From bb6453c183d3573e4fc72e07a423f70a80df2fc0 Mon Sep 17 00:00:00 2001 From: Jan Benda Date: Thu, 22 Oct 2015 15:40:29 +0200 Subject: [PATCH 1/2] Updated exercises code and added figures --- statistics/exercises/bootstrapmean.m | 17 ++++ .../exercises/bootstraptymus-datahist.pdf | 92 ++++++++++++++++++ .../exercises/bootstraptymus-meanhist.pdf | Bin 0 -> 3514 bytes .../exercises/bootstraptymus-samples.pdf | Bin 0 -> 4126 bytes statistics/exercises/bootstraptymus.m | 57 +++++++---- statistics/exercises/centrallimit-hist01.pdf | Bin 0 -> 4771 bytes statistics/exercises/centrallimit-hist02.pdf | Bin 0 -> 5053 bytes statistics/exercises/centrallimit-hist03.pdf | Bin 0 -> 5413 bytes statistics/exercises/centrallimit-hist04.pdf | Bin 0 -> 5559 bytes statistics/exercises/centrallimit-hist05.pdf | Bin 0 -> 5716 bytes statistics/exercises/centrallimit-hist06.pdf | Bin 0 -> 5811 bytes statistics/exercises/centrallimit-hist07.pdf | Bin 0 -> 5953 bytes statistics/exercises/centrallimit-hist08.pdf | Bin 0 -> 6136 bytes statistics/exercises/centrallimit-hist09.pdf | Bin 0 -> 6218 bytes statistics/exercises/centrallimit-hist10.pdf | Bin 0 -> 6258 bytes statistics/exercises/centrallimit-samples.pdf | Bin 0 -> 4792 bytes statistics/exercises/centrallimit.m | 7 +- .../exercises/correlationsignificance.m | 14 ++- statistics/exercises/die1.m | 6 +- statistics/exercises/die1.pdf | Bin 0 -> 3853 bytes statistics/exercises/die2.m | 4 +- statistics/exercises/die2.pdf | Bin 0 -> 3823 bytes statistics/exercises/diehist.m | 3 +- statistics/exercises/normprobs.m | 5 +- statistics/exercises/normprobs.pdf | Bin 0 -> 3488 bytes statistics/exercises/randomwalk-hists.pdf | Bin 0 -> 4323 bytes statistics/exercises/randomwalk-stdev.pdf | Bin 0 -> 8230 bytes statistics/exercises/randomwalk-traces.pdf | Bin 0 -> 263288 bytes statistics/exercises/randomwalkstatistics.m | 11 ++- statistics/exercises/savefigpdf.m | 28 ++++++ statistics/exercises/statistics01.tex | 3 + statistics/exercises/statistics02.tex | 8 ++ statistics/exercises/statistics03.tex | 25 ++++- statistics/exercises/tdistribution-n03.pdf | Bin 0 -> 9871 bytes statistics/exercises/tdistribution-n05.pdf | Bin 0 -> 9516 bytes statistics/exercises/tdistribution-n10.pdf | Bin 0 -> 9052 bytes statistics/exercises/tdistribution-n50.pdf | Bin 0 -> 8349 bytes statistics/exercises/tdistribution.m | 84 ++++++++++------ 38 files changed, 301 insertions(+), 63 deletions(-) create mode 100644 statistics/exercises/bootstrapmean.m create mode 100644 statistics/exercises/bootstraptymus-datahist.pdf create mode 100644 statistics/exercises/bootstraptymus-meanhist.pdf create mode 100644 statistics/exercises/bootstraptymus-samples.pdf create mode 100644 statistics/exercises/centrallimit-hist01.pdf create mode 100644 statistics/exercises/centrallimit-hist02.pdf create mode 100644 statistics/exercises/centrallimit-hist03.pdf create mode 100644 statistics/exercises/centrallimit-hist04.pdf create mode 100644 statistics/exercises/centrallimit-hist05.pdf create mode 100644 statistics/exercises/centrallimit-hist06.pdf create mode 100644 statistics/exercises/centrallimit-hist07.pdf create mode 100644 statistics/exercises/centrallimit-hist08.pdf create mode 100644 statistics/exercises/centrallimit-hist09.pdf create mode 100644 statistics/exercises/centrallimit-hist10.pdf create mode 100644 statistics/exercises/centrallimit-samples.pdf create mode 100644 statistics/exercises/die1.pdf create mode 100644 statistics/exercises/die2.pdf create mode 100644 statistics/exercises/normprobs.pdf create mode 100644 statistics/exercises/randomwalk-hists.pdf create mode 100644 statistics/exercises/randomwalk-stdev.pdf create mode 100644 statistics/exercises/randomwalk-traces.pdf create mode 100644 statistics/exercises/savefigpdf.m create mode 100644 statistics/exercises/tdistribution-n03.pdf create mode 100644 statistics/exercises/tdistribution-n05.pdf create mode 100644 statistics/exercises/tdistribution-n10.pdf create mode 100644 statistics/exercises/tdistribution-n50.pdf diff --git a/statistics/exercises/bootstrapmean.m b/statistics/exercises/bootstrapmean.m new file mode 100644 index 0000000..356c531 --- /dev/null +++ b/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 diff --git a/statistics/exercises/bootstraptymus-datahist.pdf b/statistics/exercises/bootstraptymus-datahist.pdf new file mode 100644 index 0000000..4cc71ac --- /dev/null +++ 