reorganized statistics exercises
This commit is contained in:
@@ -1,6 +1,6 @@
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TEXFILES=$(wildcard statistics??.tex)
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TEXFILES=$(wildcard exercises??.tex)
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EXERCISES=$(TEXFILES:.tex=.pdf)
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SOLUTIONS=$(EXERCISES:statistics%=solutions%)
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SOLUTIONS=$(EXERCISES:exercises%=solutions%)
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.PHONY: pdf exercises solutions watch watchexercises watchsolutions clean
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@@ -10,7 +10,7 @@ exercises : $(EXERCISES)
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solutions : $(SOLUTIONS)
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$(SOLUTIONS) : solutions%.pdf : statistics%.tex instructions.tex
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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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@@ -1,17 +0,0 @@
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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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xref
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startxref
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Binary file not shown.
Binary file not shown.
@@ -1,47 +0,0 @@
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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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@@ -1,58 +0,0 @@
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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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@@ -1,30 +0,0 @@
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% plot gamma pdfs:
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xx = 0.0:0.1:10.0;
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shapes = [ 1.0, 2.0, 3.0, 5.0];
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cc = jet(length(shapes) );
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for i=1:length(shapes)
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yy = gampdf(xx, shapes(i), 1.0);
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plot(xx, yy, '-', 'linewidth', 3, 'color', cc(i,:), ...
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'DisplayName', sprintf('s=%.0f', shapes(i)) );
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hold on;
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end
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% generate gamma distributed random numbers:
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n = 50;
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x = gamrnd(3.0, 1.0, n, 1);
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% histogram:
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[h,b] = hist(x, 15);
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h = h/sum(h)/(b(2)-b(1));
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bar(b, h, 1.0, 'DisplayName', 'data');
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% maximum likelihood estimate:
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p = mle(x, 'distribution', 'gamma');
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yy = gampdf(xx, p(1), p(2));
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plot(xx, yy, '-k', 'linewidth', 5, 'DisplayName', 'mle' );
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hold off;
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xlabel('x');
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ylabel('pdf');
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legend('show');
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savefigpdf(gcf, 'mlepdffit.pdf', 12, 8)
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Binary file not shown.
@@ -1,31 +0,0 @@
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m = 2.0; % slope
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sigma = 1.0; % standard deviation
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n = 100; % number of data pairs
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% data pairs:
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x = 5.0*rand(n, 1);
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y = m*x + sigma*randn(n, 1);
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% fit:
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slope = mleslope(x, y);
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fprintf('slopes:\n');
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fprintf('original = %.2f\n', m);
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fprintf(' fit = %.2f\n', slope);
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% lines:
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xx = 0.0:0.1:5.0; % x-axis values
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yorg = m*xx;
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yfit = slope*xx;
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% plot:
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plot(xx, yorg, '-r', 'linewidth', 5);
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hold on;
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plot(xx, yfit, '-g', 'linewidth', 2);
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plot(x, y, 'ob');
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hold off;
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legend('data', 'original', 'fit', 'Location', 'NorthWest');
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legend('boxoff')
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xlabel('x');
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ylabel('y');
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savefigpdf(gcf, 'mlepropfit.pdf', 12, 7);
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Binary file not shown.
@@ -1,6 +0,0 @@
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function slope = mleslope(x, y )
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% Compute the maximum likelihood estimate of the slope
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% of a line through the origin
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% given the data pairs in the vectors x and y.
