros( length(nsamplesrange),1);
+for n=1:length(nsamplesrange)
+    nsamples = nsamplesrange(n);
+    % [bootsems(n), mu] = bootstrapmean(x, resample);
+    bootsems(n) = bootstrapmean(thymusglandweights(1:nsamples), resample);
+end
+plot(nsamplesrange, bootsems, 'b', 'linewidth', 2);
 hold on
-hist( x, 20 );
+plot(nsamplesrange, std(x)./sqrt(nsamplesrange), 'r', 'linewidth', 1)
-hist( mu, 20 );
 hold off
-
+xlabel('sample size')
-disp(['bootstrap standard error: ', num2str(bootsem)]);
+ylabel('SEM')
-disp(['standard error: ', num2str(sem)]);
+legend('bootsrap', 'theory')
+savefigpdf( gcf, 'bootstraptymus-samples.pdf', 6, 5 );
 

statistics/exercises/centrallimit.m

@@ -2,7 +2,7 @@
 n = 10000;
 m = 10;  % number of loops
 
-%% (b) a single random number:
+%% (b) a single data set of random numbers:
 x = rand( n, 1 );
 
 %% (c) plot probability density:
@@ -41,7 +41,7 @@ for i=1:m
     %xx = min(x):0.01:max(x);
     xx = -1:0.01:i+1;   % x-axis values for plot of pdf
     p = exp(-0.5*(xx-mu).^2/sd^2)/sqrt(2*pi*sd^2);  % pdf
-    plot(xx, p, 'r', 'linewidth', 6 )
+    plot(xx, p, 'r', 'linewidth', 3 )
     ns = sprintf( 'N=%d', i );
     text( 0.1, 0.9, ns, 'units', 'normalized' )
     hold on
@@ -50,8 +50,10 @@ for i=1:m
     h = h/sum(h)/(b(2)-b(1));  % normalization
     bar(b, h)
     hold off
+    xlim([-0.5, i+0.5])
     xlabel( 'x' )
     ylabel( 'summed pdf' )
+    savefigpdf( gcf, sprintf('centrallimit-hist%02d.pdf', i), 6, 5 );
     if i < 6
         pause( 3.0 )
     end
@@ -68,5 +70,6 @@ plot( xx, sqrt(xx)*sdu, 'k' )
 legend( 'mean', 'std', 'theory' )
 xlabel('N')
 hold off
+savefigpdf( gcf, 'centrallimit-samples.pdf', 6, 5 );
 
 

statistics/exercises/correlationsignificance.m

@@ -6,22 +6,29 @@ y = randn(n, 1) + a*x;
 
 %% (b) scatter plot:
 subplot(1, 2, 1);
+plot(x, a*x, 'r', 'linewidth', 3 );
+hold on
 %scatter(x, y );   % either scatter ...
-plot(x, y, 'o' );  % ... or plot - same plot.
+plot(x, y, 'o', 'markersize', 2 );  % ... or plot - same plot.
+xlim([-4 4])
+ylim([-4 4])
+xlabel('x')
+ylabel('y')
+hold off
 
 %% (d) correlation coefficient:
+%c = corrcoef(x, y);  % returns correlation matrix
+%rd = c(1, 2);
 rd = corr(x, y);
-%rd = r(0, 1);
 fprintf('correlation coefficient = %.2f\n', rd );
 
-%% (f) permutation:
+%% (e) permutation:
 nperm = 1000;
 rs = zeros(nperm,1);
 for i=1:nperm
     xr=x(randperm(length(x)));  % shuffle x
     yr=y(randperm(length(y)));  % shuffle y
     rs(i) = corr(xr, yr);
-    %rs(i) = r(0,1);
 end
 
 %% (g) pdf of the correlation coefficients:
@@ -43,7 +50,9 @@ hold on;
 bar(b, h, 'facecolor', 'b');
 bar(b(b>=rq), h(b>=rq), 'facecolor', 'r');
 plot( [rd rd], [0 4], 'r', 'linewidth', 2 );
-xlabel('correlation coefficient');
+xlim([-0.2 0.2])
-ylabel('probability density');
+xlabel('Correlation coefficient');
+ylabel('Probability density of H0');
 hold off;
 
