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fixed many index entries

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Jan Benda 2019-12-09 20:01:27 +01:00
parent f24c14e6f5
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5
Makefile
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@ -22,9 +22,10 @@ $(BASENAME).pdf : $(BASENAME).tex header.tex $(SUBTEXS)
splitindex $(BASENAME).idx splitindex $(BASENAME).idx
index : index :
pdflatex -interaction=scrollmode $(BASENAME).tex pdflatex $(BASENAME).tex
splitindex $(BASENAME).idx splitindex $(BASENAME).idx
pdflatex -interaction=scrollmode $(BASENAME).tex | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex $(BASENAME).tex || true pdflatex $(BASENAME).tex
pdflatex $(BASENAME).tex
again : again :
pdflatex $(BASENAME).tex pdflatex $(BASENAME).tex

2
bootstrap/lecture/bootstrap.tex
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@ -171,7 +171,7 @@ A good example for the application of a
assessment of \entermde[correlation]{Korrelation}{correlations}. Given assessment of \entermde[correlation]{Korrelation}{correlations}. Given
are measured pairs of data points $(x_i, y_i)$. By calculating the are measured pairs of data points $(x_i, y_i)$. By calculating the
\entermde[correlation!correlation \entermde[correlation!correlation
coefficient]{Korrelation!Korrelationskoeffizient}{correlation coefficient]{Korrelation!-skoeffizient}{correlation
coefficient} we can quantify how strongly $y$ depends on $x$. The coefficient} we can quantify how strongly $y$ depends on $x$. The
correlation coefficient alone, however, does not tell whether the correlation coefficient alone, however, does not tell whether the
correlation is significantly different from a random correlation. The correlation is significantly different from a random correlation. The

43
codestyle/lecture/codestyle.tex
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@ -1,4 +1,4 @@
\chapter{\tr{Code style}{Programmierstil}} \chapter{Code style}
\shortquote{Any code of your own that you haven't looked at for six or \shortquote{Any code of your own that you haven't looked at for six or
more months might as well have been written by someone more months might as well have been written by someone
@ -33,7 +33,7 @@ by calling functions that work on the data and managing the
results. Applying this structure makes it easy to understand the flow results. Applying this structure makes it easy to understand the flow
of the program but two questions remain: (i) How to organize the files of the program but two questions remain: (i) How to organize the files
on the file system and (ii) how to name them that the controlling on the file system and (ii) how to name them that the controlling
script is easily identified among the other \codeterm{m-files}. script is easily identified among the other \codeterm[m-file]{m-files}.
Upon installation \matlab{} creates a folder called \file{MATLAB} in Upon installation \matlab{} creates a folder called \file{MATLAB} in
the user space (Windows: My files, Linux: Documents, MacOS: the user space (Windows: My files, Linux: Documents, MacOS:
@ -43,7 +43,7 @@ moment. Of course, any other location can specified as well. Generally
it is of great advantage to store related scripts and functions within it is of great advantage to store related scripts and functions within
the same folder on the hard drive. An easy approach is to create a the same folder on the hard drive. An easy approach is to create a
project-specific folder structure that contains sub-folders for each project-specific folder structure that contains sub-folders for each
task (analysis) and to store all related \codeterm{m-files} task (analysis) and to store all related \codeterm[m-file]{m-files}
(screenshot \ref{fileorganizationfig}). In these task-related folders (screenshot \ref{fileorganizationfig}). In these task-related folders
one may consider to create a further sub-folder to store results one may consider to create a further sub-folder to store results
(created figures, result data). On the project level a single script (created figures, result data). On the project level a single script
@ -307,8 +307,8 @@ and to briefly explain what they do. Whenever one feels tempted to do
this, one could also consider to delegate the respective task to a this, one could also consider to delegate the respective task to a
function. In most cases this is preferable. function. In most cases this is preferable.
Not delegating the tasks leads to very long \codeterm{m-files} which Not delegating the tasks leads to very long \codeterm[m-file]{m-files}
can be confusing. Sometimes such a code is called ``spaghetti which can be confusing. Sometimes such a code is called ``spaghetti
code''. It is high time to think about delegation of tasks to code''. It is high time to think about delegation of tasks to
functions. functions.
@ -323,17 +323,17 @@ functions.
\end{important} \end{important}
\subsection{Local and nested functions} \subsection{Local and nested functions}
Generally, functions live in their own \codeterm{m-files} that have Generally, functions live in their own \codeterm[m-file]{m-files} that
the same name as the function itself. Delegating tasks to functions have the same name as the function itself. Delegating tasks to
thus leads to a large set of \codeterm{m-files} which increases functions thus leads to a large set of \codeterm[m-file]{m-files}
complexity and may lead to confusion. If the delegated functionality which increases complexity and may lead to confusion. If the delegated
is used in multiple instances, it is still advisable to do so. On the functionality is used in multiple instances, it is still advisable to
other hand, when the delegated functionality is only used within the do so. On the other hand, when the delegated functionality is only
context of another function \matlab{} allows to define used within the context of another function \matlab{} allows to define
\codeterm[function!local]{local functions} and \entermde[function!local]{Funktion!lokale}{local functions} and
\codeterm[function!nested]{nested functions} within the same \entermde[function!nested]{Funktion!verschachtelte}{nested functions}
file. Listing \ref{localfunctions} shows an example of a local within the same file. Listing \ref{localfunctions} shows an example of
function definition. a local function definition.
\pagebreak[3] \lstinputlisting[label=localfunctions, caption={Example \pagebreak[3] \lstinputlisting[label=localfunctions, caption={Example
for local functions.}]{calculateSines.m} for local functions.}]{calculateSines.m}
@ -408,11 +408,12 @@ advisable to adhere to these.
Repeated tasks should (to be read as must) be delegated to Repeated tasks should (to be read as must) be delegated to
functions. In cases in which a function is only locally applied and functions. In cases in which a function is only locally applied and
not of more global interest across projects consider to define it as not of more global interest across projects consider to define it as
\codeterm[function!local]{local function} or \entermde[function!local]{Funktion!lokale}{local function} or
\codeterm[function!nested]{nested function}. Taking care to increase \entermde[function!nested]{Funktion!verschachtelte}{nested
readability and comprehensibility pays off, even to the author! function}. Taking care to increase readability and comprehensibility
\footnote{Reading tip: Robert C. Martin: \textit{Clean Code: A Handbook of pays off, even to the author! \footnote{Reading tip: Robert
Agile Software Craftmanship}, Prentice Hall} C. Martin: \textit{Clean Code: A Handbook of Agile Software
Craftmanship}, Prentice Hall}
\shortquote{Programs must be written for people to read, and only \shortquote{Programs must be written for people to read, and only
incidentally for machines to execute.}{Abelson / Sussman} incidentally for machines to execute.}{Abelson / Sussman}

78
debugging/lecture/debugging.tex
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@ -49,13 +49,14 @@ obscure logical errors! Take care when using the \codeterm{try-catch
\end{important} \end{important}
\subsection{\codeterm{Syntax errors}}\label{syntax_error} \subsection{Syntax errors}\label{syntax_error}
The most common and easiest to fix type of error. A syntax error The most common and easiest to fix type of error. A
violates the rules (spelling and grammar) of the programming \entermde[error!syntax]{Fehler!Syntax\~}{syntax error} violates the
language. For example every opening parenthesis must be matched by a rules (spelling and grammar) of the programming language. For example
closing one or every \code{for} loop has to be closed by an every opening parenthesis must be matched by a closing one or every
\code{end}. Usually, the respective error messages are clear and \code{for} loop has to be closed by an \code{end}. Usually, the
the editor will point out and highlight most \codeterm{syntax error}s. respective error messages are clear and the editor will point out and
highlight most syntax errors.
\begin{lstlisting}[label=syntaxerror, caption={Unbalanced parenthesis error.}] \begin{lstlisting}[label=syntaxerror, caption={Unbalanced parenthesis error.}]
>> mean(random_numbers >> mean(random_numbers
@ -66,8 +67,9 @@ Did you mean:
>> mean(random_numbers) >> mean(random_numbers)
\end{lstlisting} \end{lstlisting}
\subsection{\codeterm{Indexing error}}\label{index_error} \subsection{Indexing error}\label{index_error}
Second on the list of common errors are the indexing errors. Usually Second on the list of common errors are the
\entermde[error!indexing]{Fehler!Index\~}{indexing errors}. Usually
\matlab{} gives rather precise infromation about the cause, once you \matlab{} gives rather precise infromation about the cause, once you
know what they mean. Consider the following code. know what they mean. Consider the following code.
@ -111,14 +113,16 @@ to a number and uses this number to address the element in
\varcode{my\_array}. The \codeterm{char} has the ASCII code 65 and \varcode{my\_array}. The \codeterm{char} has the ASCII code 65 and
thus the 65th element of \varcode{my\_array} is returned. thus the 65th element of \varcode{my\_array} is returned.
\subsection{\codeterm{Assignment error}} \subsection{Assignment error}
Related to the Indexing error, an assignment error occurs when we want Related to the indexing error, an
to write data into a variable, that does not fit into it. Listing \entermde[error!assignment]{Fehler!Zuweisungs\~}{assignment error}
\ref{assignmenterror} shows the simple case for 1-d data but, of occurs when we want to write data into a variable, that does not fit
course, it extents to n-dimensional data. The data that is to be into it. Listing \ref{assignmenterror} shows the simple case for 1-d
filled into a matrix hat to fit in all dimensions. The command in line data but, of course, it extents to n-dimensional data. The data that
7 works due to the fact, that matlab automatically extends the matrix, is to be filled into a matrix hat to fit in all dimensions. The
if you assign values to a range outside its bounds. command in line 7 works due to the fact, that matlab automatically
extends the matrix, if you assign values to a range outside its
bounds.
\begin{lstlisting}[label=assignmenterror, caption={Assignment errors.}] \begin{lstlisting}[label=assignmenterror, caption={Assignment errors.}]
>> a = zeros(1, 100); >> a = zeros(1, 100);
@ -133,21 +137,20 @@ ans =
110 1 110 1
\end{lstlisting} \end{lstlisting}
\subsection{\codeterm{Dimension mismatch error}} \subsection{Dimension mismatch error}
Similarly, some arithmetic operations are only valid if the variables Similarly, some arithmetic operations are only valid if the variables
fulfill some size constraints. Consider the following commands fulfill some size constraints. Consider the following commands
(listing\,\ref{dimensionmismatch}). The first one (line 3) fails (listing\,\ref{dimensionmismatch}). The first one (line 3) fails
because we are trying to do al elementwise add on two vectors that because we are trying to add two vectors of different lengths
have different lengths, respectively sizes. The matrix multiplication elementwise. The matrix multiplication in line 6 also fails since for
in line 6 also fails since for this operations to succeed the inner this operations to succeed the inner matrix dimensions must agree (for
matrix dimensions must agree (for more information on the more information on the matrixmultiplication see
matrixmultiplication see box\,\ref{matrixmultiplication} in box\,\ref{matrixmultiplication} in chapter\,\ref{programming}). The
chapter\,\ref{programming}). The elementwise multiplication issued in elementwise multiplication issued in line 10 fails for the same reason
line 10 fails for the same reason as the addition we tried as the addition we tried earlier. Sometimes, however, things
earlier. Sometimes, however, things apparently work but the result may apparently work but the result may be surprising. The last operation
be surprising. The last operation in listing\,\ref{dimensionmismatch} in listing\,\ref{dimensionmismatch} does not throw an error but the
does not throw an error but the result is something else than the result is something else than the expected elementwise multiplication.
expected elementwise multiplication.
% XXX Some arithmetic operations make size constraints, violating them leads to dimension mismatch errors. % XXX Some arithmetic operations make size constraints, violating them leads to dimension mismatch errors.
\begin{lstlisting}[label=dimensionmismatch, caption={Dimension mismatch errors.}] \begin{lstlisting}[label=dimensionmismatch, caption={Dimension mismatch errors.}]
@ -174,7 +177,8 @@ expected elementwise multiplication.
\section{Logical error} \section{Logical error}
Sometimes a program runs smoothly and terminates without any Sometimes a program runs smoothly and terminates without any
complaint. This, however, does not necessarily mean that the program complaint. This, however, does not necessarily mean that the program
is correct. We may have made a \codeterm{logical error}. Logical is correct. We may have made a
\entermde[error!logical]{Fehler!logischer}{logical error}. Logical
errors are hard to find, \matlab{} has no chance to detect such errors errors are hard to find, \matlab{} has no chance to detect such errors
since they do not violate the syntax or cause the throwing of an since they do not violate the syntax or cause the throwing of an
error. Thus, we are on our own to find and fix the bug. There are a error. Thus, we are on our own to find and fix the bug. There are a
@ -283,7 +287,7 @@ validity.
Matlab offers a unit testing framework in which small scripts are Matlab offers a unit testing framework in which small scripts are
written that test the features of the program. We will follow the written that test the features of the program. We will follow the
example given in the \matlab{} help and assume that there is a example given in the \matlab{} help and assume that there is a
function \code{rightTriangle} (listing\,\ref{trianglelisting}). function \varcode{rightTriangle()} (listing\,\ref{trianglelisting}).
% XXX Slightly more readable version of the example given in the \matlab{} help system. Note: The variable name for the angles have been capitalized in order to not override the matlab defined functions \code{alpha, beta,} and \code{gamma}. % XXX Slightly more readable version of the example given in the \matlab{} help system. Note: The variable name for the angles have been capitalized in order to not override the matlab defined functions \code{alpha, beta,} and \code{gamma}.
\begin{lstlisting}[label=trianglelisting, caption={Example function for unit testing.}] \begin{lstlisting}[label=trianglelisting, caption={Example function for unit testing.}]
@ -308,7 +312,7 @@ folder that follows the following rules.
\item The name of the script file must start or end with the word \item The name of the script file must start or end with the word
'test', which is case-insensitive. 'test', which is case-insensitive.
\item Each unit test should be placed in a separate section/cell of the script. \item Each unit test should be placed in a separate section/cell of the script.
\item After the \code{\%\%} that defines the cell, a name for the \item After the \mcode{\%\%} that defines the cell, a name for the
particular unit test may be given. particular unit test may be given.
\end{enumerate} \end{enumerate}
@ -328,11 +332,11 @@ Further there are a few things that are different in tests compared to normal sc
tests. tests.
\end{enumerate} \end{enumerate}
The test script for the \code{rightTrianlge} function The test script for the \varcode{rightTriangle()} function
(listing\,\ref{trianglelisting}) may look like in (listing\,\ref{trianglelisting}) may look like in
listing\,\ref{testscript}. listing\,\ref{testscript}.
\begin{lstlisting}[label=testscript, caption={Unit test for the \code{rightTriangle} function stored in an m-file testRightTriangle.m}] \begin{lstlisting}[label=testscript, caption={Unit test for the \varcode{rightTriangle()} function stored in an m-file testRightTriangle.m}]
tolerance = 1e-10; tolerance = 1e-10;
% preconditions % preconditions
@ -372,7 +376,7 @@ assert(abs(approx - smallAngle) <= tolerance, 'Problem with small angle approxim
In a test script we can execute any code. The actual test whether or In a test script we can execute any code. The actual test whether or
not the results match our predictions is done using the not the results match our predictions is done using the
\code{assert()}{assert} function. This function basically expects a \code{assert()} function. This function basically expects a
boolean value and if this is not true, it raises an error that, in the boolean value and if this is not true, it raises an error that, in the
context of the test does not lead to a termination of the program. In context of the test does not lead to a termination of the program. In
the tests above, the argument to assert is always a boolean expression the tests above, the argument to assert is always a boolean expression
@ -392,7 +396,7 @@ result = runtests('testRightTriangle')
During the run, \matlab{} will put out error messages onto the command During the run, \matlab{} will put out error messages onto the command
line and a summary of the test results is then stored within the line and a summary of the test results is then stored within the
\varcode{result} variable. These can be displayed using the function \varcode{result} variable. These can be displayed using the function
\code{table(result)} \code[table()]{table(result)}.
\begin{lstlisting}[label=testresults, caption={The test results.}, basicstyle=\ttfamily\scriptsize] \begin{lstlisting}[label=testresults, caption={The test results.}, basicstyle=\ttfamily\scriptsize]
table(result) table(result)
@ -431,7 +435,7 @@ that help to solve the problem.
\item No idea what the error message is trying to say? Google it! \item No idea what the error message is trying to say? Google it!
\item Read the program line by line and understand what each line is \item Read the program line by line and understand what each line is
doing. doing.
\item Use \code{disp} to print out relevant information on the command \item Use \code{disp()} to print out relevant information on the command
line and compare the output with your expectations. Do this step by line and compare the output with your expectations. Do this step by
step and start at the beginning. step and start at the beginning.
\item Use the \matlab{} debugger to stop execution of the code at a \item Use the \matlab{} debugger to stop execution of the code at a

