added bootstrap of correlation coefficient

bootstrap/exercises/correlationbootstrap.m

@@ -0,0 +1,37 @@
+%% (a) bootstrap:
+nperm = 1000;
+rb = zeros(nperm,1);
+for i=1:nperm
+    % indices for resampling the data:
+    inx = randi(length(x), length(x), 1);
+    % resampled data pairs:
+    xb=x(inx);
+    yb=y(inx);
+    rb(i) = corr(xb, yb);
+end
+
+%% (b) pdf of the correlation coefficients:
+[hb,bb] = hist(rb, 20 );
+hb = hb/sum(hb)/(bb(2)-bb(1));  % normalization
+
+%% (c) significance:
+rbq = quantile(rb, 0.05);
+fprintf('correlation coefficient at 5%% significance = %.2f\n', rbq );
+if rbq > 0.0
+    fprintf('--> correlation r=%.2f is significant\n', rd);
+else
+    fprintf('--> r=%.2f is not a significant correlation\n', rd);
+end
+
+%% plot:
+hold on;
+bar(b, h, 'facecolor', [0.5 0.5 0.5]);
+bar(bb, hb, 'facecolor', 'b');
+bar(bb(bb<=rbq), hb(bb<=rbq), 'facecolor', 'r');
+plot( [rd rd], [0 4], 'r', 'linewidth', 2 );
+xlim([-0.25 0.75])
+xlabel('Correlation coefficient');
+ylabel('Probability density');
+hold off;
+
+savefigpdf( gcf, 'correlationbootstrap.pdf', 12, 6 );

bootstrap/exercises/exercises01.tex

@@ -148,32 +148,56 @@ distributed?
 
 
 \continue
-\question \qt{Permutation test}
+\question \qt{Permutation test} \label{permutationtest}
 We want to compute the significance of a correlation by means of a permutation test.
 \begin{parts}
-\part Generate 1000 correlated pairs $x$, $y$ of random numbers according to:
+  \part \label{permutationtestdata} Generate 1000 correlated pairs
+  $x$, $y$ of random numbers according to:
 \begin{verbatim}
 n = 1000
 a = 0.2;
 x = randn(n, 1);
 y = randn(n, 1) + a*x;
 \end{verbatim}
-\part Generate a scatter plot of the two variables.
-\part Why is $y$ correlated with $x$?
-\part Compute the correlation coefficient between $x$ and $y$.
-\part What do you need to do in order to destroy the correlations between the $x$-$y$ pairs?
-\part Do exactly this 1000 times and compute each time the correlation coefficient.
-\part Compute the probability density of these correlation coefficients.
-\part Is the correlation of the original data set significant?
-\part What does significance of the correlation mean?
-\part Vary the sample size \code{n} and compute in the same way the
-significance of the correlation.
+  \part Generate a scatter plot of the two variables.
+  \part Why is $y$ correlated with $x$?
+  \part Compute the correlation coefficient between $x$ and $y$.
+  \part What do you need to do in order to destroy the correlations between the $x$-$y$ pairs?
+  \part Do exactly this 1000 times and compute each time the correlation coefficient.
+  \part Compute and plot the probability density of these correlation
+  coefficients.
+  \part Is the correlation of the original data set significant?
+  \part What does significance of the correlation mean?
+  \part Vary the sample size \code{n} and compute in the same way the
+  significance of the correlation.
 \end{parts}
 \begin{solution}
   \lstinputlisting{correlationsignificance.m}
   \includegraphics[width=1\textwidth]{correlationsignificance}
 \end{solution}
 
+\question \qt{Bootstrap of the correlation coefficient} 
+The permutation test generates the distribution of the null hypothesis
+of uncorrelated data and we check whether the correlation coefficient
+of the data differs significantly from this
+distribution. Alternatively we can bootstrap the data while keeping
+the pairs and determine the confidence interval of the correlation
+coefficient of the data. If this differs significantly from a
+correlation coefficient of zero we can conclude that the correlation
+coefficient of the data quantifies indeed a correlated data.
+
+We take the same data set that we have generated in exercise
+\ref{permutationtest} (\ref{permutationtestdata}).
+\begin{parts}
+  \part Bootstrap 1000 times the correlation coefficient from the data.
+  \part Compute and plot the probability density of these correlation
+  coefficients.
+  \part Is the correlation of the original data set significant?
+\end{parts}
+\begin{solution}
+  \lstinputlisting{correlationbootstrap.m}
+  \includegraphics[width=1\textwidth]{correlationbootstrap}
+\end{solution}
 
 \end{questions}