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[Bootstrap] language fixes

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Jan Grewe 2019-10-12 12:01:18 +02:00
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bootstrap/lecture/bootstrap.tex
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@ -31,15 +31,15 @@ average length of all pickles (\figref{statisticalpopulationfig}). But
we cannot measure the lengths of all pickles in the statistical we cannot measure the lengths of all pickles in the statistical
population. Rather, we draw samples (simple random sample population. Rather, we draw samples (simple random sample
\enterm[SRS|see{simple random sample}]{SRS}, in German: \enterm[SRS|see{simple random sample}]{SRS}, in German:
\determ{Stichprobe}). We then estimate a statistical measures \determ{Stichprobe}). We then estimate a statistical measure of interest
(e.g. the average length of the pickles) within in this sample and (e.g. the average length of the pickles) within this sample and
hope that it is a good approximation of the unknown and immeasurable hope that it is a good approximation of the unknown and immeasurable
real average length of the statistical population (in German aka real average length of the statistical population (in German aka
\determ{Populationsparameter}). We apply statistical methods to find \determ{Populationsparameter}). We apply statistical methods to find
out how good this approximation is. out how precise this approximation is.
If we could draw a large number of \textit{simple random samples} we could If we could draw a large number of \textit{simple random samples} we could
estimate the statistical measure of interest for each sample and calculate the statistical measure of interest for each sample and
estimate the probability distribution using a histogram. This estimate the probability distribution using a histogram. This
distribution is called the \enterm{sampling distribution} (German: distribution is called the \enterm{sampling distribution} (German:
\determ{Stichprobenverteilung}, \determ{Stichprobenverteilung},
@ -85,16 +85,17 @@ of the statistical population. We can use the bootstrap distribution
to draw conclusion regarding the precision of our estimation (e.g. to draw conclusion regarding the precision of our estimation (e.g.
standard errors and confidence intervals). standard errors and confidence intervals).
Bootstrapping method create new SRS by resampling to estimate the Bootstrapping methods create bootstrapped samples from a SRS by
sampling distribution of a statistical measure. The bootstrapped resampling. The bootstrapped samples are used to estimate the sampling
samples have the same size as the original sample and are created by distribution of a statistical measure. The bootstrapped samples have
sampling with replacement, that is, each value of the original sample the same size as the original sample and are created by randomly drawing with
can occur once, multiple time, or not at all in a bootstrapped sample. replacement, that is, each value of the original sample can occur
once, multiple time, or not at all in a bootstrapped sample.
\section{Bootstrap of the standard error} \section{Bootstrap of the standard error}
Bootstrapping can be nicely illustrated at the example the standard Bootstrapping can be nicely illustrated at the example of the standard
error of the mean. The arithmetic mean is calculated for a simple error of the mean. The arithmetic mean is calculated for a simple
random sample. The standard error of the mean is the standard random sample. The standard error of the mean is the standard
deviation of the expected distribution of mean values around the mean deviation of the expected distribution of mean values around the mean
@ -120,13 +121,13 @@ distribution is the standard error of the mean.
\pagebreak[4] \pagebreak[4]
\begin{exercise}{bootstrapsem.m}{bootstrapsem.out} \begin{exercise}{bootstrapsem.m}{bootstrapsem.out}
Create the distribution of mean values from bootstrapped samples Create the distribution of mean values from bootstrapped samples
resampled form a single SRS. Use this distribution to estimate the resampled from a single SRS. Use this distribution to estimate the
standard error of the mean. standard error of the mean.
\begin{enumerate} \begin{enumerate}
\item Draw 1000 normally distributed random number and calculate the \item Draw 1000 normally distributed random number and calculate the
mean, the standard deviation and the standard error mean, the standard deviation, and the standard error
($\sigma/\sqrt{n}$). ($\sigma/\sqrt{n}$).
\item Resample the data 1000 times (draw and replace) and calculate \item Resample the data 1000 times (randomly draw and replace) and calculate
the mean of each bootstrapped sample. the mean of each bootstrapped sample.
\item Plot a histogram of the respective distribution and calculate its mean and \item Plot a histogram of the respective distribution and calculate its mean and
standard deviation. Compare with the standard deviation. Compare with the
@ -135,7 +136,7 @@ distribution is the standard error of the mean.
\end{exercise} \end{exercise}
\section{Permutationtests} \section{Permutation tests}
Statistical tests ask for the probability that a measured value Statistical tests ask for the probability that a measured value
originates from the null hypothesis. Is this probability smaller than originates from the null hypothesis. Is this probability smaller than
the desired significance level, the null hypothesis may be rejected. the desired significance level, the null hypothesis may be rejected.
@ -182,16 +183,16 @@ statistical significance (figure\,\ref{permutecorrelationfig}).
Estimate the statistical significance of a correlation coefficient. Estimate the statistical significance of a correlation coefficient.
\begin{enumerate} \begin{enumerate}
\item Create pairs of $(x_i, y_i)$ values. Randomly choose $x$-values \item Create pairs of $(x_i, y_i)$ values. Randomly choose $x$-values
and calculate the respective $y$-values according to $y=0.2 \cdot x$ and calculate the respective $y$-values according to $y_i =0.2 \cdot x_i + u_i$
to which you add a random value drawn from a normal distribution. where $u_i$ is a random number drawn from a normal distribution.
\item Calculate the correlation coefficient. \item Calculate the correlation coefficient.
\item Generate the distribution according to the null hypothesis by \item Generate the distribution according to the null hypothesis by
generating uncorrelated pairs. For this permute $x$- and $y$-values generating uncorrelated pairs. For this permute $x$- and $y$-values
(\matlabfun{randperm()}) 1000 times and calculate for each \matlabfun{randperm()} 1000 times and calculate for each
permutation the correlation coefficient. permutation the correlation coefficient.
\item From the resulting null hypothesis distribution the 95\,\% \item Read out the 95\,\% percentile from the resulting null
percentile and compare it with the correlation coefficient hypothesis distribution and compare it with the correlation
calculated for the original data. coefficient calculated for the original data.
\end{enumerate} \end{enumerate}
\end{exercise} \end{exercise}

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header.tex
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@ -5,7 +5,7 @@
\author{{\LARGE Jan Grewe \& Jan Benda}\\[5ex]Abteilung Neuroethologie\\[2ex]% \author{{\LARGE Jan Grewe \& Jan Benda}\\[5ex]Abteilung Neuroethologie\\[2ex]%
\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}\vspace{3ex}} \includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}\vspace{3ex}}
\date{WS 2018/2019\\\vfill% \date{WS 2019/2020\\\vfill%
\centerline{\includegraphics[width=0.7\textwidth]{announcements/correlationcartoon}% \centerline{\includegraphics[width=0.7\textwidth]{announcements/correlationcartoon}%
\rotatebox{90}{\footnotesize\url{www.xkcd.com}}}} \rotatebox{90}{\footnotesize\url{www.xkcd.com}}}}