4
header.tex
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@ -217,7 +217,7 @@
% the english index. % the english index.
\newcommand{\enterm}[2][]{\textit{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[enterm]{#2}}{\protect\sindex[enterm]{#1}}} \newcommand{\enterm}[2][]{\textit{#2}\ifthenelse{\equal{#1}{}}{\protect\sindex[enterm]{#2}}{\protect\sindex[enterm]{#1}}}
% \endeterm[english index entry]{<german index entry>}{<english term>} % \entermde[english index entry]{<german index entry>}{<english term>}
% typeset the english term in italics and add it (or the first % typeset the english term in italics and add it (or the first
% optional argument) to the english index. In addition add the german % optional argument) to the english index. In addition add the german
% index entry to the german index without printing it. % index entry to the german index without printing it.
@ -270,7 +270,7 @@
\newcommand{\pythonfun}[1]{(\tr{\python-function}{\python-Funktion} \varcode{#1})\protect\sindex[pcode]{#1}} \newcommand{\pythonfun}[1]{(\tr{\python-function}{\python-Funktion} \varcode{#1})\protect\sindex[pcode]{#1}}
% typeset '(matlab-function #1)' and add the function to the matlab index: % typeset '(matlab-function #1)' and add the function to the matlab index:
\newcommand{\matlabfun}[1]{(\tr{\matlab-function}{\matlab-Funktion} \varcode{#1})\protect\sindex[mcode]{#1}} \newcommand{\matlabfun}[1]{(function \varcode{#1})\protect\sindex[mcode]{#1}}
%%%%% shortquote and widequote commands: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% shortquote and widequote commands: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

24
likelihood/lecture/likelihood.tex
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@ -26,15 +26,16 @@ parameters $\theta$. This could be the normal distribution
defined by the mean $\mu$ and the standard deviation $\sigma$ as defined by the mean $\mu$ and the standard deviation $\sigma$ as
parameters $\theta$. If the $n$ independent observations of $x_1, parameters $\theta$. If the $n$ independent observations of $x_1,
x_2, \ldots x_n$ originate from the same probability density x_2, \ldots x_n$ originate from the same probability density
distribution (they are \enterm{i.i.d.} independent and identically distribution (they are \enterm[i.i.d.|see{independent and identically
distributed) then the conditional probability $p(x_1,x_2, \ldots distributed}]{i.i.d.}, \enterm{independent and identically
distributed}) then the conditional probability $p(x_1,x_2, \ldots
x_n|\theta)$ of observing $x_1, x_2, \ldots x_n$ given a specific x_n|\theta)$ of observing $x_1, x_2, \ldots x_n$ given a specific
$\theta$ is given by $\theta$ is given by
\begin{equation} \begin{equation}
p(x_1,x_2, \ldots x_n|\theta) = p(x_1|\theta) \cdot p(x_2|\theta) p(x_1,x_2, \ldots x_n|\theta) = p(x_1|\theta) \cdot p(x_2|\theta)
\ldots p(x_n|\theta) = \prod_{i=1}^n p(x_i|\theta) \; . \ldots p(x_n|\theta) = \prod_{i=1}^n p(x_i|\theta) \; .
\end{equation} \end{equation}
Vice versa, the \enterm{likelihood} of the parameters $\theta$ Vice versa, the \entermde{Likelihood}{likelihood} of the parameters $\theta$
given the observed data $x_1, x_2, \ldots x_n$ is given the observed data $x_1, x_2, \ldots x_n$ is
\begin{equation} \begin{equation}
{\cal L}(\theta|x_1,x_2, \ldots x_n) = p(x_1,x_2, \ldots x_n|\theta) \; . {\cal L}(\theta|x_1,x_2, \ldots x_n) = p(x_1,x_2, \ldots x_n|\theta) \; .
@ -57,7 +58,7 @@ The position of a function's maximum does not change when the values
of the function are transformed by a strictly monotonously rising of the function are transformed by a strictly monotonously rising
function such as the logarithm. For numerical and reasons that we will function such as the logarithm. For numerical and reasons that we will
discuss below, we commonly search for the maximum of the logarithm of discuss below, we commonly search for the maximum of the logarithm of
the likelihood (\enterm{log-likelihood}): the likelihood (\entermde[likelihood!log-]{Likelihood!Log-}{log-likelihood}):
\begin{eqnarray} \begin{eqnarray}
\theta_{mle} & = & \text{argmax}_{\theta}\; {\cal L}(\theta|x_1,x_2, \ldots x_n) \nonumber \\ \theta_{mle} & = & \text{argmax}_{\theta}\; {\cal L}(\theta|x_1,x_2, \ldots x_n) \nonumber \\
@ -136,9 +137,10 @@ from the data.
For non-Gaussian distributions (e.g. a Gamma-distribution), however, For non-Gaussian distributions (e.g. a Gamma-distribution), however,
such simple analytical expressions for the parameters of the such simple analytical expressions for the parameters of the
distribution do not exist, e.g. the shape parameter of a distribution do not exist, e.g. the shape parameter of a
\enterm{Gamma-distribution}. How do we fit such a distribution to \entermde[distribution!Gamma-]{Verteilung!Gamma-}{Gamma-distribution}. How
some data? That is, how should we compute the values of the parameters do we fit such a distribution to some data? That is, how should we
of the distribution, given the data? compute the values of the parameters of the distribution, given the
data?
A first guess could be to fit the probability density function by A first guess could be to fit the probability density function by
minimization of the squared difference to a histogram of the measured minimization of the squared difference to a histogram of the measured
@ -289,10 +291,10 @@ out of \eqnref{mleslope} and we get
To see what this expression is, we need to standardize the data. We To see what this expression is, we need to standardize the data. We
make the data mean free and normalize them to their standard make the data mean free and normalize them to their standard
deviation, i.e. $x \mapsto (x - \bar x)/\sigma_x$. The resulting deviation, i.e. $x \mapsto (x - \bar x)/\sigma_x$. The resulting
numbers are also called \enterm[z-values]{$z$-values} or $z$-scores and they numbers are also called \entermde[z-values]{z-Wert}{$z$-values} or
have the property $\bar x = 0$ and $\sigma_x = 1$. $z$-scores are $z$-scores and they have the property $\bar x = 0$ and $\sigma_x =
often used in Biology to make quantities that differ in their units 1$. $z$-scores are often used in Biology to make quantities that
comparable. For standardized data the variance differ in their units comparable. For standardized data the variance
\[ \sigma_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)^2 = \frac{1}{n} \sum_{i=1}^n x_i^2 = 1 \] \[ \sigma_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)^2 = \frac{1}{n} \sum_{i=1}^n x_i^2 = 1 \]
is given by the mean squared data and equals one. is given by the mean squared data and equals one.
The covariance between $x$ and $y$ also simplifies to The covariance between $x$ and $y$ also simplifies to