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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 ); diff --git a/statistics/exercises/correlationsignificance.m b/statistics/exercises/correlationsignificance.m index 7a0f5f0..97c09e7 100644 --- a/statistics/exercises/correlationsignificance.m +++ b/statistics/exercises/correlationsignificance.m @@ -6,22 +6,27 @@ y = randn(n, 1) + a*x; %% (b) scatter plot: subplot(1, 2, 1); +plot(x, a*x, 'r', 'linewidth', 4 ); +hold on %scatter(x, y ); % either scatter ... plot(x, y, 'o' ); % ... or plot - same plot. +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: @@ -44,6 +49,7 @@ 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'); -ylabel('probability density'); +ylabel('probability density of H0'); hold off; +savefigpdf( gcf, 'correlationsignificance.pdf', 12, 7 ); diff --git a/statistics/exercises/die1.m b/statistics/exercises/die1.m index d407ff1..32ddad4 100644 --- a/statistics/exercises/die1.m +++ b/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) \ No newline at end of file diff --git a/statistics/exercises/die1.pdf b/statistics/exercises/die1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bc4ed165253526afcac0cb2ed18b301112165f45 GIT binary patch literal 3853 zcmb_fc~}$I7Pmf2HN~ZB^()nS5iwxRY}sQHfe;WO1z7}(7AMJsj9Dg51_|z4wOXx9 z)wZ;#pH;pVYg?BpYC(O^icg9>!HTq^mDQc6-~(!rcPAi6{hFASE7+pzR#R0yM%-a#T2FV$76QD{ylx 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-3,10 +3,11 @@ function diehist( x ) % die. [h,b] = hist( x, 1:6 ); h = h/sum(h); % normalization - plot( [0 7], [1/6 1/6], 'r', 'linewidth', 6 ) + plot( [0 7], [1/6 1/6], 'r', 'linewidth', 3 ) hold on bar( b, h ); hold off + set(gca, 'XTick', 1:6 ); xlim( [ 0, 7 ] ); xlabel('Eyes'); ylabel('Probability'); diff --git a/statistics/exercises/normprobs.m b/statistics/exercises/normprobs.m index c7e4f37..d858b27 100644 --- a/statistics/exercises/normprobs.m +++ b/statistics/exercises/normprobs.m @@ -27,17 +27,20 @@ fprintf( 'Integral over the Gaussian pdf from -3 to 3 is %.4f\n\n', P ); %% (e) probability of small ranges nr = 50; +xmax = 3.0 xs = zeros(nr, 1); % size of integration interval Ps = zeros(nr, 1); % storage for i = 1:nr % upper limit goes from 4.0 down to 0.0: - xupper = 3.0*(nr-i)/nr; + xupper = xmax*(nr-i)/nr; xs(i) = xupper; % integral from 0 to xupper: Ps(i) = sum(pg((xx>=0.0)&(xx<=xupper)))*dx; end plot( xs, Ps, 'linewidth', 3 ) +xlim([0 xmax]) ylim([0 0.55]) xlabel('Integration interval') ylabel('Probability') 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znL_k(VL%k%3|DtJ!yS#YM`4IKlC3NnN4CYtpis6%vaGBvMjA(PAW0*TGBVOIclz&6 za-k~xKKRKU{x6CD#)UsY!0rXS3I8)UD1gBKrRTrL=wAsDC_8HfNPk0MnVnG+)zja3WdZXa2RO-5A3%8g!)P9<|(18Y}1G1KVp4;Y8VqOo8K z_!kT-iw5c`^8wTTfJx&JKtcXH3@alIlsH3DcWHgX6C|Mj1(& bhBfnWMQ0E_89$vN4eWf7xVXBW2I&6)D_{7^ literal 0 HcmV?d00001 diff --git a/statistics/exercises/tdistribution.m b/statistics/exercises/tdistribution.m index 9d8920b..223cbe5 100644 --- a/statistics/exercises/tdistribution.m +++ b/statistics/exercises/tdistribution.m @@ -1,32 +1,58 @@ -n = 100000 +%% (a) generate random numbers: +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); +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 -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; - From 381542dad0a4b66ad64da86ca65060e72df83adf Mon Sep 17 00:00:00 2001 From: Jan Benda Date: Thu, 22 Oct 2015 15:56:16 +0200 Subject: [PATCH 2/2] Added figure for correlations --- statistics/exercises/correlationsignificance.m | 13 ++++++++----- .../exercises/correlationsignificance.pdf | Bin 0 -> 59521 bytes 2 files changed, 8 insertions(+), 5 deletions(-) create mode 100644 statistics/exercises/correlationsignificance.pdf diff --git a/statistics/exercises/correlationsignificance.m b/statistics/exercises/correlationsignificance.m index 97c09e7..013b8a9 100644 --- a/statistics/exercises/correlationsignificance.m +++ b/statistics/exercises/correlationsignificance.m @@ -6,10 +6,12 @@ y = randn(n, 1) + a*x; %% (b) scatter plot: subplot(1, 2, 1); -plot(x, a*x, 'r', 'linewidth', 4 ); +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 @@ -48,8 +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'); -ylabel('probability density of H0'); +xlim([-0.2 0.2]) +xlabel('Correlation coefficient'); +ylabel('Probability density of H0'); hold off; -savefigpdf( gcf, 'correlationsignificance.pdf', 12, 7 ); +savefigpdf( gcf, 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