new command \endeterm for english terms that also make an entry into the german index - not working yet
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@ -33,10 +33,11 @@ population. Rather, we draw samples (\enterm{simple random sample}
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then estimate a statistical measure of interest (e.g. the average
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length of the pickles) within this sample and hope that it is a good
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approximation of the unknown and immeasurable true average length of
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the population (\determ{Populationsparameter}). We apply statistical
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methods to find out how precise this approximation is.
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the population (\endeterm{Populationsparameter}{population
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parameter}). We apply statistical methods to find out how precise
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this approximation is.
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If we could draw a large number of \enterm{simple random samples} we
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If we could draw a large number of simple random samples we
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could calculate the statistical measure of interest for each sample
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and estimate its probability distribution using a histogram. This
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distribution is called the \enterm{sampling distribution}
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@ -69,17 +70,18 @@ error of the mean which is the standard deviation of the sampling
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distribution of average values around the true mean of the population
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(\subfigref{bootstrapsamplingdistributionfig}{b}).
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Alternatively, we can use ``bootstrapping'' to generate new samples
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from one set of measurements (resampling). From these bootstrapped
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samples we compute the desired statistical measure and estimate their
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distribution (\enterm{bootstrap distribution},
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\subfigref{bootstrapsamplingdistributionfig}{c}). Interestingly, this
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distribution is very similar to the sampling distribution regarding
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its width. The only difference is that the bootstrapped values are
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distributed around the measure of the original sample and not the one
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of the statistical population. We can use the bootstrap distribution
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to draw conclusion regarding the precision of our estimation (e.g.
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standard errors and confidence intervals).
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Alternatively, we can use \enterm{bootstrapping}
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(\determ{Bootstrap-Verfahren}) to generate new samples from one set of
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measurements (\endeterm{Resampling}{resampling}). From these
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bootstrapped samples we compute the desired statistical measure and
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estimate their distribution (\endeterm{Bootstrapverteilung}{bootstrap
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distribution}, \subfigref{bootstrapsamplingdistributionfig}{c}).
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Interestingly, this distribution is very similar to the sampling
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distribution regarding its width. The only difference is that the
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bootstrapped values are distributed around the measure of the original
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sample and not the one of the statistical population. We can use the
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bootstrap distribution to draw conclusion regarding the precision of
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our estimation (e.g. standard errors and confidence intervals).
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Bootstrapping methods create bootstrapped samples from a SRS by
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resampling. The bootstrapped samples are used to estimate the sampling
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@ -93,11 +95,12 @@ data set using the \code{randi()} function.
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\section{Bootstrap of the standard error}
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Bootstrapping can be nicely illustrated at the example of the standard
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error of the mean. The arithmetic mean is calculated for a simple
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random sample. The standard error of the mean is the standard
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deviation of the expected distribution of mean values around the mean
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of the statistical population.
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Bootstrapping can be nicely illustrated at the example of the
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\enterm{standard error} of the mean (\determ{Standardfehler}). The
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arithmetic mean is calculated for a simple random sample. The standard
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error of the mean is the standard deviation of the expected
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distribution of mean values around the mean of the statistical
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population.
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\begin{figure}[tp]
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\includegraphics[width=1\textwidth]{bootstrapsem}
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@ -135,9 +138,10 @@ distribution is the standard error of the mean.
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\section{Permutation tests}
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Statistical tests ask for the probability of a measured value
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to originate from a null hypothesis. Is this probability smaller than
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the desired significance level, the null hypothesis may be rejected.
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Statistical tests ask for the probability of a measured value to
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originate from a null hypothesis. Is this probability smaller than the
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desired \endeterm{Signifikanz}{significance level}, the
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\endeterm{Nullhypothese}{null hypothesis} may be rejected.
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Traditionally, such probabilities are taken from theoretical
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distributions which are based on assumptions about the data. Thus the
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@ -161,22 +165,25 @@ while we conserve all other statistical properties of the data.