77
plotting/lecture/plotting.tex
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@ -112,16 +112,15 @@ the missing information ourselves. Thus, we need a second variable
that contains the respective \varcode{x} values. The length of that contains the respective \varcode{x} values. The length of
\varcode{x} and \varcode{y} must be the same otherwise the later call \varcode{x} and \varcode{y} must be the same otherwise the later call
of the \varcode{plot} function will raise an error. The respective of the \varcode{plot} function will raise an error. The respective
call will expand to \code[plot()]{plot(x, y)}. The x-axis will be now call will expand to \code[plot()]{plot(x, y)}. The x-axis will now be
be scaled from the minimum in \varcode{x} to the maximum of scaled from the minimum in \varcode{x} to the maximum of \varcode{x}
\varcode{x} and by default it will be plotted as a line plot with a and by default it will be plotted as a line plot with a solid blue
solid blue line of the linewidth 1pt. A second plot that is added to the line of the linewidth 1pt. A second plot that is added to the figure
figure will be plotted in red using the same settings. The will be plotted in red using the same settings. The order of the used
order of the used colors depends on the \enterm{colormap} settings colors depends on the \enterm{colormap} settings which can be adjusted
which can be adjusted to personal taste or to personal taste or need. Table\,\ref{plotlinestyles} shows some
need. Table\,\ref{plotlinestyles} shows some predefined values that predefined values that can be chosen for the line style, the marker,
can be chosen for the line style, the marker, or the color. For or the color. For additional options consult the help.
additional options consult the help.
\begin{table}[htp] \begin{table}[htp]
\titlecaption{Predefined line styles (left), colors (center) and \titlecaption{Predefined line styles (left), colors (center) and
@ -184,8 +183,8 @@ chosen.
\subsection{Changing the axes properties} \subsection{Changing the axes properties}
The first thing a plot needs are axis labels with correct units. By The first thing a plot needs are axis labels with correct units. By
calling the functions \code[xlabel]{xlabel('Time [ms]')} and calling the functions \code[xlabel()]{xlabel('Time [ms]')} and
\code[ylabel]{ylabel('Voltage [mV]')} these can be set. By default the \code[ylabel()]{ylabel('Voltage [mV]')} these can be set. By default the
axes will be scaled to show the full extent of the data. The extremes axes will be scaled to show the full extent of the data. The extremes
will be selected as the closest integer for small values or the next will be selected as the closest integer for small values or the next
full multiple of tens, hundreds, thousands, etc.\ depending on the full multiple of tens, hundreds, thousands, etc.\ depending on the
@ -196,8 +195,8 @@ functions expect a single argument, that is a 2-element vector
containing the minimum and maximum value. Table\,\ref{plotaxisprops} containing the minimum and maximum value. Table\,\ref{plotaxisprops}
lists some of the commonly adjusted properties of an axis. To set lists some of the commonly adjusted properties of an axis. To set
these properties, we need to have the axes object which can either be these properties, we need to have the axes object which can either be
stored in a variable when calling \varcode{plot} (\code{axes = stored in a variable when calling \varcode{plot} (\varcode{axes =
plot(x,y);}) or can be retrieved using the \code[gca]{gca} function plot(x,y);}) or can be retrieved using the \code{gca()} function
(gca stands for ``get current axes''). Changing the properties of the axes (gca stands for ``get current axes''). Changing the properties of the axes
object will update the plot (listing\,\ref{niceplotlisting}). object will update the plot (listing\,\ref{niceplotlisting}).
@ -253,8 +252,8 @@ and the placement of the axes on the
paper. Table\,\ref{plotfigureprops} lists commonly used paper. Table\,\ref{plotfigureprops} lists commonly used
properties. For a complete reference check the help. To change the properties. For a complete reference check the help. To change the
figure's appearance, we need to change the properties of the figure figure's appearance, we need to change the properties of the figure
object which can be retrieved during creation of the figure (\code{fig object which can be retrieved during creation of the figure (\code[figure()]{fig
= figure();}) or by using the \code{gcf} (``get current figure'') = figure();}) or by using the \code{gcf()} (``get current figure'')
command. command.
The script shown in the listing\,\ref{niceplotlisting} exemplifies The script shown in the listing\,\ref{niceplotlisting} exemplifies
@ -334,10 +333,10 @@ the last one defines the output format (box\,\ref{graphicsformatbox}).
properties could be read and set using the functions properties could be read and set using the functions
\code[get()]{get} and \code[set()]{set}. The first argument these \code[get()]{get} and \code[set()]{set}. The first argument these
functions expect are valid figure or axis \emph{handles} which were functions expect are valid figure or axis \emph{handles} which were
returned by the \code{figure} and \code{plot} functions, or could be returned by the \code{figure()} and \code{plot()} functions, or could be
retrieved using \code[gcf()]{gcf} or \code[gca()]{gca} for the retrieved using \code{gcf()} or \code{gca()} for the
current figure or axis handle, respectively. Subsequent arguments current figure or axis handle, respectively. Subsequent arguments
passed to \code{set} are pairs of a property's name and the desired passed to \code{set()} are pairs of a property's name and the desired
value. value.
\begin{lstlisting}[caption={Using set to change figure and axis properties.}] \begin{lstlisting}[caption={Using set to change figure and axis properties.}]
frequency = 5; % frequency of the sine wave in Hz frequency = 5; % frequency of the sine wave in Hz
@ -351,8 +350,8 @@ the last one defines the output format (box\,\ref{graphicsformatbox}).
set(figure_handle, 'PaperSize', [5.5, 5.5], 'PaperUnit', 'centimeters', ... set(figure_handle, 'PaperSize', [5.5, 5.5], 'PaperUnit', 'centimeters', ...
'PaperPosition', [0, 0, 5.5, 5.5]); 'PaperPosition', [0, 0, 5.5, 5.5]);
\end{lstlisting} \end{lstlisting}
With newer versions the handles returned by \varcode{gcf} and With newer versions the handles returned by \code{gcf()} and
\varcode{gca} are ``objects'' and setting properties became much \code{gca()} are ``objects'' and setting properties became much
easier as it is used throughout this chapter. For downward easier as it is used throughout this chapter. For downward
compatibility with older versions set and get still work in current compatibility with older versions set and get still work in current
versions of \matlab{}. versions of \matlab{}.
@ -371,7 +370,7 @@ For some types of plots we present examples in the following sections.
\subsection{Scatter} \subsection{Scatter}
For displaying events or pairs of x-y coordinates the standard line For displaying events or pairs of x-y coordinates the standard line
plot is not optimal. Rather, we use \code[scatter()]{scatter} for this plot is not optimal. Rather, we use \code{scatter()} for this
purpose. For example, we have a number of measurements of a system's purpose. For example, we have a number of measurements of a system's
response to a certain stimulus intensity. There is no dependency response to a certain stimulus intensity. There is no dependency
between the data points, drawing them with a line-plot would be between the data points, drawing them with a line-plot would be
@ -417,8 +416,8 @@ A very common scenario is to combine several plots in the same
figure. To do this we create so-called subplots figure. To do this we create so-called subplots
figures\,\ref{regularsubplotsfig},\,\ref{irregularsubplotsfig}. The figures\,\ref{regularsubplotsfig},\,\ref{irregularsubplotsfig}. The
\code[subplot()]{subplot()} command allows to place multiple axes onto \code[subplot()]{subplot()} command allows to place multiple axes onto
a single sheet of paper. Generally, \varcode{subplot} expects three argument a single sheet of paper. Generally, \code{subplot()} expects three
defining the number of rows, column, and the currently active argument defining the number of rows, column, and the currently active
plot. The currently active plot number starts with 1 and goes up to plot. The currently active plot number starts with 1 and goes up to
$rows \cdot columns$ (numbers in the subplots in $rows \cdot columns$ (numbers in the subplots in
figures\,\ref{regularsubplotsfig}, \ref{irregularsubplotsfig}). figures\,\ref{regularsubplotsfig}, \ref{irregularsubplotsfig}).
@ -439,7 +438,7 @@ figures\,\ref{regularsubplotsfig}, \ref{irregularsubplotsfig}).
By default, all subplots have the same size, if something else is By default, all subplots have the same size, if something else is
desired, e.g.\ one subplot should span a whole row, while two others desired, e.g.\ one subplot should span a whole row, while two others
are smaller and should be placed side by side in the same row, the are smaller and should be placed side by side in the same row, the
third argument of \varcode{subplot} can be a vector or numbers that third argument of \code{subplot()} can be a vector or numbers that
should be joined. These have, of course, to be adjacent numbers should be joined. These have, of course, to be adjacent numbers
(\figref{irregularsubplotsfig}, (\figref{irregularsubplotsfig},
listing\,\ref{irregularsubplotslisting}). listing\,\ref{irregularsubplotslisting}).
@ -457,7 +456,7 @@ columns, need to be used in a plot. If you want to create something
more elaborate, or have more spacing between the subplots one can more elaborate, or have more spacing between the subplots one can
create a grid with larger numbers of columns and rows, and specify the create a grid with larger numbers of columns and rows, and specify the
used cells of the grid by passing a vector as the third argument to used cells of the grid by passing a vector as the third argument to
\varcode{subplot}. \code{subplot()}.
\lstinputlisting[caption={Script for creating subplots of different \lstinputlisting[caption={Script for creating subplots of different
sizes \figref{irregularsubplotsfig}.}, sizes \figref{irregularsubplotsfig}.},
@ -498,12 +497,12 @@ more apt. Accordingly, four arguments are needed (line 12 in listing
\ref{errorbarlisting}). The first two arguments are the same, the next \ref{errorbarlisting}). The first two arguments are the same, the next
to represent the positive and negative deflections. to represent the positive and negative deflections.
By default the \code{errorbar} function does not draw a marker. In the By default the \code{errorbar()} function does not draw a marker. In the
examples shown here we provide extra arguments to define that a circle examples shown here we provide extra arguments to define that a circle
is used for that purpose. The line connecting the average values can is used for that purpose. The line connecting the average values can
be removed by passing additional arguments. The properties of the be removed by passing additional arguments. The properties of the
errorbars themselves (linestyle, linewidth, capsize, etc.) can be errorbars themselves (linestyle, linewidth, capsize, etc.) can be
changed by taking the return argument of \code{errorbar} and changing changed by taking the return argument of \code{errorbar()} and changing
its properties. See the \matlab{} help for more information. its properties. See the \matlab{} help for more information.
\begin{figure}[ht] \begin{figure}[ht]
@ -530,18 +529,18 @@ areas instead of errorbars: In case you have a lot of data points with
respective errorbars such that they would merge in the figure it is respective errorbars such that they would merge in the figure it is
cleaner and probably easier to read and handle if one uses an error cleaner and probably easier to read and handle if one uses an error
area instead. To achieve an illustration as shown in area instead. To achieve an illustration as shown in
figure\,\ref{errorbarplot} C, we use the \code{fill} command in figure\,\ref{errorbarplot} C, we use the \code{fill()} command in
combination with a standard line plot. The original purpose of combination with a standard line plot. The original purpose of
\code{fill} is to draw a filled polygon. We hence have to provide it \code{fill()} is to draw a filled polygon. We hence have to provide it
with the vertex points of the polygon. For each x-value we now have with the vertex points of the polygon. For each x-value we now have
two y-values (average minus error and average plus error). Further, we two y-values (average minus error and average plus error). Further, we
want the vertices to be connected in a defined order. One can achieve want the vertices to be connected in a defined order. One can achieve
this by going back and forth on the x-axis; we append a reversed this by going back and forth on the x-axis; we append a reversed
version of the x-values to the original x-values using \code{cat} and version of the x-values to the original x-values using \code{cat()} and
\code{fliplr} for concatenation and inversion, respectively (line 3 in \code{fliplr()} for concatenation and inversion, respectively (line 3 in
listing \ref{errorbarlisting2}; Depending on the layout of your data listing \ref{errorbarlisting2}; Depending on the layout of your data
you may need concatenate along a different dimension of the data and you may need concatenate along a different dimension of the data and
use \code{flipud} instead). The y-coordinates of the polygon vertices use \code{flipud()} instead). The y-coordinates of the polygon vertices
are concatenated in a similar way (line 4). In the example shown here are concatenated in a similar way (line 4). In the example shown here
we accept the polygon object that is returned by fill (variable p) and we accept the polygon object that is returned by fill (variable p) and
use it to change a few properties of the polygon. The \emph{FaceAlpha} use it to change a few properties of the polygon. The \emph{FaceAlpha}
@ -561,9 +560,9 @@ connecting the average values (line 12).
The \code[text()]{text()} or \code[annotation()]{annotation()} are The \code[text()]{text()} or \code[annotation()]{annotation()} are
used for highlighting certain parts of a plot or simply adding an used for highlighting certain parts of a plot or simply adding an
annotation that does not fit or does not belong into the legend. annotation that does not fit or does not belong into the legend.
While \varcode{text} simply prints out the given text string at the While \code{text()} simply prints out the given text string at the
defined position (for example line in defined position (for example line in
listing\,\ref{regularsubplotlisting}) the \varcode{annotation} listing\,\ref{regularsubplotlisting}) the \code{annotation()}
function allows to add some more advanced highlights like arrows, function allows to add some more advanced highlights like arrows,
lines, ellipses, or rectangles. Figure\,\ref{annotationsplot} shows lines, ellipses, or rectangles. Figure\,\ref{annotationsplot} shows
some examples, the respective code can be found in some examples, the respective code can be found in
@ -583,9 +582,9 @@ listing\,\ref{annotationsplotlisting}. For more options consult the
\begin{important}[Positions in data or figure coordinates.] \begin{important}[Positions in data or figure coordinates.]
A very confusing pitfall are the different coordinate systems used A very confusing pitfall are the different coordinate systems used
by \varcode{text} and \varcode{annotation}. While \varcode{text} by \varcode{text()} and \varcode{annotation()}. While \varcode{text()}
expects the positions to be in data coordinates, i.e.\,in the limits expects the positions to be in data coordinates, i.e.\,in the limits
of the x- and y-axis, \varcode{annotation} requires the positions to of the x- and y-axis, \varcode{annotation()} requires the positions to
be given in normalized figure coordinates. Normalized means that the be given in normalized figure coordinates. Normalized means that the
width and height of the figure are expressed by numbers in the range width and height of the figure are expressed by numbers in the range
0 to 1. The bottom/left corner then has the coordinates $(0,0)$ and 0 to 1. The bottom/left corner then has the coordinates $(0,0)$ and
@ -624,9 +623,9 @@ Lissajous figure. The basic steps are:
is created and opened for writing. This also implies that is has to is created and opened for writing. This also implies that is has to
be closed after the whole process (line 31). be closed after the whole process (line 31).
\item For each frame of the video, we plot the appropriate data (we \item For each frame of the video, we plot the appropriate data (we
use \code[scatter]{scatter} for this purpose, line 20) and ``grab'' use \code{scatter()} for this purpose, line 20) and ``grab''
the frame (line 28). Grabbing is similar to making a screenshot of the frame (line 28). Grabbing is similar to making a screenshot of
the figure. The \code{drawnow}{drawnow} command (line 27) is used to the figure. The \code{drawnow()} command (line 27) is used to
stop the excution of the for loop until the drawing process is stop the excution of the for loop until the drawing process is
finished. finished.
\item Write the frame to file (line 29). \item Write the frame to file (line 29).

