Updated maximum likelihood chapter.
Common latex header file.
This commit is contained in:
parent
91f1f3663e
commit
40a5343fa8
@ -1,217 +1,8 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
\author{Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
|
||||
\date{WS 15/16}
|
||||
\input{../../header}
|
||||
|
||||
%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \newcommand{\tr}[2]{#1} % en
|
||||
% \usepackage[english]{babel}
|
||||
\newcommand{\tr}[2]{#2} % de
|
||||
\usepackage[german]{babel}
|
||||
|
||||
%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{pslatex} % nice font for pdf file
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
|
||||
\setcounter{tocdepth}{1}
|
||||
|
||||
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[sf,bf,it,big,clearempty]{titlesec}
|
||||
\setcounter{secnumdepth}{1}
|
||||
|
||||
|
||||
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
|
||||
|
||||
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{graphicx}
|
||||
\usepackage{xcolor}
|
||||
\pagecolor{white}
|
||||
|
||||
\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
|
||||
\put(0,4){\line(1,0){170}}%
|
||||
\multiput(0,2)(10,0){18}{\line(0,1){4}}%
|
||||
\multiput(0,3)(1,0){170}{\line(0,1){2}}%
|
||||
\put(0,0){\makebox(0,0){{\tiny 0}}}%
|
||||
\put(10,0){\makebox(0,0){{\tiny 1}}}%
|
||||
\put(20,0){\makebox(0,0){{\tiny 2}}}%
|
||||
\put(30,0){\makebox(0,0){{\tiny 3}}}%
|
||||
\put(40,0){\makebox(0,0){{\tiny 4}}}%
|
||||
\put(50,0){\makebox(0,0){{\tiny 5}}}%
|
||||
\put(60,0){\makebox(0,0){{\tiny 6}}}%
|
||||
\put(70,0){\makebox(0,0){{\tiny 7}}}%
|
||||
\put(80,0){\makebox(0,0){{\tiny 8}}}%
|
||||
\put(90,0){\makebox(0,0){{\tiny 9}}}%
|
||||
\put(100,0){\makebox(0,0){{\tiny 10}}}%
|
||||
\put(110,0){\makebox(0,0){{\tiny 11}}}%
|
||||
\put(120,0){\makebox(0,0){{\tiny 12}}}%
|
||||
\put(130,0){\makebox(0,0){{\tiny 13}}}%
|
||||
\put(140,0){\makebox(0,0){{\tiny 14}}}%
|
||||
\put(150,0){\makebox(0,0){{\tiny 15}}}%
|
||||
\put(160,0){\makebox(0,0){{\tiny 16}}}%
|
||||
\put(170,0){\makebox(0,0){{\tiny 17}}}%
|
||||
\end{picture}\par}
|
||||
|
||||
% figures:
|
||||
\setlength{\fboxsep}{0pt}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
|
||||
%\newcommand{\texpicture}[1]{}
|
||||
\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
|
||||
|
||||
% maximum number of floats:
|
||||
\setcounter{topnumber}{2}
|
||||
\setcounter{bottomnumber}{0}
|
||||
\setcounter{totalnumber}{2}
|
||||
|
||||
% float placement fractions:
|
||||
\renewcommand{\textfraction}{0.2}
|
||||
\renewcommand{\topfraction}{0.8}
|
||||
\renewcommand{\bottomfraction}{0.0}
|
||||
\renewcommand{\floatpagefraction}{0.5}
|
||||
|
||||
% spacing for floats:
|
||||
\setlength{\floatsep}{12pt plus 2pt minus 2pt}
|
||||
\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
|
||||
\setlength{\intextsep}{12pt plus 2pt minus 2pt}
|
||||
|
||||
% spacing for a floating page:
|
||||
\makeatletter
|
||||
\setlength{\@fptop}{0pt}
|
||||
\setlength{\@fpsep}{8pt plus 2.0fil}
|
||||
\setlength{\@fpbot}{0pt plus 1.0fil}
|
||||
\makeatother
|
||||
|
||||
% rules for floats:
|
||||
\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
|
||||
\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
|
||||
|
||||
% captions:
|
||||
\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
|
||||
|
||||
% put caption on separate float:
|
||||
\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
|
||||
|
||||
% references to panels of a figure within the caption:
|
||||
\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
|
||||
% references to figures:
|
||||
\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
|
||||
\newcommand{\fref}[1]{\textup{\ref{#1}}}
|
||||
\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
|
||||
% references to figures in normal text:
|
||||
\newcommand{\fig}{Fig.}
|
||||
\newcommand{\Fig}{Figure}
|
||||
\newcommand{\figs}{Figs.}
|
||||
\newcommand{\Figs}{Figures}
|
||||
\newcommand{\figref}[1]{\fig~\fref{#1}}
|
||||
\newcommand{\Figref}[1]{\Fig~\fref{#1}}
|
||||
\newcommand{\figsref}[1]{\figs~\fref{#1}}
|
||||
\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
|
||||
\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
|
||||
% references to figures within bracketed text:
|
||||
\newcommand{\figb}{Fig.}
|
||||
\newcommand{\figsb}{Figs.}
|
||||
\newcommand{\figrefb}[1]{\figb~\fref{#1}}
|
||||
\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
|
||||
\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
|
||||
|
||||
% references to tables:
|
||||