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slope = sum(x.*y)/sum(x.*x);
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end
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@@ -1,30 +0,0 @@
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% draw random numbers:
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n = 50;
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mu = 3.0;
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sigma =2.0;
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x = randn(n,1)*sigma+mu;
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fprintf(' mean of the data is %.2f\n', mean(x))
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fprintf('standard deviation of the data is %.2f\n', std(x))
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% standard deviation as parameter:
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psigs = 1.0:0.01:3.0;
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% matrix with the probabilities for each x and psigs:
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lms = zeros(length(x), length(psigs));
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for i=1:length(psigs)
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psig = psigs(i);
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p = exp(-0.5*((x-mu)/psig).^2.0)/sqrt(2.0*pi)/psig;
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lms(:,i) = p;
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end
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lm = prod(lms, 1); % likelihood
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loglm = sum(log(lms), 1); % log likelihood
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% plot likelihood of standard deviation:
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subplot(1, 2, 1);
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plot(psigs, lm );
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xlabel('standard deviation')
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ylabel('likelihood')
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subplot(1, 2, 2);
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plot(psigs, loglm);
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xlabel('standard deviation')
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ylabel('log likelihood')
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savefigpdf(gcf, 'mlestd.pdf', 15, 5);
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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{\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}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\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}
|
||||
|
||||
%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\qt}[1]{\textbf{#1}\\}
|
||||
\newcommand{\pref}[1]{(\ref{#1})}
|
||||
\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\input{instructions}
|
||||
|
||||
|
||||
\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}
|
||||
\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, 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}
|
||||
@@ -1,192 +0,0 @@
|
||||
\documentclass[12pt,a4paper,pdftex]{exam}
|
||||
|
||||
\usepackage[german]{babel}
|
||||
\usepackage{pslatex}
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
\usepackage{xcolor}
|
||||
\usepackage{graphicx}
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
|
||||
\pagestyle{headandfoot}
|
||||
\ifprintanswers
|
||||
\newcommand{\stitle}{: L\"osungen}
|
||||
\else
|
||||
\newcommand{\stitle}{}
|
||||
\fi
|
||||
\header{{\bfseries\large \"Ubung 4\stitle}}{{\bfseries\large Statistik}}{{\bfseries\large 26. 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}
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
language=Matlab,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
numberstyle=\tiny,
|
||||
title=\lstname,
|
||||
showstringspaces=false,
|
||||
commentstyle=\itshape\color{darkgray},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\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}
|
||||
|
||||
%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\qt}[1]{\textbf{#1}\\}
|
||||
\newcommand{\pref}[1]{(\ref{#1})}
|
||||
\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\input{instructions}
|
||||
|
||||
|
||||
\begin{questions}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\question \qt{Maximum Likelihood der Standardabweichung}
|
||||
Wir wollen uns die Likelihood und die Log-Likelihood am Beispiel der
|
||||
Absch\"atzung der Standardabweichung verdeutlichen.
|
||||
\begin{parts}
|
||||
\part Ziehe $n=50$ normalverteilte Zufallsvariablen mit Mittelwert $\mu=3$
|
||||
und einer Standardabweichung $\sigma=2$.
|
||||
|
||||
\part
|
||||
Plotte die Likelihood (aus dem Produkt der Wahrscheinlichkeiten) und
|
||||
die Log-Likelihood (aus der Summe der logarithmierten
|
||||
Wahrscheinlichkeiten) f\"ur die Standardabweichung als Parameter. Vergleiche die
|
||||
Position der Maxima mit der aus den Daten berechneten Standardabweichung.
|
||||
|
||||
\part
|
||||
Erh\"ohe $n$ auf 1000. Was passiert mit der Likelihood, was mit der Log-Likelihood? Warum?
|
||||
\end{parts}
|
||||
\begin{solution}
|
||||
\lstinputlisting{mlestd.m}
|
||||
\includegraphics[width=1\textwidth]{mlestd}
|
||||
\end{solution}
|
||||
|
||||
\continue
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\question \qt{Maximum-Likelihood-Sch\"atzer einer Ursprungsgeraden}
|
||||
In der Vorlesung haben wir folgende Formel f\"ur die Maximum-Likelihood
|
||||
Absch\"atzung der Steigung $\theta$ einer Ursprungsgeraden durch $n$ Datenpunkte $(x_i|y_i)$ mit Standardabweichung $\sigma_i$ hergeleitet:
|
||||
\[\theta = \frac{\sum_{i=1}^n \frac{x_iy_i}{\sigma_i^2}}{ \sum_{i=1}^n
|
||||
\frac{x_i^2}{\sigma_i^2}} \]
|
||||
\begin{parts}
|
||||
\part \label{mleslopefunc} Schreibe eine Funktion, die in einem $x$ und einem
|
||||
$y$ Vektor die Datenpaare \"uberreicht bekommt und die Steigung der
|
||||
Ursprungsgeraden, die die Likelihood maximiert, zur\"uckgibt
|
||||
($\sigma=\text{const}$).