30
pointprocesses/lecture/pointprocesses.tex
View File

@ -73,10 +73,10 @@ number of observed events within a certain time window $n_i$
(\figref{pointprocessscetchfig}). (\figref{pointprocessscetchfig}).
\begin{exercise}{rasterplot.m}{} \begin{exercise}{rasterplot.m}{}
Implement a function \code{rasterplot()} that displays the times of Implement a function \varcode{rasterplot()} that displays the times of
action potentials within the first \code{tmax} seconds in a raster action potentials within the first \varcode{tmax} seconds in a raster
plot. The spike times (in seconds) recorded in the individual trials plot. The spike times (in seconds) recorded in the individual trials
are stored as vectors of times within a \codeterm{cell-array}. are stored as vectors of times within a \codeterm{cell array}.
\end{exercise} \end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -95,10 +95,10 @@ describing the statistics of stochastic real-valued variables:
\end{figure} \end{figure}
\begin{exercise}{isis.m}{} \begin{exercise}{isis.m}{}
Implement a function \code{isis()} that calculates the interspike Implement a function \varcode{isis()} that calculates the interspike
intervals from several spike trains. The function should return a intervals from several spike trains. The function should return a
single vector of intervals. The spike times (in seconds) of each single vector of intervals. The spike times (in seconds) of each
trial are stored as vectors within a \codeterm{cell-array}. trial are stored as vectors within a cell-array.
\end{exercise} \end{exercise}
%\subsection{First order interval statistics} %\subsection{First order interval statistics}
@ -117,7 +117,7 @@ describing the statistics of stochastic real-valued variables:
\end{itemize} \end{itemize}
\begin{exercise}{isihist.m}{} \begin{exercise}{isihist.m}{}
Implement a function \code{isiHist()} that calculates the normalized Implement a function \varcode{isiHist()} that calculates the normalized
interspike interval histogram. The function should take two input interspike interval histogram. The function should take two input
arguments; (i) a vector of interspike intervals and (ii) the width arguments; (i) a vector of interspike intervals and (ii) the width
of the bins used for the histogram. It further returns the of the bins used for the histogram. It further returns the
@ -126,7 +126,7 @@ describing the statistics of stochastic real-valued variables:
\begin{exercise}{plotisihist.m}{} \begin{exercise}{plotisihist.m}{}
Implement a function that takes the return values of Implement a function that takes the return values of
\code{isiHist()} as input arguments and then plots the data. The \varcode{isiHist()} as input arguments and then plots the data. The
plot should show the histogram with the x-axis scaled to plot should show the histogram with the x-axis scaled to
milliseconds and should be annotated with the average ISI, the milliseconds and should be annotated with the average ISI, the
standard deviation and the coefficient of variation. standard deviation and the coefficient of variation.
@ -167,7 +167,7 @@ $\rho_k$ is usually plotted against the lag $k$
with itself and is always 1. with itself and is always 1.
\begin{exercise}{isiserialcorr.m}{} \begin{exercise}{isiserialcorr.m}{}
Implement a function \code{isiserialcorr()} that takes a vector of Implement a function \varcode{isiserialcorr()} that takes a vector of
interspike intervals as input argument and calculates the serial interspike intervals as input argument and calculates the serial
correlation. The function should further plot the serial correlation. The function should further plot the serial
correlation. \pagebreak[4] correlation. \pagebreak[4]
@ -213,12 +213,12 @@ time interval , \determ{Feuerrate}) that is given in Hertz
% \end{figure} % \end{figure}
\begin{exercise}{counthist.m}{} \begin{exercise}{counthist.m}{}
Implement a function \code{counthist()} that calculates and plots Implement a function \varcode{counthist()} that calculates and plots
the distribution of spike counts observed in a certain time the distribution of spike counts observed in a certain time
window. The function should take two input arguments: (i) a window. The function should take two input arguments: (i) a
\codeterm{cell-array} of vectors containing the spike times in cell-array of vectors containing the spike times in seconds observed
seconds observed in a number of trials, and (ii) the duration of the in a number of trials, and (ii) the duration of the time window that
time window that is used to evaluate the counts.\pagebreak[4] is used to evaluate the counts.\pagebreak[4]
\end{exercise} \end{exercise}
@ -244,7 +244,7 @@ In an \enterm[Poisson process!inhomogeneous]{inhomogeneous Poisson
\lambda(t)$. \lambda(t)$.
\begin{exercise}{poissonspikes.m}{} \begin{exercise}{poissonspikes.m}{}
Implement a function \code{poissonspikes()} that uses a homogeneous Implement a function \varcode{poissonspikes()} that uses a homogeneous
Poisson process to generate events at a given rate for a certain Poisson process to generate events at a given rate for a certain
duration and a number of trials. The rate should be given in Hertz duration and a number of trials. The rate should be given in Hertz
and the duration of the trials is given in seconds. The function and the duration of the trials is given in seconds. The function
@ -293,7 +293,7 @@ The homogeneous Poisson process has the following properties:
\end{itemize} \end{itemize}
\begin{exercise}{hompoissonspikes.m}{} \begin{exercise}{hompoissonspikes.m}{}
Implement a function \code{hompoissonspikes()} that uses a Implement a function \varcode{hompoissonspikes()} that uses a
homogeneous Poisson process to generate spike events at a given rate homogeneous Poisson process to generate spike events at a given rate
for a certain duration and a number of trials. The rate should be for a certain duration and a number of trials. The rate should be
given in Hertz and the duration of the trials is given in given in Hertz and the duration of the trials is given in
@ -422,7 +422,7 @@ potentials (\figref{binpsthfig} top). The resulting histogram is then
normalized with the bin width $W$ to yield the firing rate shown in normalized with the bin width $W$ to yield the firing rate shown in
the bottom trace of figure \ref{binpsthfig}. The above sketched the bottom trace of figure \ref{binpsthfig}. The above sketched
process is equivalent to estimating the probability density. It is process is equivalent to estimating the probability density. It is
possible to estimate the PSTH using the \code{hist()} method possible to estimate the PSTH using the \code{hist()} function
\sindex[term]{Feuerrate!Binningmethode} \sindex[term]{Feuerrate!Binningmethode}
The estimated firing rate is valid for the total duration of each The estimated firing rate is valid for the total duration of each