\newcommand{\tref}[1]{\textup{\ref{#1}}}
|
||||
% references to tables in normal text:
|
||||
\newcommand{\tab}{Tab.}
|
||||
\newcommand{\Tab}{Table}
|
||||
\newcommand{\tabs}{Tabs.}
|
||||
\newcommand{\Tabs}{Tables}
|
||||
\newcommand{\tabref}[1]{\tab~\tref{#1}}
|
||||
\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
|
||||
\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
|
||||
\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
|
||||
% references to tables within bracketed text:
|
||||
\newcommand{\tabb}{Tab.}
|
||||
\newcommand{\tabsb}{Tab.}
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\newcommand{\eqref}[1]{(\ref{#1})}
|
||||
\newcommand{\eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\Eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\Eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
|
||||
\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
|
||||
\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
|
||||
\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
|
||||
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
inputpath=../code,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{red!60!black},
|
||||
keywordstyle=\color{blue!50!black},
|
||||
stringstyle=\color{green!50!black},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{9} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
\lstset{inputpath=../code}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
@ -1,217 +1,8 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
\author{Jan Grewe \& Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
|
||||
\date{WS 15/16}
|
||||
\input{../../header}
|
||||
|
||||
%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \newcommand{\tr}[2]{#1} % en
|
||||
% \usepackage[english]{babel}
|
||||
\newcommand{\tr}[2]{#2} % de
|
||||
\usepackage[german]{babel}
|
||||
|
||||
%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{pslatex} % nice font for pdf file
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
|
||||
\setcounter{tocdepth}{1}
|
||||
|
||||
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[sf,bf,it,big,clearempty]{titlesec}
|
||||
\setcounter{secnumdepth}{1}
|
||||
|
||||
|
||||
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
|
||||
|
||||
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{graphicx}
|
||||
\usepackage{xcolor}
|
||||
\pagecolor{white}
|
||||
|
||||
\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
|
||||
\put(0,4){\line(1,0){170}}%
|
||||
\multiput(0,2)(10,0){18}{\line(0,1){4}}%
|
||||
\multiput(0,3)(1,0){170}{\line(0,1){2}}%
|
||||
\put(0,0){\makebox(0,0){{\tiny 0}}}%
|
||||
\put(10,0){\makebox(0,0){{\tiny 1}}}%
|
||||
\put(20,0){\makebox(0,0){{\tiny 2}}}%
|
||||
\put(30,0){\makebox(0,0){{\tiny 3}}}%
|
||||
\put(40,0){\makebox(0,0){{\tiny 4}}}%
|
||||
\put(50,0){\makebox(0,0){{\tiny 5}}}%
|
||||
\put(60,0){\makebox(0,0){{\tiny 6}}}%
|
||||
\put(70,0){\makebox(0,0){{\tiny 7}}}%
|
||||
\put(80,0){\makebox(0,0){{\tiny 8}}}%
|
||||
\put(90,0){\makebox(0,0){{\tiny 9}}}%
|
||||
\put(100,0){\makebox(0,0){{\tiny 10}}}%
|
||||
\put(110,0){\makebox(0,0){{\tiny 11}}}%
|
||||
\put(120,0){\makebox(0,0){{\tiny 12}}}%
|
||||
\put(130,0){\makebox(0,0){{\tiny 13}}}%
|
||||
\put(140,0){\makebox(0,0){{\tiny 14}}}%
|
||||
\put(150,0){\makebox(0,0){{\tiny 15}}}%
|
||||
\put(160,0){\makebox(0,0){{\tiny 16}}}%
|
||||
\put(170,0){\makebox(0,0){{\tiny 17}}}%
|
||||
\end{picture}\par}
|
||||
|
||||
% figures:
|
||||
\setlength{\fboxsep}{0pt}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
|
||||
%\newcommand{\texpicture}[1]{}
|
||||
\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
|
||||
|
||||
% maximum number of floats:
|
||||
\setcounter{topnumber}{2}
|
||||
\setcounter{bottomnumber}{0}
|
||||
\setcounter{totalnumber}{2}
|
||||
|
||||
% float placement fractions:
|
||||
\renewcommand{\textfraction}{0.2}
|
||||
\renewcommand{\topfraction}{0.8}
|
||||
\renewcommand{\bottomfraction}{0.0}
|
||||
\renewcommand{\floatpagefraction}{0.5}
|
||||
|
||||
% spacing for floats:
|
||||
\setlength{\floatsep}{12pt plus 2pt minus 2pt}
|
||||
\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
|
||||
\setlength{\intextsep}{12pt plus 2pt minus 2pt}
|
||||
|
||||
% spacing for a floating page:
|
||||
\makeatletter
|
||||
\setlength{\@fptop}{0pt}
|
||||
\setlength{\@fpsep}{8pt plus 2.0fil}
|
||||
\setlength{\@fpbot}{0pt plus 1.0fil}
|
||||
\makeatother
|
||||
|
||||
% rules for floats:
|
||||
\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
|
||||
\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
|