+savefigpdf( gcf, 'correlationsignificance.pdf', 12, 6 );

statistics/exercises/die1.m

@@ -18,10 +18,11 @@ for i =1:6
     P(i) = sum(x == i)/length(x);
 end
 subplot( 1, 2, 1 )
-plot( [0 7], [1/6 1/6], 'r', 'linewidth', 6 )
+plot( [0 7], [1/6 1/6], 'r', 'linewidth', 3 )
 hold on
 bar( P );
 hold off
+set(gca, 'XTick', 1:6 );
 xlim( [ 0 7 ] );
 xlabel('Eyes');
 ylabel('Probability');
@@ -35,5 +36,4 @@ diehist( x );
 x = randi( 8, 1, n );   % random numbers from 1 to 8
 x(x>6) = 6;             % set numbers 7 and 8 to 6
 diehist( x );
-
+savefigpdf(gcf, 'die1.pdf', 12, 5)
-

statistics/exercises/die2.m

@@ -17,8 +17,10 @@ s = std(P, 1);
 bar(m, 'facecolor', [0.8 0 0]);   % darker red
 hold on;
 errorbar(m, s, '.k', 'linewidth', 2 );   % k is black
+set(gca, 'XTick', 1:6 );
 xlim( [ 0, 7 ] );
+ylim( [ 0, 0.25])
 xlabel('Eyes');
 ylabel('Probability');
 hold off;
-
+savefigpdf(gcf, 'die2.pdf', 6, 5)

statistics/exercises/diehist.m

@@ -3,10 +3,11 @@ function diehist( x )
 % die.
     [h,b] = hist( x, 1:6 );
     h = h/sum(h);   % normalization
-    plot( [0 7], [1/6 1/6], 'r', 'linewidth', 6 )
+    plot( [0 7], [1/6 1/6], 'r', 'linewidth', 3 )
     hold on
     bar( b, h );
     hold off
+    set(gca, 'XTick', 1:6 );
     xlim( [ 0, 7 ] );
     xlabel('Eyes');
     ylabel('Probability');

statistics/exercises/normprobs.m

@@ -27,17 +27,20 @@ fprintf( 'Integral over the Gaussian pdf from -3 to 3 is %.4f\n\n', P );
 
 %% (e) probability of small ranges
 nr = 50;
+xmax = 3.0
 xs = zeros(nr, 1);  % size of integration interval
 Ps = zeros(nr, 1);  % storage
 for i = 1:nr
     % upper limit goes from 4.0 down to 0.0:
-    xupper = 3.0*(nr-i)/nr;
+    xupper = xmax*(nr-i)/nr;
     xs(i) = xupper;
     % integral from 0 to xupper:
     Ps(i) = sum(pg((xx>=0.0)&(xx<=xupper)))*dx;
 end
 plot( xs, Ps, 'linewidth', 3 )
+xlim([0 xmax])
 ylim([0 0.55])
 xlabel('Integration interval')
 ylabel('Probability')
 fprintf('The probability P(0.1234) = %.4f\n\n', sum(x == 0.1234)/length(x) );
+savefigpdf(gcf, 'normprobs.pdf', 12, 8);

statistics/exercises/randomwalkstatistics.m

@@ -11,9 +11,10 @@ for i=1:length(nwalks)
     end
     text( 0.05, 0.8, sprintf( 'N=%d', nwalks(i)), 'units', 'normalized' )
     xlabel( 'Number of steps' );
-    ylabel( 'Position of walker' )
+    ylabel( 'Position' )
     hold off;
 end
+savefigpdf( gcf, 'randomwalk-traces.pdf', 12, 16 );
 pause( 5.0 )
 
 nsteps = 100;
@@ -34,7 +35,10 @@ xx = 0:0.01:nsteps;
 plot( xx, sqrt(xx), 'k' )
 plot( xx, zeros(length(xx),1), 'k' )
 legend( 'mean', 'std', 'theory' )
+xlabel('Steps')
+ylabel('Position')
 hold off
+savefigpdf( gcf, 'randomwalk-stdev.pdf', 6, 5 );
 pause( 3.0 );
 