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statistically significant.}
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\end{figure}
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A good example for the application of a permutaion test is the
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statistical assessment of correlations. Given are measured pairs of
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data points $(x_i, y_i)$. By calculating the correlation coefficient
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we can quantify how strongly $y$ depends on $x$. The correlation
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coefficient alone, however, does not tell whether the correlation is
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significantly different from a random correlation. The null hypothesis
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for such a situation is that $y$ does not depend on $x$. In
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order to perform a permutation test, we need to destroy the
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correlation by permuting the $(x_i, y_i)$ pairs, i.e. we rearrange the
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$x_i$ and $y_i$ values in a random fashion. Generating many sets of
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random pairs and computing the resulting correlation coefficients
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yields a distribution of correlation coefficients that result
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randomly from uncorrelated data. By comparing the actually measured
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correlation coefficient with this distribution we can directly assess
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the significance of the correlation
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(figure\,\ref{permutecorrelationfig}).
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A good example for the application of a
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\endeterm{Permutationstest}{permutaion test} is the statistical
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assessment of \endeterm[correlation]{Korrelation}{correlations}. Given
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are measured pairs of data points $(x_i, y_i)$. By calculating the
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\endeterm[correlation!correlation
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coefficient]{Korrelation!Korrelationskoeffizient}{correlation
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coefficient} we can quantify how strongly $y$ depends on $x$. The
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correlation coefficient alone, however, does not tell whether the
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correlation is significantly different from a random correlation. The
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\endeterm[]{Nullhypothese}{null hypothesis} for such a situation is that
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$y$ does not depend on $x$. In order to perform a permutation test, we
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need to destroy the correlation by permuting the $(x_i, y_i)$ pairs,
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i.e. we rearrange the $x_i$ and $y_i$ values in a random
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fashion. Generating many sets of random pairs and computing the
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resulting correlation coefficients yields a distribution of
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correlation coefficients that result randomly from uncorrelated
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data. By comparing the actually measured correlation coefficient with
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this distribution we can directly assess the significance of the
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correlation (figure\,\ref{permutecorrelationfig}).
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\begin{exercise}{correlationsignificance.m}{correlationsignificance.out}
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Estimate the statistical significance of a correlation coefficient.
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10
header.tex
10
header.tex
@ -212,9 +212,19 @@
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%%%%% english, german, code and file terms: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{ifthen}
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% \enterm[english index entry]{<english term>}
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\newcommand{\enterm}[2][]{\tr{\textit{#2}}{``#2''}\ifthenelse{\equal{#1}{}}{\tr{\protect\sindex[term]{#2}}{\protect\sindex[enterm]{#2}}}{\tr{\protect\sindex[term]{#1}}{\protect\sindex[enterm]{#1}}}}
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% \endeterm[english index entry]{<german index entry>}{<english term>}
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\newcommand{\endeterm}[3][]{\tr{\textit{#3}}{``#3''}\ifthenelse{\equal{#1}{}}{\tr{\protect\sindex[term]{#3}}{\protect\sindex[enterm]{#3}}}{\tr{\protect\sindex[term]{#1}}{\protect\sindex[enterm]{#1}}}\protect\sindex[determ]{#2}}
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% \determ[index entry]{<german term>}
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\newcommand{\determ}[2][]{\tr{``#2''}{\textit{#2}}\ifthenelse{\equal{#1}{}}{\tr{\protect\sindex[determ]{#2}}{\protect\sindex[term]{#2}}}{\tr{\protect\sindex[determ]{#1}}{\protect\sindex[term]{#1}}}}
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% \codeterm[index entry]{<code>}
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\newcommand{\codeterm}[2][]{\textit{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[term]{#2}}{\protect\sindex[term]{#1}}}
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\newcommand{\file}[1]{\texttt{#1}}
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% for escaping special characters into the index:
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@ -455,7 +455,7 @@ bivariate or multivariate data sets where we have pairs or tuples of
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data values (e.g. size and weight of elephants) we want to analyze
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dependencies between the variables.
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The \enterm{correlation coefficient}
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The \enterm[correlation!correlation coefficient]{correlation coefficient}
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\begin{equation}
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\label{correlationcoefficient}
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r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle
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@ -465,7 +465,7 @@ The \enterm{correlation coefficient}
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\end{equation}
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quantifies linear relationships between two variables
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\matlabfun{corr()}. The correlation coefficient is the
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\determ{covariance} normalized by the standard deviations of the
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\enterm{covariance} normalized by the standard deviations of the
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single variables. Perfectly correlated variables result in a
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correlation coefficient of $+1$, anit-correlated or negatively
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correlated data in a correlation coefficient of $-1$ and un-correlated
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