||||
|
||||
% captions:
|
||||
\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
|
||||
|
||||
% put caption on separate float:
|
||||
\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
|
||||
|
||||
% references to panels of a figure within the caption:
|
||||
\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
|
||||
% references to figures:
|
||||
\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
|
||||
\newcommand{\fref}[1]{\textup{\ref{#1}}}
|
||||
\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
|
||||
% references to figures in normal text:
|
||||
\newcommand{\fig}{Fig.}
|
||||
\newcommand{\Fig}{Figure}
|
||||
\newcommand{\figs}{Figs.}
|
||||
\newcommand{\Figs}{Figures}
|
||||
\newcommand{\figref}[1]{\fig~\fref{#1}}
|
||||
\newcommand{\Figref}[1]{\Fig~\fref{#1}}
|
||||
\newcommand{\figsref}[1]{\figs~\fref{#1}}
|
||||
\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
|
||||
\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
|
||||
% references to figures within bracketed text:
|
||||
\newcommand{\figb}{Fig.}
|
||||
\newcommand{\figsb}{Figs.}
|
||||
\newcommand{\figrefb}[1]{\figb~\fref{#1}}
|
||||
\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
|
||||
\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
|
||||
|
||||
% references to tables:
|
||||
\newcommand{\tref}[1]{\textup{\ref{#1}}}
|
||||
% references to tables in normal text:
|
||||
\newcommand{\tab}{Tab.}
|
||||
\newcommand{\Tab}{Table}
|
||||
\newcommand{\tabs}{Tabs.}
|
||||
\newcommand{\Tabs}{Tables}
|
||||
\newcommand{\tabref}[1]{\tab~\tref{#1}}
|
||||
\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
|
||||
\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
|
||||
\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
|
||||
% references to tables within bracketed text:
|
||||
\newcommand{\tabb}{Tab.}
|
||||
\newcommand{\tabsb}{Tab.}
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\newcommand{\eqref}[1]{(\ref{#1})}
|
||||
\newcommand{\eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\Eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\Eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
|
||||
\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
|
||||
\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
|
||||
\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
|
||||
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
inputpath=../code,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{red!60!black},
|
||||
keywordstyle=\color{blue!50!black},
|
||||
stringstyle=\color{green!50!black},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{9} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
\lstset{inputpath=../code}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
@ -11,10 +11,10 @@ einige dieser ``Design pattern'' zusammen.
|
||||
\section{Plotten einer mathematischen Funktion}
|
||||
Eine mathematische Funktion ordnet einem beliebigen $x$-Wert einen
|
||||
$y$-Wert zu. Um eine solche Funktion zeichnen zu k\"onnen, m\"ussen
|
||||
wir uns eine Wertetabelle aus vielen $x$-Werten und den
|
||||
wir eine Wertetabelle aus vielen $x$-Werten und den
|
||||
dazugeh\"origen Funktionswerten $y=f(x)$ erstellen.
|
||||
|
||||
Wir erstellen uns dazu einen Vektor mit geeigneten $x$-Werten, die von
|
||||
Wir erstellen dazu einen Vektor mit geeigneten $x$-Werten, die von
|
||||
dem kleinsten bis zu dem gr\"o{\ss}ten $x$-Wert laufen, den wir
|
||||
plotten wollen. Die Schrittweite f\"ur die $x$-Werte w\"ahlen wir
|
||||
klein genug, um eine sch\"one glatte Kurve zu bekommen. F\"ur jeden
|
||||
@ -88,7 +88,7 @@ end
|
||||
\end{lstlisting}
|
||||
|
||||
Wenn in der Schleife das Ergebnis in einen Vektor gespeichert werden soll,
|
||||
sollten wir uns vor der Schleife schon einen Vektor f\"ur die Ergebnisse
|
||||
sollten wir vor der Schleife schon einen Vektor f\"ur die Ergebnisse
|
||||
erstellen:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
@ -125,9 +125,9 @@ einzigen, durchgehenden Vektor zusammengestellt werden:
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
y = []; % Leerer Vektor fuer die Ergebnisse
|
||||
for i=1:length(x)
|
||||
% Die Funktion get_something gibt uns einen Vektor zurueck:
|
||||
% Die Funktion get_something gibt einen Vektor zurueck:
|
||||
z = get_something( x(i) );
|
||||
% dessen Inhalt h\"angen wir an unseren Ergebnissvektor an:
|
||||
% dessen Inhalt h\"angen wir an den Ergebnissvektor an:
|
||||
y = [y; z(:)];
|
||||
% z(:) stellt sicher, das wir auf jeden Fall einen Spaltenvektoren aneinanderreihen.
