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new command \endeterm for english terms that also make an entry into the german index - not working yet

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Jan Benda
2019-12-05 09:16:24 +01:00
parent 006fa998cc
commit 4d2bedd78c
3 changed files with 57 additions and 40 deletions
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bootstrap/lecture/bootstrap.tex
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@@ -33,10 +33,11 @@ population. Rather, we draw samples (\enterm{simple random sample}
then estimate a statistical measure of interest (e.g. the average
length of the pickles) within this sample and hope that it is a good
approximation of the unknown and immeasurable true average length of
the population (\determ{Populationsparameter}). We apply statistical
methods to find out how precise this approximation is.
the population (\endeterm{Populationsparameter}{population
parameter}). We apply statistical methods to find out how precise
this approximation is.
If we could draw a large number of \enterm{simple random samples} we
If we could draw a large number of simple random samples we
could calculate the statistical measure of interest for each sample
and estimate its probability distribution using a histogram. This
distribution is called the \enterm{sampling distribution}
@@ -69,17 +70,18 @@ error of the mean which is the standard deviation of the sampling
distribution of average values around the true mean of the population
(\subfigref{bootstrapsamplingdistributionfig}{b}).
Alternatively, we can use ``bootstrapping'' to generate new samples
from one set of measurements (resampling). From these bootstrapped
samples we compute the desired statistical measure and estimate their
distribution (\enterm{bootstrap distribution},
\subfigref{bootstrapsamplingdistributionfig}{c}). Interestingly, this
distribution is very similar to the sampling distribution regarding
its width. The only difference is that the bootstrapped values are
distributed around the measure of the original sample and not the one
of the statistical population. We can use the bootstrap distribution
to draw conclusion regarding the precision of our estimation (e.g.
standard errors and confidence intervals).
Alternatively, we can use \enterm{bootstrapping}
(\determ{Bootstrap-Verfahren}) to generate new samples from one set of
measurements (\endeterm{Resampling}{resampling}). From these
bootstrapped samples we compute the desired statistical measure and
estimate their distribution (\endeterm{Bootstrapverteilung}{bootstrap
distribution}, \subfigref{bootstrapsamplingdistributionfig}{c}).
Interestingly, this distribution is very similar to the sampling
distribution regarding its width. The only difference is that the
bootstrapped values are distributed around the measure of the original
sample and not the one of the statistical population. We can use the
bootstrap distribution to draw conclusion regarding the precision of
our estimation (e.g. standard errors and confidence intervals).
Bootstrapping methods create bootstrapped samples from a SRS by
resampling. The bootstrapped samples are used to estimate the sampling
@@ -93,11 +95,12 @@ data set using the \code{randi()} function.
\section{Bootstrap of the standard error}
Bootstrapping can be nicely illustrated at the example of the standard
error of the mean. The arithmetic mean is calculated for a simple
random sample. The standard error of the mean is the standard
deviation of the expected distribution of mean values around the mean
of the statistical population.
Bootstrapping can be nicely illustrated at the example of the
\enterm{standard error} of the mean (\determ{Standardfehler}). The
arithmetic mean is calculated for a simple random sample. The standard
error of the mean is the standard deviation of the expected
distribution of mean values around the mean of the statistical
population.
\begin{figure}[tp]
\includegraphics[width=1\textwidth]{bootstrapsem}
@@ -135,9 +138,10 @@ distribution is the standard error of the mean.
\section{Permutation tests}
Statistical tests ask for the probability of a measured value
to originate from a null hypothesis. Is this probability smaller than
the desired significance level, the null hypothesis may be rejected.
Statistical tests ask for the probability of a measured value to
originate from a null hypothesis. Is this probability smaller than the
desired \endeterm{Signifikanz}{significance level}, the
\endeterm{Nullhypothese}{null hypothesis} may be rejected.
Traditionally, such probabilities are taken from theoretical
distributions which are based on assumptions about the data. Thus the
@@ -161,22 +165,25 @@ while we conserve all other statistical properties of the data.
statistically significant.}
\end{figure}
A good example for the application of a permutaion test is the
statistical assessment of correlations. Given are measured pairs of
data points $(x_i, y_i)$. By calculating the correlation coefficient
we can quantify how strongly $y$ depends on $x$. The correlation
coefficient alone, however, does not tell whether the correlation is
significantly different from a random correlation. The null hypothesis
for such a situation is that $y$ does not depend on $x$. In
order to perform a permutation test, we need to destroy the
correlation by permuting the $(x_i, y_i)$ pairs, i.e. we rearrange the
$x_i$ and $y_i$ values in a random fashion. Generating many sets of
random pairs and computing the resulting correlation coefficients
yields a distribution of correlation coefficients that result
randomly from uncorrelated data. By comparing the actually measured
correlation coefficient with this distribution we can directly assess
the significance of the correlation
(figure\,\ref{permutecorrelationfig}).
A good example for the application of a
\endeterm{Permutationstest}{permutaion test} is the statistical
assessment of \endeterm[correlation]{Korrelation}{correlations}. Given
are measured pairs of data points $(x_i, y_i)$. By calculating the
\endeterm[correlation!correlation
coefficient]{Korrelation!Korrelationskoeffizient}{correlation
coefficient} we can quantify how strongly $y$ depends on $x$. The
correlation coefficient alone, however, does not tell whether the
correlation is significantly different from a random correlation. The
\endeterm[]{Nullhypothese}{null hypothesis} for such a situation is that
$y$ does not depend on $x$. In order to perform a permutation test, we
need to destroy the correlation by permuting the $(x_i, y_i)$ pairs,
i.e. we rearrange the $x_i$ and $y_i$ values in a random
fashion. Generating many sets of random pairs and computing the
resulting correlation coefficients yields a distribution of
correlation coefficients that result randomly from uncorrelated
data. By comparing the actually measured correlation coefficient with
this distribution we can directly assess the significance of the
correlation (figure\,\ref{permutecorrelationfig}).
\begin{exercise}{correlationsignificance.m}{correlationsignificance.out}
Estimate the statistical significance of a correlation coefficient.