|
||||
|
||||
\part
|
||||
Schreibe ein Skript, das Datenpaare erzeugt, die um eine
|
||||
Ursprungsgerade mit vorgegebener Steigung streuen. Berechne mit der
|
||||
Funktion aus \pref{mleslopefunc} die Steigung aus den Daten,
|
||||
vergleiche mit der wahren Steigung, und plotte die urspr\"ungliche
|
||||
sowie die gefittete Gerade zusammen mit den Daten.
|
||||
|
||||
\part
|
||||
Ver\"andere die Anzahl der Datenpunkte, die Steigung, sowie die
|
||||
Streuung der Daten um die Gerade.
|
||||
\end{parts}
|
||||
\begin{solution}
|
||||
\lstinputlisting{mleslope.m}
|
||||
\lstinputlisting{mlepropfit.m}
|
||||
\includegraphics[width=1\textwidth]{mlepropfit}
|
||||
\end{solution}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\question \qt{Maximum-Likelihood-Sch\"atzer einer Wahrscheinlichkeitsdichtefunktion}
|
||||
Verschiedene Wahrscheinlichkeitsdichtefunktionen haben Parameter, die
|
||||
nicht so einfach wie der Mittelwert und die Standardabweichung einer
|
||||
Normalverteilung direkt aus den Daten berechnet werden k\"onnen. Solche Parameter
|
||||
m\"ussen dann aus den Daten mit der Maximum-Likelihood-Methode gefittet werden.
|
||||
|
||||
Um dies zu veranschaulichen ziehen wir uns diesmal nicht normalverteilte Zufallszahlen, sondern Zufallszahlen aus der Gamma-Verteilung.
|
||||
\begin{parts}
|
||||
\part
|
||||
Finde heraus welche \code{matlab} Funktion die
|
||||
Wahrscheinlichkeitsdichtefunktion (probability density function) der
|
||||
Gamma-Verteilung berechnet.
|
||||
|
||||
\part
|
||||
Plotte mit Hilfe dieser Funktion die Wahrscheinlichkeitsdichtefunktion
|
||||
der Gamma-Verteilung f\"ur verschiedene Werte des (positiven) ``shape'' Parameters.
|
||||
Den ``scale'' Parameter setzen wir auf Eins.
|
||||
|
||||
\part
|
||||
Finde heraus mit welcher Funktion Gammaverteilte Zufallszahlen in
|
||||
\code{matlab} gezogen werden k\"onnen. Erzeuge mit dieser Funktion
|
||||
50 Zufallszahlen mit einem der oben geplotteten ``shape'' Parameter.
|
||||
|
||||
\part
|
||||
Berechne und plotte ein normiertes Histogramm dieser Zufallszahlen.
|
||||
|
||||
\part
|
||||
Finde heraus mit welcher \code{matlab}-Funktion eine beliebige
|
||||
Verteilung (``distribution'') an die Zufallszahlen nach der
|
||||
Maximum-Likelihood Methode gefittet werden kann. Wie wird diese
|
||||
Funktion benutzt, um die Gammaverteilung an die Daten zu fitten?
|
||||
|
||||
\part
|
||||
Bestimme mit dieser Funktion die Parameter der Gammaverteilung aus
|
||||
den Zufallszahlen.
|
||||
|
||||
\part
|
||||
Plotte anschlie{\ss}end die Gammaverteilung mit den gefitteten
|
||||
Parametern.
|
||||
\end{parts}
|
||||
\begin{solution}
|
||||
\lstinputlisting{mlepdffit.m}
|
||||
\includegraphics[width=1\textwidth]{mlepdffit}
|
||||
\end{solution}
|
||||
|
||||
\end{questions}
|
||||
|
||||
\end{document}
|
||||
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@@ -1,58 +0,0 @@
|
||||
%% (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
|
||||
File diff suppressed because it is too large
Load Diff
Reference in New Issue
Block a user