|
||||
end
|
||||
@ -136,7 +136,6 @@ mean(y)
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Normierung von Histogrammen}
|
||||
Meistens sollten Histogramme normiert werden, damit sie vergleichbar
|
||||
|
||||
@ -1,4 +1,3 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
@ -23,46 +22,18 @@
|
||||
\usepackage[sf,bf,it,big,clearempty]{titlesec}
|
||||
\setcounter{secnumdepth}{1}
|
||||
|
||||
|
||||
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
|
||||
|
||||
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{graphicx}
|
||||
\usepackage{xcolor}
|
||||
\pagecolor{white}
|
||||
|
||||
\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
|
||||
\put(0,4){\line(1,0){170}}%
|
||||
\multiput(0,2)(10,0){18}{\line(0,1){4}}%
|
||||
\multiput(0,3)(1,0){170}{\line(0,1){2}}%
|
||||
\put(0,0){\makebox(0,0){{\tiny 0}}}%
|
||||
\put(10,0){\makebox(0,0){{\tiny 1}}}%
|
||||
\put(20,0){\makebox(0,0){{\tiny 2}}}%
|
||||
\put(30,0){\makebox(0,0){{\tiny 3}}}%
|
||||
\put(40,0){\makebox(0,0){{\tiny 4}}}%
|
||||
\put(50,0){\makebox(0,0){{\tiny 5}}}%
|
||||
\put(60,0){\makebox(0,0){{\tiny 6}}}%
|
||||
\put(70,0){\makebox(0,0){{\tiny 7}}}%
|
||||
\put(80,0){\makebox(0,0){{\tiny 8}}}%
|
||||
\put(90,0){\makebox(0,0){{\tiny 9}}}%
|
||||
\put(100,0){\makebox(0,0){{\tiny 10}}}%
|
||||
\put(110,0){\makebox(0,0){{\tiny 11}}}%
|
||||
\put(120,0){\makebox(0,0){{\tiny 12}}}%
|
||||
\put(130,0){\makebox(0,0){{\tiny 13}}}%
|
||||
\put(140,0){\makebox(0,0){{\tiny 14}}}%
|
||||
\put(150,0){\makebox(0,0){{\tiny 15}}}%
|
||||
\put(160,0){\makebox(0,0){{\tiny 16}}}%
|
||||
\put(170,0){\makebox(0,0){{\tiny 17}}}%
|
||||
\end{picture}\par}
|
||||
|
||||
% figures:
|
||||
\setlength{\fboxsep}{0pt}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
|
||||
%\newcommand{\texpicture}[1]{}
|
||||
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% gnuplot figures:
|
||||
\newcommand{\texinputpath}{}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{\texinputpath#1.tex}}}
|
||||
\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
|
||||
|
||||
% maximum number of floats:
|
||||
@ -98,6 +69,7 @@
|
||||
% put caption on separate float:
|
||||
\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
|
||||
|
||||
%%%%% figure references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% references to panels of a figure within the caption:
|
||||
\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
|
||||
% references to figures:
|
||||
@ -105,10 +77,10 @@
|
||||
\newcommand{\fref}[1]{\textup{\ref{#1}}}
|
||||
\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
|
||||
% references to figures in normal text:
|
||||
\newcommand{\fig}{Fig.}
|
||||
\newcommand{\Fig}{Figure}
|
||||
\newcommand{\figs}{Figs.}
|
||||
\newcommand{\Figs}{Figures}
|
||||
\newcommand{\fig}{\tr{Fig.}{Abb.}}
|
||||
\newcommand{\Fig}{\tr{Figure}{Abb.}}
|
||||
\newcommand{\figs}{\tr{Figs.}{Abb.}}
|
||||
\newcommand{\Figs}{\tr{Figures}{Abb.}}
|
||||
\newcommand{\figref}[1]{\fig~\fref{#1}}
|
||||
\newcommand{\Figref}[1]{\Fig~\fref{#1}}
|
||||
\newcommand{\figsref}[1]{\figs~\fref{#1}}
|
||||
@ -118,31 +90,30 @@
|
||||
\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
|
||||
% references to figures within bracketed text:
|
||||
\newcommand{\figb}{Fig.}
|
||||
\newcommand{\figsb}{Figs.}
|
||||
\newcommand{\figb}{\tr{Fig.}{Abb.}}
|
||||
\newcommand{\figsb}{\tr{Figs.}{Abb.}}
|
||||
\newcommand{\figrefb}[1]{\figb~\fref{#1}}
|
||||
\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
|
||||
\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
|
||||
|
||||
% references to tables:
|
||||
%%%%% table references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\tref}[1]{\textup{\ref{#1}}}
|
||||
% references to tables in normal text:
|
||||
\newcommand{\tab}{Tab.}
|
||||
\newcommand{\Tab}{Table}
|
||||
\newcommand{\tabs}{Tabs.}
|
||||
\newcommand{\Tabs}{Tables}
|
||||
\newcommand{\tab}{\tr{Tab.}{Tabelle}}
|
||||
\newcommand{\Tab}{\tr{Tabel}{Tabelle}}
|
||||
\newcommand{\tabs}{\tr{Tabs.}{Tabellen}}
|
||||
\newcommand{\Tabs}{\tr{Tabels.}{Tabellen}}
|
||||
\newcommand{\tabref}[1]{\tab~\tref{#1}}
|
||||
\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
|
||||
\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
|
||||
\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
|
||||
% references to tables within bracketed text:
|
||||
\newcommand{\tabb}{Tab.}
|
||||
\newcommand{\tabsb}{Tab.}
|
||||
\newcommand{\tabb}{\tr{Tab.}{Tabelle}}
|
||||
\newcommand{\tabsb}{\tr{Tab.}{Tabelle}}
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\newcommand{\eqref}[1]{(\ref{#1})}
|
||||
\newcommand{\eqn}{\tr{Eq}{Gl}.}
|
||||
@ -154,18 +125,16 @@
|
||||
\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
|
||||
\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
|
||||
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
inputpath=../code,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{darkgray},
|
||||
keywordstyle=\color{blue},
|
||||
stringstyle=\color{green},
|
||||
commentstyle=\itshape\color{red!60!black},
|
||||
keywordstyle=\color{blue!50!black},
|
||||
stringstyle=\color{green!50!black},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
@ -173,9 +142,10 @@