 %% (d) histograms:
@@ -47,5 +51,8 @@ for i = 1:length(tinx)
     hold on;
 end
 hold off;
-xlabel('Position of walker');
+xlabel('Position');
 ylabel('Probability density');
+xlim([-30 30])
+ylim([0 0.3])
+savefigpdf( gcf, 'randomwalk-hists.pdf', 6, 5 );

statistics/exercises/savefigpdf.m

@@ -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
+

statistics/exercises/statistics01.tex

@@ -115,6 +115,7 @@ Der Computer kann auch als W\"urfel verwendet werden!
   \lstinputlisting{rollthedie.m}
   \lstinputlisting{diehist.m}
   \lstinputlisting{die1.m}
+  \includegraphics[width=1\textwidth]{die1}
 \end{solution}
 
 
@@ -132,6 +133,7 @@ Wir werten nun das Verhalten mehrerer W\"urfel aus.
 \end{parts}
 \begin{solution}
   \lstinputlisting{die2.m}
+  \includegraphics[width=0.5\textwidth]{die2}
 \end{solution}
 
 
@@ -173,6 +175,7 @@ Mittelwert enthalten ist.
 \end{parts}
 \begin{solution}
   \lstinputlisting{normprobs.m}
+  \includegraphics[width=1\textwidth]{normprobs}
 \end{solution}
 
 

statistics/exercises/statistics02.tex

@@ -121,6 +121,11 @@ Den Zentralen Grenzwertsatz wollen wir uns im Folgenden veranschaulichen.
 \end{parts}
 \begin{solution}
   \lstinputlisting{centrallimit.m}
+  \includegraphics[width=0.5\textwidth]{centrallimit-hist01}
+  \includegraphics[width=0.5\textwidth]{centrallimit-hist02}
+  \includegraphics[width=0.5\textwidth]{centrallimit-hist03}
+  \includegraphics[width=0.5\textwidth]{centrallimit-hist05}
+  \includegraphics[width=0.5\textwidth]{centrallimit-samples}
 \end{solution}
 
 
@@ -147,6 +152,9 @@ Im folgenden wollen wir einige Eigenschaften des Random Walks bestimmen.
 \begin{solution}
   \lstinputlisting{randomwalk.m}
   \lstinputlisting{randomwalkstatistics.m}
+  \includegraphics[width=0.8\textwidth]{randomwalk-traces}\\
+  \includegraphics[width=0.5\textwidth]{randomwalk-stdev}
+  \includegraphics[width=0.5\textwidth]{randomwalk-hists}
 \end{solution}
 
 

statistics/exercises/statistics03.tex

@@ -103,19 +103,33 @@ jan.benda@uni-tuebingen.de}
   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}
+\begin{solution}
+  \lstinputlisting{bootstrapmean.m}
+  \lstinputlisting{bootstraptymus.m}
+  \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, 50).
+\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 $t=\bar x/(\sigma_x/\sqrt{m}$
+\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}
@@ -135,8 +149,13 @@ 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.
+\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}

statistics/exercises/tdistribution.m

@@ -1,32 +1,58 @@
-n = 100000
+%% (a) generate random numbers:
+n = 100000;
 x=randn(n, 1);
 
-nsamples = 3;
+for nsamples=[3 5 10 50]
-nmeans = 10000;
+    nsamples
-means = zeros( nmeans, 1 );
+    %% compute mean, standard deviation and t:
-sdevs = zeros( nmeans, 1 );
+    nmeans = 10000;
-students = zeros( nmeans, 1 );
+    means = zeros( nmeans, 1 );
-for i=1:nmeans
+    sdevs = zeros( nmeans, 1 );
-  sample = x(randi(n, nsamples, 1));
+    students = zeros( nmeans, 1 );
-  means(i) = mean(sample);
+    for i=1:nmeans
-  sdevs(i) = std(sample);
+        sample = x(randi(n, nsamples, 1));
-  students(i) = mean(sample)/std(sample)*sqrt(nsamples);
+        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
-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;
-