154
programming/lecture/programming.tex
View File

@ -112,7 +112,7 @@ x y z
\begin{important}[Naming conventions] \begin{important}[Naming conventions]
There are a few rules regarding variable names. \matlab{} is There are a few rules regarding variable names. \matlab{} is
case-sensitive, i.e. \code{x} and \code{X} are two different case-sensitive, i.e. \varcode{x} and \varcode{X} are two different
names. Names must begin with an alphabetic character. German (or names. Names must begin with an alphabetic character. German (or
other) umlauts, special characters and spaces are forbidden in other) umlauts, special characters and spaces are forbidden in
variable names. variable names.
@ -689,9 +689,11 @@ then compare it to the elements on each page, and so on. An
alternative way is to make use of the so called \emph{linear indexing} alternative way is to make use of the so called \emph{linear indexing}
in which each element of the matrix is addressed by a single in which each element of the matrix is addressed by a single
number. The linear index thus ranges from 1 to number. The linear index thus ranges from 1 to
\code{numel(matrix)}. The linear index increases first along the 1st, \code[numel()]{numel(matrix)}. The linear index increases first along
2nd, 3rd etc. dimension (figure~\ref{matrixlinearindexingfig}). It is the 1st, 2nd, 3rd etc. dimension
not as intuitive since one would need to know the shape of the matrix and perform a remapping, but can be really helpful (figure~\ref{matrixlinearindexingfig}). It is not as intuitive since
one would need to know the shape of the matrix and perform a
remapping, but can be really helpful
(listing~\ref{matrixLinearIndexing}). (listing~\ref{matrixLinearIndexing}).
@ -882,10 +884,11 @@ table~\ref{logicaloperators}) which are introduced in the following
sections. sections.
\subsection{Relational operators} \subsection{Relational operators}
With \codeterm[Operator!relational]{relational operators} (table~\ref{relationaloperators}) With \codeterm[Operator!relational]{relational operators}
we can ask questions such as: ''Is the value of variable \code{a} (table~\ref{relationaloperators}) we can ask questions such as: ''Is
larger than the value of \code{b}?'' or ``Is the value in \code{a} the value of variable \varcode{a} larger than the value of
equal to the one stored in variable \code{b}?''. \varcode{b}?'' or ``Is the value in \varcode{a} equal to the one
stored in variable \varcode{b}?''.
\begin{table}[h!] \begin{table}[h!]
\titlecaption{\label{relationaloperators} \titlecaption{\label{relationaloperators}
@ -930,13 +933,13 @@ Testing the relations between numbers and scalar variables is straight
forward. When comparing vectors, the relational operator will be forward. When comparing vectors, the relational operator will be
applied element-wise and compare the respective elements of the applied element-wise and compare the respective elements of the
left-hand-side and right-hand-side vectors. Note: vectors must have left-hand-side and right-hand-side vectors. Note: vectors must have
the same length and orientation. The result of \code{[2 0 0 5 0] == [1 the same length and orientation. The result of \varcode{[2 0 0 5 0] == [1
0 3 2 0]'} in which the second vector is transposed to give a 0 3 2 0]'} in which the second vector is transposed to give a
column vector is a matrix! column vector is a matrix!
\subsection{Logical operators} \subsection{Logical operators}
With the relational operators we could for example test whether a With the relational operators we could for example test whether a
number is greater than a certain threshold (\code{x > 0.25}). But what number is greater than a certain threshold (\varcode{x > 0.25}). But what
if we wanted to check whether the number falls into the range greater if we wanted to check whether the number falls into the range greater
than 0.25 but less than 0.75? Numbers that fall into this range must than 0.25 but less than 0.75? Numbers that fall into this range must
satisfy the one and the other condition. With satisfy the one and the other condition. With
@ -1068,13 +1071,13 @@ values stored in a vector or matrix. It is very powerful and, once
understood, very intuitive. understood, very intuitive.
The basic concept is that applying a Boolean operation on a vector The basic concept is that applying a Boolean operation on a vector
results in a \code{logical} vector of the same size (see results in a \codeterm{logical} vector of the same size (see
listing~\ref{logicaldatatype}). This logical vector is then used to listing~\ref{logicaldatatype}). This logical vector is then used to
select only those values for which the logical vector is true. Line 14 select only those values for which the logical vector is true. Line 14
in listing~\ref{logicalindexing1} can be read: ``Select all those in listing~\ref{logicalindexing1} can be read: ``Select all those
elements of \varcode{x} where the Boolean expression \varcode{x < 0} elements of \varcode{x} where the Boolean expression \varcode{x < 0}
evaluates to true and store the result in the variable evaluates to true and store the result in the variable
\emph{x\_smaller\_zero}''. \varcode{x\_smaller\_zero}''.
\begin{lstlisting}[caption={Logical indexing.}, label=logicalindexing1] \begin{lstlisting}[caption={Logical indexing.}, label=logicalindexing1]
>> x = randn(1, 6) % a vector with 6 random numbers >> x = randn(1, 6) % a vector with 6 random numbers
@ -1154,13 +1157,14 @@ segment of data of a certain time span (the stimulus was on,
data and metadata in a single variable. data and metadata in a single variable.
\textbf{Cell arrays} Arrays of variables that contain different \textbf{Cell arrays} Arrays of variables that contain different
types. Unlike structures, the entries of a \codeterm{Cell array} are types. Unlike structures, the entries of a \codeterm{cell array} are
not named. Indexing in \codeterm{Cell arrays} requires a special not named. Indexing in \codeterm[cell array]{cell arrays} requires a
operator the \code{\{\}}. \matlab{} uses \codeterm{Cell arrays} for special operator the \code{\{\}}. \matlab{} uses \codeterm[cell
example when strings of different lengths should be stored in the array]{cell arrays} for example when strings of different lengths
same variable: \varcode{months = \{'Januar', 'February', 'March', should be stored in the same variable: \varcode{months = \{'Januar',
'April', 'May', 'Jun'\};}. Note the curly braces that are used to 'February', 'March', 'April', 'May', 'Jun'\};}. Note the curly
create the array and are also used for indexing. braces that are used to create the array and are also used for
indexing.
\textbf{Tables} Tabular structure that allows to have columns of \textbf{Tables} Tabular structure that allows to have columns of
varying type combined with a header (much like a spreadsheet). varying type combined with a header (much like a spreadsheet).
@ -1170,8 +1174,8 @@ segment of data of a certain time span (the stimulus was on,
irregular intervals togehter with the measurement time in a single irregular intervals togehter with the measurement time in a single
variable. Without the \codeterm{Timetable} data type at least two variable. Without the \codeterm{Timetable} data type at least two
variables (one storing the time, the other the measurement) would be variables (one storing the time, the other the measurement) would be
required. \codeterm{Timetables} offer specific convenience functions required. \codeterm[Timetable]{Timetables} offer specific
to work with timestamps. convenience functions to work with timestamps.
\textbf{Maps} In a \codeterm{map} a \codeterm{value} is associated \textbf{Maps} In a \codeterm{map} a \codeterm{value} is associated
with an arbitrary \codeterm{key}. The \codeterm{key} is not with an arbitrary \codeterm{key}. The \codeterm{key} is not
@ -1246,7 +1250,7 @@ All imperative programming languages offer a solution: the loop. It is
used whenever the same commands have to be repeated. used whenever the same commands have to be repeated.
\subsubsection{The \code{for} --- loop} \subsubsection{The \varcode{for} --- loop}
The most common type of loop is the \codeterm{for-loop}. It The most common type of loop is the \codeterm{for-loop}. It
consists of a \codeterm[Loop!head]{head} and the consists of a \codeterm[Loop!head]{head} and the
\codeterm[Loop!body]{body}. The head defines how often the code in the \codeterm[Loop!body]{body}. The head defines how often the code in the
@ -1258,7 +1262,7 @@ next value of this vector. In the body of the loop any code can be
executed which may or may not use the running variable for a certain executed which may or may not use the running variable for a certain
purpose. The \code{for} loop is closed with the keyword purpose. The \code{for} loop is closed with the keyword
\code{end}. Listing~\ref{looplisting} shows a simple version of such a \code{end}. Listing~\ref{looplisting} shows a simple version of such a
\code{for} loop. \codeterm{for-loop}.
\begin{lstlisting}[caption={Example of a \varcode{for}-loop.}, label=looplisting] \begin{lstlisting}[caption={Example of a \varcode{for}-loop.}, label=looplisting]
>> for x = 1:3 % head >> for x = 1:3 % head
@ -1273,15 +1277,15 @@ purpose. The \code{for} loop is closed with the keyword
\begin{exercise}{factorialLoop.m}{factorialLoop.out} \begin{exercise}{factorialLoop.m}{factorialLoop.out}
Can we solve the factorial with a for-loop? Implement a for loop that Can we solve the factorial with a \varcode{for}-loop? Implement a
calculates the factorial of a number \varcode{n}. for loop that calculates the factorial of a number \varcode{n}.
\end{exercise} \end{exercise}
\subsubsection{The \varcode{while} --- loop} \subsubsection{The \varcode{while} --- loop}
The \code{while}--loop is the second type of loop that is available in The \codeterm{while-loop} is the second type of loop that is available in
almost all programming languages. Other, than the \code{for} -- loop, almost all programming languages. Other, than the \codeterm{for-loop},
that iterates with the running variable over a vector, the while loop that iterates with the running variable over a vector, the while loop
uses a Boolean expression to determine when to execute the code in uses a Boolean expression to determine when to execute the code in
it's body. The head of the loop starts with the keyword \code{while} it's body. The head of the loop starts with the keyword \code{while}
@ -1289,22 +1293,22 @@ that is followed by a Boolean expression. If this can be evaluated to
true, the code in the body is executed. The loop is closed with an true, the code in the body is executed. The loop is closed with an
\code{end}. \code{end}.
\begin{lstlisting}[caption={Basic structure of a \code{while} loop.}, label=whileloop] \begin{lstlisting}[caption={Basic structure of a \varcode{while} loop.}, label=whileloop]
while x == true % head with a Boolean expression while x == true % head with a Boolean expression
% execute this code if the expression yields true % execute this code if the expression yields true
end end
\end{lstlisting} \end{lstlisting}
\begin{exercise}{factorialWhileLoop.m}{} \begin{exercise}{factorialWhileLoop.m}{}
Implement the factorial of a number \varcode{n} using a \code{while} Implement the factorial of a number \varcode{n} using a \varcode{while}-loop.
-- loop.
\end{exercise} \end{exercise}
\begin{exercise}{neverendingWhile.m}{} \begin{exercise}{neverendingWhile.m}{}
Implement a \code{while}--loop that is never-ending. Hint: the body Implement a \varcode{while}-loop that is never-ending. Hint: the
is executed as long as the Boolean expression in the head is body is executed as long as the Boolean expression in the head is
\code{true}. You can escape the loop by pressing \keycode{Ctrl+C}. \varcode{true}. You can escape the loop by pressing
\keycode{Ctrl+C}.
\end{exercise} \end{exercise}
@ -1312,15 +1316,15 @@ end
\begin{itemize} \begin{itemize}
\item Both execute the code in the body iterative. \item Both execute the code in the body iterative.
\item When using a \code{for} -- loop the body of the loop is executed \item When using a \code{for}-loop the body of the loop is executed
at least once (except when the vector used in the head is empty). at least once (except when the vector used in the head is empty).
\item In a \code{while} -- loop, the body is not necessarily \item In a \code{while}-loop, the body is not necessarily
executed. It is entered only if the Boolean expression in the head executed. It is entered only if the Boolean expression in the head
yields true. yields true.
\item The \code{for} -- loop is best suited for cases in which the \item The \code{for}-loop is best suited for cases in which the
elements of a vector have to be used for a computation or when the elements of a vector have to be used for a computation or when the
number of iterations is known. number of iterations is known.
\item The \code{while} -- loop is best suited for cases when it is not \item The \code{while}-loop is best suited for cases when it is not
known in advance how often a certain piece of code has to be known in advance how often a certain piece of code has to be
executed. executed.
\item Any problem that can be solved with one type can also be solve \item Any problem that can be solved with one type can also be solve
