205 lines
7.4 KiB
TeX
205 lines
7.4 KiB
TeX
\documentclass[12pt,a4paper,pdftex]{exam}
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\usepackage[english]{babel}
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\usepackage{pslatex}
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\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
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\usepackage{xcolor}
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\usepackage{graphicx}
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\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
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%%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
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\pagestyle{headandfoot}
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\ifprintanswers
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\newcommand{\stitle}{: Solutions}
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\else
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\newcommand{\stitle}{}
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\fi
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\header{{\bfseries\large Exercise 9\stitle}}{{\bfseries\large Bootstrap}}{{\bfseries\large December 9th, 2019}}
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\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
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jan.benda@uni-tuebingen.de}
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\runningfooter{}{\thepage}{}
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\setlength{\baselineskip}{15pt}
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\setlength{\parindent}{0.0cm}
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\setlength{\parskip}{0.3cm}
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\renewcommand{\baselinestretch}{1.15}
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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\lstset{
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language=Matlab,
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basicstyle=\ttfamily\footnotesize,
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numbers=left,
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numberstyle=\tiny,
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title=\lstname,
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showstringspaces=false,
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commentstyle=\itshape\color{darkgray},
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breaklines=true,
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breakautoindent=true,
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columns=flexible,
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frame=single,
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xleftmargin=1em,
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xrightmargin=1em,
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aboveskip=10pt
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}
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%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{bm}
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\usepackage{dsfont}
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\newcommand{\naZ}{\mathds{N}}
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\newcommand{\gaZ}{\mathds{Z}}
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\newcommand{\raZ}{\mathds{Q}}
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\newcommand{\reZ}{\mathds{R}}
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\newcommand{\reZp}{\mathds{R^+}}
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\newcommand{\reZpN}{\mathds{R^+_0}}
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\newcommand{\koZ}{\mathds{C}}
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%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\continue}{\ifprintanswers%
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\else
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\vfill\hspace*{\fill}$\rightarrow$\newpage%
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\fi}
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\newcommand{\continuepage}{\ifprintanswers%
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\newpage
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\else
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\vfill\hspace*{\fill}$\rightarrow$\newpage%
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\fi}
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\newcommand{\newsolutionpage}{\ifprintanswers%
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\newpage%
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\else
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\fi}
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%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\qt}[1]{\textbf{#1}\\}
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\newcommand{\pref}[1]{(\ref{#1})}
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\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
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\newcommand{\code}[1]{\texttt{#1}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\input{instructions}
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\begin{questions}
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\question \qt{Bootstrap the standard error of the mean}
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We want to compute the standard error of the mean of a data set by
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means of the bootstrap method and compare the result with the formula
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``standard deviation divided by the square-root of $n$''.
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\begin{parts}
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\part Download the file \code{thymusglandweights.dat} from Ilias.
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This is a data set of the weights of the thymus glands of 14-day old chicken embryos
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measured in milligram.
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\part Load the data into Matlab (\code{load} function).
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\part Compute histogram, mean, and standard error of the mean of the first 80 data points.
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\part Compute the standard error of the mean of the first 80 data
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points by means of 500 times bootstrapping. Write a function that
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bootstraps the standard error of the mean of a given data set. The
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function should also return a vector with the bootstrapped means.
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\part Compute the 95\,\% confidence interval for the mean from the
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bootstrap distribution (\code{quantile()} function) --- the
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interval that contains the true mean with 95\,\% probability.
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\part Use the whole data set and the bootstrap method for computing
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the dependence of the standard error of the mean from the sample
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size $n$.
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\part Compare your result with the formula for the standard error
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$\sigma/\sqrt{n}$.
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\end{parts}
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\begin{solution}
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\lstinputlisting{bootstrapmean.m}
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\lstinputlisting{bootstraptymus.m}
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\includegraphics[width=0.5\textwidth]{bootstraptymus-datahist}
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\includegraphics[width=0.5\textwidth]{bootstraptymus-meanhist}
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\includegraphics[width=0.5\textwidth]{bootstraptymus-samples}
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\end{solution}
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\question \qt{Student t-distribution}
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The distribution of Student's t, $t=\bar x/(\sigma_x/\sqrt{n})$, the
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estimated mean $\bar x$ of a data set of size $n$ divided by the
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estimated standard error of the mean $\sigma_x/\sqrt{n}$, where
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$\sigma_x$ is the estimated standard deviation, is not a normal
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distribution but a Student-t distribution. We want to compute the
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Student-t distribution and compare it with the normal distribution.
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\begin{parts}
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\part Generate 100000 normally distributed random numbers.
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\part Draw from these data 1000 samples of size $n=3$, 5, 10, and
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50. For each sample size $n$ ...
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\part ... compute the mean $\bar x$ of the samples and plot the
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probability density of these means.
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\part ... compare the resulting probability densities with corresponding
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normal distributions.
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\part ... compute Student's $t=\bar x/(\sigma_x/\sqrt{n})$ and compare its
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distribution with the normal distribution with standard deviation of
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one. Is $t$ normally distributed? Under which conditions is $t$
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normally distributed?
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\end{parts}
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\newsolutionpage
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\begin{solution}
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\lstinputlisting{tdistribution.m}
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\includegraphics[width=1\textwidth]{tdistribution-n03}\\
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\includegraphics[width=1\textwidth]{tdistribution-n05}\\
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\includegraphics[width=1\textwidth]{tdistribution-n10}\\
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\includegraphics[width=1\textwidth]{tdistribution-n50}
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\end{solution}
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\continue
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\question \qt{Permutation test} \label{permutationtest}
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We want to compute the significance of a correlation by means of a permutation test.
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\begin{parts}
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\part \label{permutationtestdata} Generate 1000 correlated pairs
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$x$, $y$ of random numbers according to:
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\begin{verbatim}
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n = 1000
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a = 0.2;
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x = randn(n, 1);
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y = randn(n, 1) + a*x;
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\end{verbatim}
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\part Generate a scatter plot of the two variables.
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\part Why is $y$ correlated with $x$?
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\part Compute the correlation coefficient between $x$ and $y$.
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\part What do you need to do in order to destroy the correlations between the $x$-$y$ pairs?
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\part Do exactly this 1000 times and compute each time the correlation coefficient.
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\part Compute and plot the probability density of these correlation
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coefficients.
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\part Is the correlation of the original data set significant?
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\part What does ``significance of the correlation'' mean?
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% \part Vary the sample size \code{n} and compute in the same way the
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% significance of the correlation.
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\end{parts}
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\begin{solution}
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\lstinputlisting{correlationsignificance.m}
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\includegraphics[width=1\textwidth]{correlationsignificance}
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\end{solution}
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\question \qt{Bootstrap the correlation coefficient}
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The permutation test generates the distribution of the null hypothesis
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of uncorrelated data and we check whether the correlation coefficient
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of the data differs significantly from this
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distribution. Alternatively we can bootstrap the data while keeping
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the pairs and determine the confidence interval of the correlation
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coefficient of the data. If this differs significantly from a
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correlation coefficient of zero we can conclude that the correlation
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coefficient of the data indeed quantifies correlated data.
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We take the same data set that we have generated in exercise
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\ref{permutationtest} (\ref{permutationtestdata}).
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\begin{parts}
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\part Bootstrap 1000 times the correlation coefficient from the data.
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\part Compute and plot the probability density of these correlation
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coefficients.
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\part Is the correlation of the original data set significant?
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\end{parts}
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\begin{solution}
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\lstinputlisting{correlationbootstrap.m}
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\includegraphics[width=1\textwidth]{correlationbootstrap}
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\end{solution}
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\end{questions}
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\end{document} |