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
xleftmargin=2.5em,
|
||||
xrightmargin=1mm,
|
||||
aboveskip=1ex,
|
||||
belowskip=2ex
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -190,37 +160,30 @@
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
%%%%% code/matlab commands: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcommand{\code}[1]{\setlength{\fboxsep}{0.5ex}\colorbox{blue!10}{\texttt{#1}}}
|
||||
\newcommand{\matlab}{MATLAB}
|
||||
\newcommand{\matlabfun}[1]{(\tr{\matlab{}-function}{\matlab-Funktion} \setlength{\fboxsep}{0.5ex}\colorbox{blue!10}{\texttt{#1}})}
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%% definition environment: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
%%%%% exercises environment: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{framed}
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{9} % show listings up to exercise maxexercise
|
||||
\setcounter{maxexercise}{10000} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\include{psth_sta}
|
||||
|
||||
\end{document}
|
||||
|
||||
\newcommand{\codepath}{}
|
||||
\newenvironment{exercise}[1][]%
|
||||
{\newcommand{\exercisesource}{#1}%
|
||||
\setlength{\fboxsep}{2mm}%
|
||||
\renewcommand{\FrameCommand}{\colorbox{yellow!15}}%
|
||||
\MakeFramed{\advance\hsize-\width \FrameRestore}%
|
||||
\noindent\textbf{\tr{Exercise}{\"Ubung} \arabic{theexercise}:}\newline}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}%
|
||||
{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}%
|
||||
{\lstinputlisting[belowskip=0pt]{\codepath\exercisesource}}}%
|
||||
\endMakeFramed\stepcounter{theexercise}}
|
||||
12
likelihood/code/mlegammafit.m
Normal file
12
likelihood/code/mlegammafit.m
Normal file
@ -0,0 +1,12 @@
|
||||
% generate gamma distributed random numbers:
|
||||
n = 50;
|
||||
shape = 2.0;
|
||||
scale = 1.0;
|
||||
x = gamrnd(shape, scale, n, 1);
|
||||
|
||||
% maximum likelihood estimate:
|
||||
p = mle(x, 'distribution', 'gamma');
|
||||
|
||||
% report results:
|
||||
fprintf('shape=%.2f\n', p(1));
|
||||
fprintf('scale=%.2f\n', p(2));
|
||||
@ -3,8 +3,8 @@ n = 100;
|
||||
mu = 3.0;
|
||||
sigma =2.0;
|
||||
x = randn(n,1)*sigma+mu;
|
||||
fprintf(' mean of the data is %.2f\n', mean(x))
|
||||
fprintf('standard deviation of the data is %.2f\n', std(x))
|
||||
fprintf(' mean of the data is %.2f\n', mean(x))
|
||||
|
||||
% mean as parameter:
|
||||
pmus = 2.0:0.01:4.0;
|
||||
@ -18,12 +18,19 @@ end
|
||||
lm = prod(lms, 1); % likelihood
|
||||
loglm = sum(log(lms), 1); % log likelihood
|
||||
|
||||
% position of maxima:
|
||||
maxlm = max(lm); % height of the maximum
|
||||
pmumaxlm = pmus(lm==maxlm); % pmu where lm is at maximum
|
||||
pmumaxloglm = pmus(loglm==max(loglm)); % same for loglm in one line
|
||||
fprintf(' maximum of likelihood is at %.2f\n', pmumaxlm)
|
||||
fprintf(' maximum of log-likelihood is at %.2f\n', pmumaxloglm)
|
||||
|
||||
% plot likelihood of mean:
|
||||
subplot(1, 2, 1);
|
||||
plot(pmus, lm );
|
||||
plot(pmus, lm);
|
||||
xlabel('mean')
|
||||
ylabel('likelihood')
|
||||
subplot(1, 2, 2);
|
||||
plot(pmus, loglm );
|
||||
plot(pmus, loglm);
|
||||
xlabel('mean')
|
||||
ylabel('log likelihood')
|
||||
|
||||
@ -1,217 +1,8 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
\author{Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
|
||||
\date{WS 15/16}
|
||||
\input{../../header}
|
||||
|
||||
%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \newcommand{\tr}[2]{#1} % en
|
||||
% \usepackage[english]{babel}
|
||||
\newcommand{\tr}[2]{#2} % de
|
||||
\usepackage[german]{babel}
|
||||
|
||||
%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{pslatex} % nice font for pdf file
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
|
||||
\setcounter{tocdepth}{1}
|
||||
|
||||
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[sf,bf,it,big,clearempty]{titlesec}
|
||||
\setcounter{secnumdepth}{1}
|
||||
|
||||
|
||||
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
|
||||
|
||||
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{graphicx}
|
||||
\usepackage{xcolor}
|
||||
\pagecolor{white}
|
||||
|
||||
\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
|
||||
\put(0,4){\line(1,0){170}}%
|
||||
\multiput(0,2)(10,0){18}{\line(0,1){4}}%
|
||||
\multiput(0,3)(1,0){170}{\line(0,1){2}}%
|
||||
\put(0,0){\makebox(0,0){{\tiny 0}}}%
|
||||
\put(10,0){\makebox(0,0){{\tiny 1}}}%
|
||||
\put(20,0){\makebox(0,0){{\tiny 2}}}%
|
||||
\put(30,0){\makebox(0,0){{\tiny 3}}}%
|
||||
\put(40,0){\makebox(0,0){{\tiny 4}}}%
|
||||
\put(50,0){\makebox(0,0){{\tiny 5}}}%
|
||||
\put(60,0){\makebox(0,0){{\tiny 6}}}%
|
||||
\put(70,0){\makebox(0,0){{\tiny 7}}}%
|
||||
\put(80,0){\makebox(0,0){{\tiny 8}}}%
|
||||
\put(90,0){\makebox(0,0){{\tiny 9}}}%
|
||||
\put(100,0){\makebox(0,0){{\tiny 10}}}%
|
||||
\put(110,0){\makebox(0,0){{\tiny 11}}}%
|
||||
\put(120,0){\makebox(0,0){{\tiny 12}}}%
|
||||
\put(130,0){\makebox(0,0){{\tiny 13}}}%
|
||||
\put(140,0){\makebox(0,0){{\tiny 14}}}%
|
||||
\put(150,0){\makebox(0,0){{\tiny 15}}}%
|
||||
\put(160,0){\makebox(0,0){{\tiny 16}}}%
|
||||
\put(170,0){\makebox(0,0){{\tiny 17}}}%
|
||||
\end{picture}\par}
|
||||
|
||||
% figures:
|
||||