@ -1336,8 +1340,8 @@ is only executed under a certain condition.
\subsubsection{The \varcode{if} -- statement} \subsubsection{The \varcode{if} -- statement}
The most prominent representative of the conditional expressions is The most prominent representative of the conditional expressions is
the \code{if} statement (sometimes also called \code{if - else} the \codeterm{if statement} (sometimes also called \codeterm{if - else
statement). It constitutes a kind of branching point. It allows to statement}). It constitutes a kind of branching point. It allows to
control which branch of the code is executed. control which branch of the code is executed.
Again, the statement consists of the head and the body. The head Again, the statement consists of the head and the body. The head
@ -1346,11 +1350,11 @@ that controls whether or not the body is entered. Optionally, the body
can be either ended by the \code{end} keyword or followed by can be either ended by the \code{end} keyword or followed by
additional statements \code{elseif}, which allows to add another additional statements \code{elseif}, which allows to add another
Boolean expression and to catch another condition or the \code{else} Boolean expression and to catch another condition or the \code{else}
the provide a default case. The last body of the \code{if - elseif - the provide a default case. The last body of the \varcode{if - elseif -
else} statement has to be finished with the \code{end} else} statement has to be finished with the \code{end}
(listing~\ref{ifelselisting}). (listing~\ref{ifelselisting}).
\begin{lstlisting}[label=ifelselisting, caption={Structure of an \code{if} statement.}] \begin{lstlisting}[label=ifelselisting, caption={Structure of an \varcode{if} statement.}]
if x < y % head if x < y % head
% body I, executed only if x < y % body I, executed only if x < y
elseif x > y elseif x > y
@ -1361,7 +1365,7 @@ end
\end{lstlisting} \end{lstlisting}
\begin{exercise}{ifelse.m}{} \begin{exercise}{ifelse.m}{}
Draw a random number and check with an appropriate \code{if} Draw a random number and check with an appropriate \varcode{if}
statement whether it is statement whether it is
\begin{enumerate} \begin{enumerate}
\item less than 0.5. \item less than 0.5.
@ -1373,9 +1377,9 @@ end
\subsubsection{The \varcode{switch} -- statement} \subsubsection{The \varcode{switch} -- statement}
The \code{switch} statement is used whenever a set of conditions The \codeterm{switch statement} is used whenever a set of conditions
requires separate treatment. The statement is initialized with the requires separate treatment. The statement is initialized with the
\code{switch} keyword that is followed by \emph{switch expression} (a \code{switch} keyword that is followed by a \emph{switch expression} (a
number or string). It is followed by a set of \emph{case expressions} number or string). It is followed by a set of \emph{case expressions}
which start with the keyword \code{case} followed by the condition which start with the keyword \code{case} followed by the condition
that defines against which the \emph{switch expression} is tested. It that defines against which the \emph{switch expression} is tested. It
@ -1412,7 +1416,7 @@ end
\end{itemize} \end{itemize}
\subsection{The keywords \code{break} and \code{continue}} \subsection{The keywords \varcode{break} and \varcode{continue}}
Whenever the execution of a loop should be ended or if you want to Whenever the execution of a loop should be ended or if you want to
skip the execution of the body under certain circumstances, one can skip the execution of the body under certain circumstances, one can
@ -1458,7 +1462,7 @@ end
has passed between the calls of \code{tic} and \code{toc}. has passed between the calls of \code{tic} and \code{toc}.
\begin{enumerate} \begin{enumerate}
\item Use a \code{for} loop to select matching values. \item Use a \varcode{for} loop to select matching values.
\item Use logical indexing. \item Use logical indexing.
\end{enumerate} \end{enumerate}
\end{exercise} \end{exercise}
@ -1486,12 +1490,12 @@ and executed line-by-line from top to bottom.
\matlab{} knows three types of programs: \matlab{} knows three types of programs:
\begin{enumerate} \begin{enumerate}
\item \codeterm[Script]{Scripts} \item \entermde[script]{Skripte}{Scripts}
\item \codeterm[Function]{Functions} \item \entermde[function]{Funktion}{Functions}
\item \codeterm[Object]{Objects} (not covered here) \item \entermde[Object]{Objekte}{Objects} (not covered here)
\end{enumerate} \end{enumerate}
Programs are stored in so called \codeterm{m-files} Programs are stored in so called \codeterm[m-file]{m-files}
(e.g. \file{myProgram.m}). To use them they have to be \emph{called} (e.g. \file{myProgram.m}). To use them they have to be \emph{called}
from the command line or from within another program. Storing your code in from the command line or from within another program. Storing your code in
programs increases the re-usability. So far we have used programs increases the re-usability. So far we have used
@ -1507,13 +1511,13 @@ and if it now wants to read the previously stored variable, it will
contain a different value than expected. Bugs like this are hard to contain a different value than expected. Bugs like this are hard to
find since each of the programs alone is perfectly fine and works as find since each of the programs alone is perfectly fine and works as
intended. A solution for this problem are the intended. A solution for this problem are the
\codeterm[Function]{functions}. \entermde[function]{Funktion}{functions}.
\subsection{Functions} \subsection{Functions}
Functions in \matlab{} are similar to mathematical functions Functions in \matlab{} are similar to mathematical functions
\[ y = f(x) \] Here, the mathematical function has the name $f$ and it \[ y = f(x) \] Here, the mathematical function has the name $f$ and it
has one \codeterm{argument} $x$ that is transformed into the has one \entermde{Argument}{argument} $x$ that is transformed into the
function's output value $y$. In \matlab{} the syntax of a function function's output value $y$. In \matlab{} the syntax of a function
declaration is very similar (listing~\ref{functiondefinitionlisting}). declaration is very similar (listing~\ref{functiondefinitionlisting}).
@ -1524,12 +1528,12 @@ function [y] = functionName(arg_1, arg_2)
\end{lstlisting} \end{lstlisting}
The keyword \code{function} is followed by the return value(s) (it can The keyword \code{function} is followed by the return value(s) (it can
be a list \code{[]} of values), the function name and the be a list \varcode{[]} of values), the function name and the
argument(s). The function head is then followed by the function's argument(s). The function head is then followed by the function's
body. A function is ended by and \code{end} (this is in fact optional body. A function is ended by and \code{end} (this is in fact optional
but we will stick to this). Each function that should be directly used but we will stick to this). Each function that should be directly used
by the user (or called from other programs) should reside in an by the user (or called from other programs) should reside in an
individual \code{m-file} that has the same name as the function. By individual \codeterm{m-file} that has the same name as the function. By
using functions instead of scripts we gain several advantages: using functions instead of scripts we gain several advantages:
\begin{itemize} \begin{itemize}
\item Encapsulation of program code that solves a certain task. It can \item Encapsulation of program code that solves a certain task. It can
@ -1566,10 +1570,9 @@ function myFirstFunction() % function head
end end
\end{lstlisting} \end{lstlisting}
\code{myFirstFunction} (listing~\ref{badsinewavelisting}) is a \varcode{myFirstFunction} (listing~\ref{badsinewavelisting}) is a
prime-example of a bad function. There are several issues with it's prime-example of a bad function. There are several issues with it's
design: design:
\begin{itemize} \begin{itemize}
\item The function's name does not tell anything about it's purpose. \item The function's name does not tell anything about it's purpose.
\item The function is made for exactly one use-case (frequency of \item The function is made for exactly one use-case (frequency of
@ -1594,7 +1597,7 @@ defined:
(e.g. the user of another program that calls a function)? (e.g. the user of another program that calls a function)?
\end{enumerate} \end{enumerate}
As indicated above the \code{myFirstFunction} does three things at As indicated above the \varcode{myFirstFunction} does three things at
once, it seems natural, that the task should be split up into three once, it seems natural, that the task should be split up into three
parts. (i) Calculation of the individual sine waves defined by the parts. (i) Calculation of the individual sine waves defined by the
frequency and the amplitudes (ii) graphical display of the data and frequency and the amplitudes (ii) graphical display of the data and
@ -1607,17 +1610,18 @@ define (i) how to name the function, (ii) which information it needs
(arguments), and (iii) what it should return to the caller. (arguments), and (iii) what it should return to the caller.
\begin{enumerate} \begin{enumerate}
\item \codeterm[Function!Name]{Name}: the name should be descriptive \item \entermde[function!name]{Funktion!-sname}{Name}: the name should be descriptive
of the function's purpose, i.e. the calculation of a sine wave. A of the function's purpose, i.e. the calculation of a sine wave. A
appropriate name might be \code{sinewave()}. appropriate name might be \varcode{sinewave()}.
\item \codeterm[Function!Arguments]{Arguments}: What information does \item \entermde[function!arguments]{Funktion!-sargument}{Arguments}:
the function need to do the calculation? There are obviously the What information does the function need to do the calculation? There
frequency as well as the amplitude. Further we may want to be able are obviously the frequency as well as the amplitude. Further we may
to define the duration of the sine wave and the temporal want to be able to define the duration of the sine wave and the
resolution. We thus need four arguments which should also named to temporal resolution. We thus need four arguments which should also
describe their content: \code{amplitude, frequency, t\_max,} and named to describe their content: \varcode{amplitude},
\code{t\_step} might be good names. \varcode{frequency}, \varcode{t\_max}, and \varcode{t\_step} might
\item \codeterm[Function!Return values]{Return values}: For a correct be good names.
\item \entermde[function!return values]{Funktion!R\"uckgabewerte}{Return values}: For a correct
display of the data we need two vectors. The time, and the sine wave display of the data we need two vectors. The time, and the sine wave
itself. We just need two return values: \varcode{time}, \varcode{sine} itself. We just need two return values: \varcode{time}, \varcode{sine}
\end{enumerate} \end{enumerate}
@ -1657,11 +1661,11 @@ specification of the function:
\begin{enumerate} \begin{enumerate}
\item It should plot a single sine wave. But it is not limited to sine \item It should plot a single sine wave. But it is not limited to sine
waves. It's name is thus: \code{plotFunction()}. waves. It's name is thus: \varcode{plotFunction()}.
\item What information does it need to solve the task? The \item What information does it need to solve the task? The
to-be-plotted data as there is the values \code{y\_data} and the to-be-plotted data as there is the values \varcode{y\_data} and the
corresponding \code{x\_data}. As we want to plot series of sine corresponding \varcode{x\_data}. As we want to plot series of sine
waves we might want to have a \code{name} for each function to be waves we might want to have a \varcode{name} for each function to be
displayed in the figure legend. displayed in the figure legend.
\item Are there any return values? No, this function is just made for \item Are there any return values? No, this function is just made for
plotting, we do not need to return anything. plotting, we do not need to return anything.
@ -1699,11 +1703,11 @@ Again, we need to specify what needs to be done:
appropriate name for the script (that is the name of the m-file) appropriate name for the script (that is the name of the m-file)
might be \file{plotMultipleSinewaves.m}. might be \file{plotMultipleSinewaves.m}.
\item What information do we need? we need to define the \item What information do we need? we need to define the
\code{frequency}, the range of \code{amplitudes}, the \varcode{frequency}, the range of \varcode{amplitudes}, the
\code{duration} of the sine waves, and the temporal resolution given \varcode{duration} of the sine waves, and the temporal resolution given
as the time between to points in time, i.e. the \code{stepsize}. as the time between to points in time, i.e. the \varcode{stepsize}.
\item We then need to create an empty figure, and work through the \item We then need to create an empty figure, and work through the
rang of \code{amplitudes}. We must not forget to switch \code{hold rang of \varcode{amplitudes}. We must not forget to switch \varcode{hold
on} if we want to see all the sine waves in one plot. on} if we want to see all the sine waves in one plot.
\end{enumerate} \end{enumerate}

