Updated some files

statistics/code/bootstrapsem.m

@@ -1,23 +1,24 @@
-nsamples = 1000
+nsamples = 100;
-resample = 500
+nresamples = 1000;
-
+
-  x = randn( nsamples, 1 );
+% draw a SRS (simple random sample, "Stichprobe") from the population:
-sem = std(x)/sqrt(nsamples);
+x = randn( 1, nsamples );
-
+fprintf('%-30s  %-5s  %-5s  %-5s\n', '', 'mean', 'stdev', 'sem' )
-mu = zeros( resample, 1 );
+fprintf('%30s  %5.2f  %5.2f  %5.2f\n', 'single SRS', mean( x ), std( x ), std( x )/sqrt(nsamples) )
-for i = 1:resample
+
-  % resample:
+% bootstrap the mean:
-  xr = x(randi(nsamples, nsamples, 1));
+mus = zeros(nresamples,1);  % vector for storing the means
-  % compute statistics on sample:
+for i = 1:nresamples        % loop for generating the bootstraps
-  mu(i) = mean(xr);
+    inx = randi(nsamples, 1, nsamples);  % range, 1D-vector, number
-end
+    xr = x(inx);            % resample the original SRS
-bootsem = std( mu );
+    mus(i) = mean(xr);      % compute statistic of the resampled SRS
-
+end
-hold on
+fprintf('%30s  %5.2f  %5.2f  -\n', 'bootstrapped distribution', mean( mus ), std( mus ) )
-hist( x, 20 );
+
-hist( mu, 20 );
+% many SRS (we can do that with the random number generator, but not in real life!):
-hold off
+musrs = zeros(nresamples,1);      % vector for the means of each SRS
-
+for i = 1:nresamples
-disp(['bootstrap standard error: ', num2str(bootsem)]);
+    x = randn( 1, nsamples );     % draw a new SRS
-disp(['standard error: ', num2str(sem)]);
+    musrs(i) = mean( x );         % compute its mean
-
+end
+fprintf('%30s  %5.2f  %5.2f  -\n', 'sampling distribution', mean( musrs ), std( musrs ) )

statistics/code/gaussianbins.m

@@ -1,18 +1,26 @@
 x = randn( 100, 1 );   % generate some data
-bins1 = -4:2:4;        % large bins
+db1=2;
-bins2 = -4:0.5:4;      % small bins
+db2 = 0.5;
+bins1 = -4:db1:4;        % large bins
+bins2 = -4:db2:4;        % small bins
+[h1,b1] = hist(x,bins1);
+[h2,b2] = hist(x,bins2);
+
 subplot( 1, 2, 1 );
-hold on;
+bar(b1,hn1)
-hist( x, bins1 );
+hold on 
-hist( x, bins2 );
+bar(b2,hn2, 'facecolor', 'r' )
 xlabel('x')
-ylabel('Frequeny')
+ylabel('Frequency')
-hold off;
+hold off
-subplot( 1, 2, 2 );
+
-hold on;
+% normalize:
-% normalize to the rigtht bin size:
+hn1 = h1/sum(h1)/db1;
-hist( x, bins1, 1.0/(bins1(2)-bins1(1)) );
+hn2 = h2/sum(h2)/db2;
-hist( x, bins2, 1.0/(bins2(2)-bins2(1)) );
+subplot(  1, 2, 2 )
+bar(b1,hn1)
+hold on 
+bar(b2,hn2, 'facecolor', 'r' )
 xlabel('x')
 ylabel('Probability density')
-hold off;
+hold off

statistics/code/sprintfexamples.m

@@ -0,0 +1,48 @@
+% sprintf returns a string.
+% This string can be used to annotate plots using the text() function.
+s = sprintf( 'x=%f', pi )
+
+% fprintf writes directly to console (or into files).
+% for fprintf you usually want to add the line break '\n':
+
+% '%f' formats floating point numbers:
+fprintf( 'x=%f\n', pi )
+% The '%f' formatting string can be anywhere in the string:
+fprintf( 'x=%fms\n', pi )
+% There can be arbitrary many '%' formatting strings:
+fprintf( 'x=%fms, y=%fkHz\n', pi, 2*pi )
+% The '%' itself is generated by '%%':
+fprintf( 'x=%f%%\n', pi )
+% A point followed by a number sets the number of digits after the point:
+fprintf( 'x=%.2fms\n', pi )
+% The numbers are appropriately rounded:
+fprintf( 'x=%.3fms\n', pi )
+% A number right before the point sets the width of the generated output:
+fprintf( 'x=%10.3fms\n', pi )
+% '%e' also formats floating point numbers but forces to write in
+% exponential style:
+fprintf( 'x=%e\n', pi )
+% again, a point and number set the number of digits after the point.
+fprintf( 'x=%.1e\n', pi )
+% '%g% formats the floating point number to a given number of valid digits
+% (default is 5):
+fprintf( 'x=%g\n', pi )
+% The number of valid digits is not the number of digits after the point:
+fprintf( 'x=%.2g\n', pi )
+fprintf( 'x=%.2g\n', 10.123 )
+fprintf( 'x=%.2g\n', 18765.123 )
+fprintf( 'x=%.5g\n', 18765.123 )
+
+% '%d' formats integers:
+fprintf( 'x=%d\n', 5 )
+% the number defines the width of the output:
+fprintf( 'x=%3d\n', 5 )
+% precedig the width with a '0' fills up the space with leading zeros:
+fprintf( 'x=%03d\n', 5 )
+
+% '%s' formats a string:
+fprintf( 'x=%s\n', 'hallo' )
+% ... aligned to the right:
+fprintf( 'x=%10s\n', 'hallo' )
+% ... unless the width is negative:
+fprintf( 'x=%-10s!\n', 'hallo' )