\setlength{\fboxsep}{0pt}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
|
||||
%\newcommand{\texpicture}[1]{}
|
||||
\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
|
||||
|
||||
% maximum number of floats:
|
||||
\setcounter{topnumber}{2}
|
||||
\setcounter{bottomnumber}{0}
|
||||
\setcounter{totalnumber}{2}
|
||||
|
||||
% float placement fractions:
|
||||
\renewcommand{\textfraction}{0.2}
|
||||
\renewcommand{\topfraction}{0.8}
|
||||
\renewcommand{\bottomfraction}{0.0}
|
||||
\renewcommand{\floatpagefraction}{0.5}
|
||||
|
||||
% spacing for floats:
|
||||
\setlength{\floatsep}{12pt plus 2pt minus 2pt}
|
||||
\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
|
||||
\setlength{\intextsep}{12pt plus 2pt minus 2pt}
|
||||
|
||||
% spacing for a floating page:
|
||||
\makeatletter
|
||||
\setlength{\@fptop}{0pt}
|
||||
\setlength{\@fpsep}{8pt plus 2.0fil}
|
||||
\setlength{\@fpbot}{0pt plus 1.0fil}
|
||||
\makeatother
|
||||
|
||||
% rules for floats:
|
||||
\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
|
||||
\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
|
||||
|
||||
% captions:
|
||||
\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
|
||||
|
||||
% put caption on separate float:
|
||||
\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
|
||||
|
||||
% references to panels of a figure within the caption:
|
||||
\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
|
||||
% references to figures:
|
||||
\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
|
||||
\newcommand{\fref}[1]{\textup{\ref{#1}}}
|
||||
\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
|
||||
% references to figures in normal text:
|
||||
\newcommand{\fig}{Fig.}
|
||||
\newcommand{\Fig}{Figure}
|
||||
\newcommand{\figs}{Figs.}
|
||||
\newcommand{\Figs}{Figures}
|
||||
\newcommand{\figref}[1]{\fig~\fref{#1}}
|
||||
\newcommand{\Figref}[1]{\Fig~\fref{#1}}
|
||||
\newcommand{\figsref}[1]{\figs~\fref{#1}}
|
||||
\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
|
||||
\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
|
||||
% references to figures within bracketed text:
|
||||
\newcommand{\figb}{Fig.}
|
||||
\newcommand{\figsb}{Figs.}
|
||||
\newcommand{\figrefb}[1]{\figb~\fref{#1}}
|
||||
\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
|
||||
\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
|
||||
|
||||
% references to tables:
|
||||
\newcommand{\tref}[1]{\textup{\ref{#1}}}
|
||||
% references to tables in normal text:
|
||||
\newcommand{\tab}{Tab.}
|
||||
\newcommand{\Tab}{Table}
|
||||
\newcommand{\tabs}{Tabs.}
|
||||
\newcommand{\Tabs}{Tables}
|
||||
\newcommand{\tabref}[1]{\tab~\tref{#1}}
|
||||
\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
|
||||
\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
|
||||
\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
|
||||
% references to tables within bracketed text:
|
||||
\newcommand{\tabb}{Tab.}
|
||||
\newcommand{\tabsb}{Tab.}
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\newcommand{\eqref}[1]{(\ref{#1})}
|
||||
\newcommand{\eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\Eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\Eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
|
||||
\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
|
||||
\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
|
||||
\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
|
||||
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
inputpath=../code,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{red!60!black},
|
||||
keywordstyle=\color{blue!50!black},
|
||||
stringstyle=\color{green!50!black},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{9} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
\lstset{inputpath=../code}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
@ -1,3 +1,4 @@
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\chapter{\tr{Maximum likelihood estimation}{Maximum-Likelihood-Sch\"atzer}}
|
||||
@ -5,9 +6,9 @@
|
||||
In vielen Situationen wollen wir einen oder mehrere Parameter $\theta$
|
||||
einer Wahrscheinlichkeitsverteilung sch\"atzen, so dass die Verteilung
|
||||
die Daten $x_1, x_2, \ldots x_n$ am besten beschreibt.
|
||||
Maximum-Likelihood-Sch\"atzer w\"ahlen die Parameter so, dass die
|
||||
Wahrscheinlichkeit, dass die Daten aus der Verteilung stammen, am
|
||||
gr\"o{\ss}ten ist.
|
||||
Maximum-Likelihood-Sch\"atzer (maximum likelihood estimate, mle)
|
||||
w\"ahlen die Parameter so, dass die Wahrscheinlichkeit, dass die Daten
|
||||
aus der Verteilung stammen, am gr\"o{\ss}ten ist.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Maximum Likelihood}
|
||||
@ -34,9 +35,14 @@ den Parameter $\theta$ zu haben, gegeben die Me{\ss}werte $x_1, x_2, \ldots x_n$
|
||||
\begin{equation}
|
||||
{\cal L}(\theta|x_1,x_2, \ldots x_n) = p(x_1,x_2, \ldots x_n|\theta)
|
||||
\end{equation}
|
||||
Hinter dieser Umformung steht eigentlich der Satz von Bayes. Bei der
|
||||
einfachen Gleichsetzung von ${\cal L}$ mit $p$ fehlen
|
||||
Normierungsfaktoren, so dass ${\cal L}$ sich nicht auf Eins
|
||||
aufintegriert, und ${\cal L}$ deshalb keine
|
||||
Wahrscheinlichkeit(sdichte) ist.