45
regression/lecture/regression.tex
View File

@ -33,7 +33,7 @@ fitting approaches. We will apply this method to find the combination
of slope and intercept that best describes the system. of slope and intercept that best describes the system.
\section{The error function --- mean square error} \section{The error function --- mean squared error}
Before the optimization can be done we need to specify what is Before the optimization can be done we need to specify what is
considered an optimal fit. In our example we search the parameter considered an optimal fit. In our example we search the parameter
@ -57,25 +57,23 @@ $\sum_{i=1}^N |y_i - y^{est}_i|$. The total error can only be small if
all deviations are indeed small no matter if they are above or below all deviations are indeed small no matter if they are above or below
the prediced line. Instead of the sum we could also ask for the the prediced line. Instead of the sum we could also ask for the
\emph{average} \emph{average}
\begin{equation} \begin{equation}
\label{meanabserror} \label{meanabserror}
f_{dist}(\{(x_i, y_i)\}|\{y^{est}_i\}) = \frac{1}{N} \sum_{i=1}^N |y_i - y^{est}_i| f_{dist}(\{(x_i, y_i)\}|\{y^{est}_i\}) = \frac{1}{N} \sum_{i=1}^N |y_i - y^{est}_i|
\end{equation} \end{equation}
should be small. Commonly, the \enterm{mean squared distance} oder should be small. Commonly, the \enterm{mean squared distance} oder
\enterm{mean squared error} \enterm[square error!mean]{mean square error} (\determ[quadratischer Fehler!mittlerer]{mittlerer quadratischer Fehler})
\begin{equation} \begin{equation}
\label{meansquarederror} \label{meansquarederror}
f_{mse}(\{(x_i, y_i)\}|\{y^{est}_i\}) = \frac{1}{N} \sum_{i=1}^N (y_i - y^{est}_i)^2 f_{mse}(\{(x_i, y_i)\}|\{y^{est}_i\}) = \frac{1}{N} \sum_{i=1}^N (y_i - y^{est}_i)^2
\end{equation} \end{equation}
is used (\figref{leastsquareerrorfig}). Similar to the absolute is used (\figref{leastsquareerrorfig}). Similar to the absolute
distance, the square of the error($(y_i - y_i^{est})^2$) is always distance, the square of the error($(y_i - y_i^{est})^2$) is always
positive error values do not cancel out. The square further punishes positive error values do not cancel out. The square further punishes
large deviations. large deviations.
\begin{exercise}{meanSquareError.m}{}\label{mseexercise}% \begin{exercise}{meanSquareError.m}{}\label{mseexercise}%
Implement a function \code{meanSquareError()}, that calculates the Implement a function \varcode{meanSquareError()}, that calculates the
\emph{mean square distance} between a vector of observations ($y$) \emph{mean square distance} between a vector of observations ($y$)
and respective predictions ($y^{est}$). and respective predictions ($y^{est}$).
\end{exercise} \end{exercise}
@ -84,18 +82,19 @@ large deviations.
\section{\tr{Objective function}{Zielfunktion}} \section{\tr{Objective function}{Zielfunktion}}
$f_{cost}(\{(x_i, y_i)\}|\{y^{est}_i\})$ is a so called $f_{cost}(\{(x_i, y_i)\}|\{y^{est}_i\})$ is a so called
\enterm{objective function} or \enterm{cost function}. We aim to adapt \enterm{objective function} or \enterm{cost function}
the model parameters to minimize the error (mean square error) and (\determ{Kostenfunktion}). We aim to adapt the model parameters to
thus the \emph{objective function}. In Chapter~\ref{maximumlikelihoodchapter} minimize the error (mean square error) and thus the \emph{objective
we will show that the minimization of the mean square error is function}. In Chapter~\ref{maximumlikelihoodchapter} we will show
equivalent to maximizing the likelihood that the observations that the minimization of the mean square error is equivalent to
originate from the model (assuming a normal distribution of the data maximizing the likelihood that the observations originate from the
around the model prediction). model (assuming a normal distribution of the data around the model
prediction).
\begin{figure}[t] \begin{figure}[t]
\includegraphics[width=1\textwidth]{linear_least_squares} \includegraphics[width=1\textwidth]{linear_least_squares}
\titlecaption{Estimating the \emph{mean square error}.} {The \titlecaption{Estimating the \emph{mean square error}.} {The
deviation (\enterm{error}, orange) between the prediction (red deviation error, orange) between the prediction (red
line) and the observations (blue dots) is calculated for each data line) and the observations (blue dots) is calculated for each data
point (left). Then the deviations are squared and the aveage is point (left). Then the deviations are squared and the aveage is
calculated (right).} calculated (right).}
@ -119,11 +118,13 @@ Replacing $y^{est}$ with the linear equation (the model) in
That is, the mean square error is given the pairs $(x_i, y_i)$ and the That is, the mean square error is given the pairs $(x_i, y_i)$ and the
parameters $m$ and $b$ of the linear equation. The optimization parameters $m$ and $b$ of the linear equation. The optimization
process will not try to optimize $m$ and $b$ to lead to the smallest process tries to optimize $m$ and $b$ such that the error is
error, the method of the \enterm{least square error}. minimized, the method of the \enterm[square error!least]{least square
error} (\determ[quadratischer Fehler!kleinster]{Methode der
kleinsten Quadrate}).
\begin{exercise}{lsqError.m}{} \begin{exercise}{lsqError.m}{}
Implement the objective function \code{lsqError()} that applies the Implement the objective function \varcode{lsqError()} that applies the
linear equation as a model. linear equation as a model.
\begin{itemize} \begin{itemize}
\item The function takes three arguments. The first is a 2-element \item The function takes three arguments. The first is a 2-element
@ -131,7 +132,7 @@ error, the method of the \enterm{least square error}.
\varcode{b}. The second is a vector of x-values the third contains \varcode{b}. The second is a vector of x-values the third contains
the measurements for each value of $x$, the respecive $y$-values. the measurements for each value of $x$, the respecive $y$-values.
\item The function returns the mean square error \eqnref{mseline}. \item The function returns the mean square error \eqnref{mseline}.
\item The function should call the function \code{meanSquareError()} \item The function should call the function \varcode{meanSquareError()}
defined in the previouos exercise to calculate the error. defined in the previouos exercise to calculate the error.
\end{itemize} \end{itemize}
\end{exercise} \end{exercise}
@ -165,7 +166,7 @@ third dimension is used to indicate the error value
\varcode{y}). Implement a script \file{errorSurface.m}, that \varcode{y}). Implement a script \file{errorSurface.m}, that
calculates the mean square error between data and a linear model and calculates the mean square error between data and a linear model and
illustrates the error surface using the \code{surf()} function illustrates the error surface using the \code{surf()} function
(consult the help to find out how to use \code{surf}.). (consult the help to find out how to use \code{surf()}.).
\end{exercise} \end{exercise}
By looking at the error surface we can directly see the position of By looking at the error surface we can directly see the position of
@ -257,7 +258,7 @@ way to the minimum of the objective function. The ball will always
follow the steepest slope. Thus we need to figure out the direction of follow the steepest slope. Thus we need to figure out the direction of
the steepest slope at the position of the ball. the steepest slope at the position of the ball.
The \enterm{gradient} (Box~\ref{partialderivativebox}) of the The \entermde{Gradient}{gradient} (Box~\ref{partialderivativebox}) of the
objective function is the vector objective function is the vector
\[ \nabla f_{cost}(m,b) = \left( \frac{\partial f(m,b)}{\partial m}, \[ \nabla f_{cost}(m,b) = \left( \frac{\partial f(m,b)}{\partial m},
@ -296,7 +297,7 @@ choose the opposite direction.
\end{figure} \end{figure}
\begin{exercise}{lsqGradient.m}{}\label{gradientexercise}% \begin{exercise}{lsqGradient.m}{}\label{gradientexercise}%
Implement a function \code{lsqGradient()}, that takes the set of Implement a function \varcode{lsqGradient()}, that takes the set of
parameters $(m, b)$ of the linear equation as a two-element vector parameters $(m, b)$ of the linear equation as a two-element vector
and the $x$- and $y$-data as input arguments. The function should and the $x$- and $y$-data as input arguments. The function should
return the gradient at that position. return the gradient at that position.
@ -316,8 +317,8 @@ choose the opposite direction.
Finally, we are able to implement the optimization itself. By now it Finally, we are able to implement the optimization itself. By now it
should be obvious why it is called the gradient descent method. All should be obvious why it is called the gradient descent method. All
ingredients are already there. We need: 1. The error function ingredients are already there. We need: 1. The error function
(\code{meanSquareError}), 2. the objective function (\varcode{meanSquareError}), 2. the objective function
(\code{lsqError()}), and 3. the gradient (\code{lsqGradient()}). The (\varcode{lsqError()}), and 3. the gradient (\varcode{lsqGradient()}). The
algorithm of the gradient descent is: algorithm of the gradient descent is:
\begin{enumerate} \begin{enumerate}

2
scientificcomputing-script.tex
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@ -9,6 +9,8 @@
\include{#1/lecture/#1}% \include{#1/lecture/#1}%
} }
%\includeonly{regression/lecture/regression}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document} \begin{document}

174
statistics/lecture/statistics.tex
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@ -7,17 +7,16 @@ Descriptive statistics characterizes data sets by means of a few measures.
In addition to histograms that estimate the full distribution of the data, In addition to histograms that estimate the full distribution of the data,
the following measures are used for characterizing univariate data: the following measures are used for characterizing univariate data:
\begin{description} \begin{description}
\item[Location, central tendency] (``Lagema{\ss}e''): \item[Location, central tendency] (\determ{Lagema{\ss}e}):
arithmetic mean, median, mode. \entermde[mean!arithmetic]{Mittel!arithmetisches}{arithmetic mean}, \entermde{Median}{median}, \enterm{mode}.
\item[Spread, dispersion] (``Streuungsma{\ss}e''): variance, \item[Spread, dispersion] (\determ{Streuungsma{\ss}e}): \entermde{Varianz}{variance},
standard deviation, inter-quartile range,\linebreak coefficient of variation \entermde{Standardabweichung}{standard deviation}, inter-quartile range,\linebreak \enterm{coefficient of variation} (\determ{Variationskoeffizient}).
(``Variationskoeffizient''). \item[Shape]: \enterm{skewness} (\determ{Schiefe}), \enterm{kurtosis} (\determ{W\"olbung}).
\item[Shape]: skewness (``Schiefe''), kurtosis (``W\"olbung'').
\end{description} \end{description}
For bivariate and multivariate data sets we can also analyse their For bivariate and multivariate data sets we can also analyse their
\begin{description} \begin{description}
\item[Dependence, association] (``Zusammenhangsma{\ss}e''): Pearson's correlation coefficient, \item[Dependence, association] (\determ{Zusammenhangsma{\ss}e}): \entermde[correlation!coefficient!Pearson's]{Korrelation!Pearson}{Pearson's correlation coefficient},
Spearman's rank correlation coefficient. \entermde[correlation!coefficient!Spearman's rank]{{Rangkorrelationskoeffizient!Spearman'scher}}{Spearman's rank correlation coefficient}.
\end{description} \end{description}
The following is in no way a complete introduction to descriptive The following is in no way a complete introduction to descriptive
@ -26,15 +25,16 @@ daily data-analysis problems.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mean, variance, and standard deviation} \section{Mean, variance, and standard deviation}
The \enterm{arithmetic mean} is a measure of location. For $n$ data values The \entermde[mean!arithmetic]{Mittel!arithmetisches}{arithmetic mean}
$x_i$ the arithmetic mean is computed by is a measure of location. For $n$ data values $x_i$ the arithmetic
mean is computed by
\[ \bar x = \langle x \rangle = \frac{1}{N}\sum_{i=1}^n x_i \; . \] \[ \bar x = \langle x \rangle = \frac{1}{N}\sum_{i=1}^n x_i \; . \]
This computation (summing up all elements of a vector and dividing by This computation (summing up all elements of a vector and dividing by
the length of the vector) is provided by the function \mcode{mean()}. the length of the vector) is provided by the function \mcode{mean()}.
The mean has the same unit as the data values. The mean has the same unit as the data values.
The dispersion of the data values around the mean is quantified by The dispersion of the data values around the mean is quantified by
their \enterm{variance} their \entermde{Varianz}{variance}
\[ \sigma^2_x = \langle (x-\langle x \rangle)^2 \rangle = \frac{1}{N}\sum_{i=1}^n (x_i - \bar x)^2 \; . \] \[ \sigma^2_x = \langle (x-\langle x \rangle)^2 \rangle = \frac{1}{N}\sum_{i=1}^n (x_i - \bar x)^2 \; . \]
The variance is computed by the function \mcode{var()}. The variance is computed by the function \mcode{var()}.
The unit of the variance is the unit of the data values squared. The unit of the variance is the unit of the data values squared.
@ -42,14 +42,15 @@ Therefore, variances cannot be compared to the mean or the data values
themselves. In particular, variances cannot be used for plotting error themselves. In particular, variances cannot be used for plotting error
bars along with the mean. bars along with the mean.
The standard deviation In contrast to the variance, the
\[ \sigma_x = \sqrt{\sigma^2_x} \; , \] \entermde{Standardabweichung}{standard deviation}
as computed by the function \mcode{std()}, however, has the same unit \[ \sigma_x = \sqrt{\sigma^2_x} \; , \]
as the data values and can (and should) be used to display the as computed by the function \mcode{std()} has the same unit as the
dispersion of the data together with their mean. data values and can (and should) be used to display the dispersion of
the data together with their mean.
The mean of a data set can be displayed by a bar-plot The mean of a data set can be displayed by a bar-plot
\matlabfun{bar()}. Additional errorbars \matlabfun{errobar()} can be \matlabfun{bar()}. Additional errorbars \matlabfun{errorbar()} can be
used to illustrate the standard deviation of the data used to illustrate the standard deviation of the data
(\figref{displayunivariatedatafig} (2)). (\figref{displayunivariatedatafig} (2)).
@ -90,18 +91,18 @@ used to illustrate the standard deviation of the data
identical with the mode.} identical with the mode.}
\end{figure} \end{figure}
The \enterm{mode} is the most frequent value, i.e. the position of the maximum of the probability distribution. The \enterm{mode} (\determ{Modus}) is the most frequent value,
i.e. the position of the maximum of the probability distribution.
The \enterm{median} separates a list of data values into two halves The \entermde{Median}{median} separates a list of data values into two
such that one half of the data is not greater and the other half is halves such that one half of the data is not greater and the other
not smaller than the median (\figref{medianfig}). half is not smaller than the median (\figref{medianfig}). The
function \mcode{median()} computes the median.
\begin{exercise}{mymedian.m}{} \begin{exercise}{mymedian.m}{}
Write a function \varcode{mymedian()} that computes the median of a vector. Write a function \varcode{mymedian()} that computes the median of a vector.
\end{exercise} \end{exercise}
\matlab{} provides the function \code{median()} for computing the median.
\begin{exercise}{checkmymedian.m}{} \begin{exercise}{checkmymedian.m}{}
Write a script that tests whether your median function really Write a script that tests whether your median function really
returns a median above which are the same number of data than returns a median above which are the same number of data than
@ -122,9 +123,9 @@ not smaller than the median (\figref{medianfig}).
\end{figure} \end{figure}
The distribution of data can be further characterized by the position The distribution of data can be further characterized by the position
of its \enterm[quartile]{quartiles}. Neighboring quartiles are of its \entermde[quartile]{Quartil}{quartiles}. Neighboring quartiles are
separated by 25\,\% of the data (\figref{quartilefig}). separated by 25\,\% of the data (\figref{quartilefig}).
\enterm[percentile]{Percentiles} allow to characterize the \entermde[percentile]{Perzentil}{Percentiles} allow to characterize the
distribution of the data in more detail. The 3$^{\rm rd}$ quartile distribution of the data in more detail. The 3$^{\rm rd}$ quartile
corresponds to the 75$^{\rm th}$ percentile, because 75\,\% of the corresponds to the 75$^{\rm th}$ percentile, because 75\,\% of the
data are smaller than the 3$^{\rm rd}$ quartile. data are smaller than the 3$^{\rm rd}$ quartile.
@ -147,11 +148,12 @@ data are smaller than the 3$^{\rm rd}$ quartile.
% from a normal distribution.} % from a normal distribution.}
% \end{figure} % \end{figure}
\enterm[box-whisker plots]{Box-whisker plots} are commonly used to \entermde[box-whisker plots]{Box-Whisker-Plot}{Box-whisker plots}, or
visualize and compare the distribution of unimodal data. A box is \entermde{Box-Plot}{box plot} are commonly used to visualize and
drawn around the median that extends from the 1$^{\rm st}$ to the compare the distribution of unimodal data. A box is drawn around the
3$^{\rm rd}$ quartile. The whiskers mark the minimum and maximum value median that extends from the 1$^{\rm st}$ to the 3$^{\rm rd}$
of the data set (\figref{displayunivariatedatafig} (3)). quartile. The whiskers mark the minimum and maximum value of the data
set (\figref{displayunivariatedatafig} (3)).
\begin{exercise}{univariatedata.m}{} \begin{exercise}{univariatedata.m}{}
Generate 40 normally distributed random numbers with a mean of 2 and Generate 40 normally distributed random numbers with a mean of 2 and
@ -170,13 +172,14 @@ of the data set (\figref{displayunivariatedatafig} (3)).
% \end{exercise} % \end{exercise}
\section{Distributions} \section{Distributions}
The distribution of values in a data set is estimated by histograms The \enterm{distribution} (\determ{Verteilung}) of values in a data
(\figref{displayunivariatedatafig} (4)). set is estimated by histograms (\figref{displayunivariatedatafig}
(4)).
\subsection{Histograms} \subsection{Histograms}
\enterm[histogram]{Histograms} count the frequency $n_i$ of \entermde[histogram]{Histogramm}{Histograms} count the frequency $n_i$
$N=\sum_{i=1}^M n_i$ measurements in each of $M$ bins $i$ of $N=\sum_{i=1}^M n_i$ measurements in each of $M$ bins $i$
(\figref{diehistogramsfig} left). The bins tile the data range (\figref{diehistogramsfig} left). The bins tile the data range
usually into intervals of the same size. The width of the bins is usually into intervals of the same size. The width of the bins is
called the bin width. The frequencies $n_i$ plotted against the called the bin width. The frequencies $n_i$ plotted against the
@ -194,13 +197,14 @@ categories $i$ is the \enterm{histogram}, or the \enterm{frequency
\end{figure} \end{figure}
Histograms are often used to estimate the Histograms are often used to estimate the
\enterm[probability!distribution]{probability distribution} of the \enterm[probability!distribution]{probability distribution}
data values. (\determ[Wahrscheinlichkeits!-verteilung]{Wahrscheinlichkeitsverteilung}) of the data values.
\subsection{Probabilities} \subsection{Probabilities}
In the frequentist interpretation of probability, the probability of In the frequentist interpretation of probability, the
an event (e.g. getting a six when rolling a die) is the relative \enterm{probability} (\determ{Wahrscheinlichkeit}) of an event
occurrence of this event in the limit of a large number of trials. (e.g. getting a six when rolling a die) is the relative occurrence of
this event in the limit of a large number of trials.
For a finite number of trials $N$ where the event $i$ occurred $n_i$ For a finite number of trials $N$ where the event $i$ occurred $n_i$
times, the probability $P_i$ of this event is estimated by times, the probability $P_i$ of this event is estimated by
@ -212,15 +216,16 @@ the sum of the probabilities of all possible events is one:
i.e. the probability of getting any event is one. i.e. the probability of getting any event is one.
\subsection{Probability distributions of categorial data} \subsection{Probability distributions of categorical data}
For categorial data values (e.g. the faces of a die (as integer For \entermde[data!categorical]{Daten!kategorische}{categorical} data
numbers or as colors)) a bin can be defined for each category $i$. values (e.g. the faces of a die (as integer numbers or as colors)) a
The histogram is normalized by the total number of measurements to bin can be defined for each category $i$. The histogram is normalized
make it independent of the size of the data set by the total number of measurements to make it independent of the size
(\figref{diehistogramsfig}). After this normalization the height of of the data set (\figref{diehistogramsfig}). After this normalization
each histogram bar is an estimate of the probability $P_i$ of the the height of each histogram bar is an estimate of the probability
category $i$, i.e. of getting a data value in the $i$-th bin. $P_i$ of the category $i$, i.e. of getting a data value in the $i$-th
bin.
\begin{exercise}{rollthedie.m}{} \begin{exercise}{rollthedie.m}{}
Write a function that simulates rolling a die $n$ times. Write a function that simulates rolling a die $n$ times.
@ -236,12 +241,14 @@ category $i$, i.e. of getting a data value in the $i$-th bin.
\subsection{Probability densities functions} \subsection{Probability densities functions}
In cases where we deal with data sets of measurements of a real In cases where we deal with
quantity (e.g. lengths of snakes, weights of elephants, times \entermde[data!continuous]{Daten!kontinuierliche}{continuous data},
between succeeding spikes) there is no natural bin width for computing (measurements of real-valued quantities, e.g. lengths of snakes,
a histogram. In addition, the probability of measuring a data value that weights of elephants, times between succeeding spikes) there is no
equals exactly a specific real number like, e.g., 0.123456789 is zero, because natural bin width for computing a histogram. In addition, the
there are uncountable many real numbers. probability of measuring a data value that equals exactly a specific
real number like, e.g., 0.123456789 is zero, because there are
uncountable many real numbers.
We can only ask for the probability to get a measurement value in some We can only ask for the probability to get a measurement value in some
range. For example, we can ask for the probability $P(1.2<x<1.3)$ to range. For example, we can ask for the probability $P(1.2<x<1.3)$ to
@ -254,14 +261,14 @@ probability can also be expressed as $P(x_0<x<x_0 + \Delta x)$.
In the limit to very small ranges $\Delta x$ the probability of In the limit to very small ranges $\Delta x$ the probability of
getting a measurement between $x_0$ and $x_0+\Delta x$ scales down to getting a measurement between $x_0$ and $x_0+\Delta x$ scales down to
zero with $\Delta x$: zero with $\Delta x$:
\[ P(x_0<x<x_0+\Delta x) \approx p(x_0) \cdot \Delta x \; . \] \[ P(x_0<x<x_0+\Delta x) \approx p(x_0) \cdot \Delta x \; . \] In here
In here the quantity $p(x_00)$ is a so called the quantity $p(x_00)$ is a so called
\enterm[probability!density]{probability density} that is larger than \enterm[probability!density]{probability density}
zero and that describes the distribution of the data values. The (\determ[Wahrscheinlichkeits!-dichte]{Wahrscheinlichkeitsdichte}) that is larger than zero and that
probability density is not a unitless probability with values between describes the distribution of the data values. The probability density
0 and 1, but a number that takes on any positive real number and has is not a unitless probability with values between 0 and 1, but a
as a unit the inverse of the unit of the data values --- hence the number that takes on any positive real number and has as a unit the
name ``density''. inverse of the unit of the data values --- hence the name ``density''.
\begin{figure}[t] \begin{figure}[t]
\includegraphics[width=1\textwidth]{pdfprobabilities} \includegraphics[width=1\textwidth]{pdfprobabilities}
@ -282,17 +289,18 @@ the probability density over the whole real axis must be one:
\end{equation} \end{equation}
The function $p(x)$, that assigns to every $x$ a probability density, The function $p(x)$, that assigns to every $x$ a probability density,
is called \enterm[probability!density function]{probability density function}, is called \enterm[probability!density function]{probability density
\enterm[pdf|see{probability density function}]{pdf}, or just function}, \enterm[pdf|see{probability density function}]{pdf}, or
\enterm[density|see{probability density function}]{density} just \enterm[density|see{probability density function}]{density}
(\determ{Wahrscheinlichkeitsdichtefunktion}). The well known (\determ[Wahrscheinlichkeits!-dichtefunktion]{Wahrscheinlichkeitsdichtefunktion},
\enterm{normal distribution} (\determ{Normalverteilung}) is an example of a \determ[Wahrscheinlichkeits!-dichte]{Wahrscheinlichkeitsdichte}). The
probability density function well known \entermde{Normalverteilung}{normal distribution} is an
example of a probability density function
\[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \] \[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
--- the \enterm{Gaussian distribution} --- the \enterm{Gaussian distribution}
(\determ{Gau{\ss}sche-Glockenkurve}) with mean $\mu$ and standard (\determ{Gau{\ss}sche-Glockenkurve}) with mean $\mu$ and standard
deviation $\sigma$. deviation $\sigma$.
The factor in front of the exponential function ensures the normalization to The factor in front of the exponential function ensures normalization to
$\int p_g(x) \, dx = 1$, \eqnref{pdfnorm}. $\int p_g(x) \, dx = 1$, \eqnref{pdfnorm}.
\begin{exercise}{gaussianpdf.m}{gaussianpdf.out} \begin{exercise}{gaussianpdf.m}{gaussianpdf.out}
@ -322,13 +330,15 @@ values fall within each bin (\figref{pdfhistogramfig} left).
To turn such histograms to estimates of probability densities they To turn such histograms to estimates of probability densities they
need to be normalized such that according to \eqnref{pdfnorm} their need to be normalized such that according to \eqnref{pdfnorm} their
integral equals one. While histograms of categorial data are integral equals one. While histograms of categorical data are
normalized such that their sum equals one, here we need to integrate normalized such that their sum equals one, here we need to integrate
over the histogram. The integral is the area (not the height) of the over the histogram. The integral is the area (not the height) of the
histogram bars. Each bar has the height $n_i$ and the width $\Delta histogram bars. Each bar has the height $n_i$ and the width $\Delta
x$. The total area $A$ of the histogram is thus x$. The total area $A$ of the histogram is thus
\[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i = N \, \Delta x \] \[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i = N \, \Delta x \]
and the normalized histogram has the heights and the
\entermde[histogram!normalized]{Histogramm!normiertes}{normalized
histogram} has the heights
\[ p(x_i) = \frac{n_i}{A} = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} = \[ p(x_i) = \frac{n_i}{A} = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} =
\frac{n_i}{N \Delta x} \; .\] \frac{n_i}{N \Delta x} \; .\]
A histogram needs to be divided by both the sum of the frequencies A histogram needs to be divided by both the sum of the frequencies
@ -375,14 +385,14 @@ shape histogram depends on the exact position of its bins
(here Gaussian kernels with standard deviation of $\sigma=0.2$).} (here Gaussian kernels with standard deviation of $\sigma=0.2$).}
\end{figure} \end{figure}
To avoid this problem one can use so called \enterm{kernel densities} To avoid this problem so called \entermde[kernel
for estimating probability densities from data. Here every data point density]{Kerndichte}{kernel densities} can be used for estimating
is replaced by a kernel (a function with integral one, like for probability densities from data. Here every data point is replaced by
example the Gaussian) that is moved exactly to the position a kernel (a function with integral one, like for example the Gaussian)
indicated by the data value. Then all the kernels of all the data that is moved exactly to the position indicated by the data
values are summed up, the sum is divided by the number of data values, value. Then all the kernels of all the data values are summed up, the
and we get an estimate of the probability density sum is divided by the number of data values, and we get an estimate of
(\figref{kerneldensityfig} right). the probability density (\figref{kerneldensityfig} right).
As for the histogram, where we need to choose a bin width, we need to As for the histogram, where we need to choose a bin width, we need to
choose the width of the kernels appropriately. choose the width of the kernels appropriately.
@ -457,7 +467,9 @@ bivariate or multivariate data sets where we have pairs or tuples of
data values (e.g. size and weight of elephants) we want to analyze data values (e.g. size and weight of elephants) we want to analyze
dependencies between the variables. dependencies between the variables.
The \enterm[correlation!correlation coefficient]{correlation coefficient} The
\entermde[correlation!coefficient]{Korrelation!-skoeffizient}{correlation
coefficient}
\begin{equation} \begin{equation}
\label{correlationcoefficient} \label{correlationcoefficient}
r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle
@ -467,8 +479,8 @@ The \enterm[correlation!correlation coefficient]{correlation coefficient}
\end{equation} \end{equation}
quantifies linear relationships between two variables quantifies linear relationships between two variables
\matlabfun{corr()}. The correlation coefficient is the \matlabfun{corr()}. The correlation coefficient is the
\enterm{covariance} normalized by the standard deviations of the \entermde{Kovarianz}{covariance} normalized by the standard deviations
single variables. Perfectly correlated variables result in a of the single variables. Perfectly correlated variables result in a
correlation coefficient of $+1$, anit-correlated or negatively correlation coefficient of $+1$, anit-correlated or negatively
correlated data in a correlation coefficient of $-1$ and un-correlated correlated data in a correlation coefficient of $-1$ and un-correlated
data in a correlation coefficient close to zero data in a correlation coefficient close to zero