|
||||
|
||||
Wir sind nun an dem Wert des Parameters $\theta_{mle}$ interessiert, der die
|
||||
Likelihood maximiert (``mle'': Maximum-Likelihood Estimate):
|
||||
Likelihood maximiert (Maximum-Likelihood Estimate ``mle''):
|
||||
\begin{equation}
|
||||
\theta_{mle} = \text{argmax}_{\theta} {\cal L}(\theta|x_1,x_2, \ldots x_n)
|
||||
\end{equation}
|
||||
@ -46,8 +52,9 @@ bei dem die Likelihood ${\cal L}(\theta)$ ihr Maximum hat.
|
||||
|
||||
An der Stelle eines Maximums einer Funktion \"andert sich nichts, wenn
|
||||
man die Funktionswerte mit einer streng monoton steigenden Funktion
|
||||
transformiert. Aus gleich ersichtlichen mathematischen Gr\"unden wird meistens
|
||||
das Maximum der logarithmierten Likelihood (``Log-Likelihood'') gesucht:
|
||||
transformiert. Aus numerischen und gleich ersichtlichen mathematischen
|
||||
Gr\"unden wird meistens das Maximum der logarithmierten Likelihood
|
||||
(``Log-Likelihood'') gesucht:
|
||||
\begin{eqnarray}
|
||||
\theta_{mle} & = & \text{argmax}_{\theta}\; {\cal L}(\theta|x_1,x_2, \ldots x_n) \nonumber \\
|
||||
& = & \text{argmax}_{\theta}\; \log {\cal L}(\theta|x_1,x_2, \ldots x_n) \nonumber \\
|
||||
@ -79,19 +86,24 @@ Die Log-Likelihood \eqnref{loglikelihood} ist
|
||||
\begin{eqnarray*}
|
||||
\log {\cal L}(\theta|x_1,x_2, \ldots x_n)
|
||||
& = & \sum_{i=1}^n \log \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x_i-\theta)^2}{2\sigma^2}} \\
|
||||
& = & \sum_{i=1}^n - \log \sqrt{2\pi \sigma^2} -\frac{(x_i-\theta)^2}{2\sigma^2}
|
||||
& = & \sum_{i=1}^n - \log \sqrt{2\pi \sigma^2} -\frac{(x_i-\theta)^2}{2\sigma^2} \; .
|
||||
\end{eqnarray*}
|
||||
Der Logarithmus hat die sch\"one Eigenschaft die Exponentialfunktion
|
||||
der Normalverteilung auszul\"oschen, da der Logarithmus die
|
||||
Umkehrfunktion der Exponentialfunktion ist ($\log(e^x)=x$).
|
||||
|
||||
Zur Bestimmung des Maximums der Log-Likelihood berechnen wir deren Ableitung
|
||||
nach dem Parameter $\theta$ und setzen diese gleich Null:
|
||||
\begin{eqnarray*}
|
||||
\frac{\text{d}}{\text{d}\theta} \log {\cal L}(\theta|x_1,x_2, \ldots x_n) & = & \sum_{i=1}^n \frac{2(x_i-\theta)}{2\sigma^2} \;\; = \;\; 0 \\
|
||||
\Leftrightarrow \quad \sum_{i=1}^n x_i - \sum_{i=1}^n x_i \theta & = & 0 \\
|
||||
\Leftrightarrow \quad n \theta & = & \sum_{i=1}^n x_i \\
|
||||
\Leftrightarrow \quad \theta & = & \frac{1}{n} \sum_{i=1}^n x_i
|
||||
\Leftrightarrow \quad \theta & = & \frac{1}{n} \sum_{i=1}^n x_i \;\; = \;\; \bar x
|
||||
\end{eqnarray*}
|
||||
Der Maximum-Likelihood-Sch\"atzer ist das arithmetische Mittel der Daten. D.h.
|
||||
das arithmetische Mittel maximiert die Wahrscheinlichkeit, dass die Daten aus einer
|
||||
Normalverteilung mit diesem Mittelwert gezogen worden sind.
|
||||
Der Maximum-Likelihood-Sch\"atzer ist das arithmetische Mittel $\bar
|
||||
x$ der Daten. D.h. das arithmetische Mittel maximiert die
|
||||
Wahrscheinlichkeit, dass die Daten aus einer Normalverteilung mit
|
||||
diesem Mittelwert gezogen worden sind (\figref{mlemeanfig}).
|
||||
|
||||
\begin{exercise}[mlemean.m]
|
||||
Ziehe $n=50$ normalverteilte Zufallsvariablen mit einem Mittelwert $\ne 0$
|
||||
@ -100,8 +112,9 @@ Normalverteilung mit diesem Mittelwert gezogen worden sind.
|
||||
Plotte die Likelihood (aus dem Produkt der Wahrscheinlichkeiten) und
|
||||
die Log-Likelihood (aus der Summe der logarithmierten
|
||||
Wahrscheinlichkeiten) f\"ur den Mittelwert als Parameter. Vergleiche
|
||||
die Position der Maxima mit den aus den Daten berechneten
|
||||
die Position der Maxima mit dem aus den Daten berechneten
|
||||
Mittelwert.
|
||||
\newpage
|
||||
\end{exercise}
|
||||
|
||||
|
||||
@ -114,7 +127,7 @@ entsprechenden Funktionswerte $f(x_i;\theta)$ mit einer
|
||||
Standardabweichung $\sigma_i$ normalverteilt streuen, dann lautet die
|
||||
Log-Likelihood
|
||||
\begin{eqnarray*}
|
||||
\log {\cal L}(\theta|x_1,x_2, \ldots x_n)
|
||||
\log {\cal L}(\theta|(x_1,y_1,\sigma_1), \ldots, (x_n,y_n,\sigma_n))
|
||||
& = & \sum_{i=1}^n \log \frac{1}{\sqrt{2\pi \sigma_i^2}}e^{-\frac{(y_i-f(x_i;\theta))^2}{2\sigma_i^2}} \\
|
||||
& = & \sum_{i=1}^n - \log \sqrt{2\pi \sigma_i^2} -\frac{(x_i-f(y_i;\theta))^2}{2\sigma_i^2} \\
|
||||
\end{eqnarray*}
|
||||
@ -140,9 +153,12 @@ Standardabweichungen wird auch mit $\chi^2$ bezeichnet. Der Wert des
|
||||
Parameters $\theta$, welcher den quadratischen Abstand minimiert, ist
|
||||
also identisch mit der Maximierung der Wahrscheinlichkeit, dass die
|
||||
Daten tats\"achlich aus der Funktion stammen k\"onnen. Minimierung des
|
||||
$\chi^2$ ist also eine Maximum-Likelihood Sch\"atzung. Aber nur, wenn
|
||||
die Daten normalverteilt um die Funktion streuen! Bei anderen
|
||||
Verteilungen m\"usste man die Log-Likelihood entsprechend
|
||||
$\chi^2$ ist also eine Maximum-Likelihood Sch\"atzung.
|
||||
|
||||
An der Herleitung sehen wir aber auch, dass die Minimierung des
|
||||
quadratischen Abstands nur dann eine Maximum-Likelihood Absch\"atzung
|
||||
ist, wenn die Daten normalverteilt um die Funktion streuen. Bei
|
||||
anderen Verteilungen m\"usste man die Log-Likelihood entsprechend
|
||||
\eqnref{loglikelihood} ausrechnen und maximieren.
|
||||
|
||||
\begin{figure}[t]
|
||||
@ -168,21 +184,32 @@ und setzen diese gleich Null:
|
||||
\Leftrightarrow \quad \theta & = & \frac{\sum_{i=1}^n \frac{x_iy_i}{\sigma_i^2}}{ \sum_{i=1}^n \frac{x_i^2}{\sigma_i^2}} \label{mleslope}
|
||||
\end{eqnarray}
|
||||
Damit haben wir nun einen anlytischen Ausdruck f\"ur die Bestimmung
|
||||
der Steigung $\theta$ des Regressionsgeraden gewonnen. Ein
|
||||
Gradientenabstieg ist f\"ur das Fitten der Geradensteigung also gar
|
||||
nicht n\"otig. Das gilt allgemein f\"ur das Fitten von Koeffizienten
|
||||
von linear kombinierten Basisfunktionen. Parameter, die nichtlinear in
|
||||
einer Funktion enthalten sind, k\"onnen im Gegensatz dazu nicht
|
||||
analytisch aus den Daten berechnet werden. F\"ur diesen Fall bleibt
|
||||
dann nur auf numerische Verfahren zur Optimierung der Kostenfunktion,
|
||||
wie z.B. der Gradientenabstieg, zur\"uckzugreifen.
|
||||
der Steigung $\theta$ des Regressionsgeraden gewonnen
|
||||
(\figref{mleproplinefig}).
|
||||
|
||||
Ein Gradientenabstieg ist f\"ur das Fitten der Geradensteigung also
|
||||
gar nicht n\"otig. Das gilt allgemein f\"ur das Fitten von
|
||||
Koeffizienten von linear kombinierten Basisfunktionen. Wie z.B. die
|
||||
die Steigung $m$ und der y-Achsenabschnitt $b$ einer Geradengleichung
|
||||
\[ y = m \cdot x +b \]
|
||||
oder allgemeiner die Koeffizienten $a_k$ eines Polynoms
|
||||
\[ y = \sum_{k=0}^N a_k x^k = a_o + a_1x + a_2x^2 + a_3x^4 + \ldots \]
|
||||
\matlabfun{polyfit}.
|
||||
|
||||
Parameter, die nichtlinear in einer Funktion enthalten sind, k\"onnen
|
||||
im Gegensatz dazu nicht analytisch aus den Daten berechnet
|
||||
werden. z.B. die Rate $\lambda$ eines exponentiellen Zerfalls
|
||||
\[ y = c \cdot e^{\lambda x} \quad , \quad c, \lambda \in \reZ \; . \]
|
||||
F\"ur diesen Fall bleibt dann nur auf numerische Verfahren zur
|
||||
Optimierung der Kostenfunktion, wie z.B. der Gradientenabstieg,
|
||||
zur\"uckzugreifen \matlabfun{lsqcurvefit}.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Fits von Wahrscheinlichkeitsverteilungen}
|
||||
Zum Abschluss betrachten wir noch den Fall, bei dem wir die Parameter
|
||||
einer Wahrscheinlichkeitsdichtefunktion (z.B. Mittelwert und
|
||||
Standardabweichung der Normalverteilung) an ein Datenset fitten wollen.
|
||||
Jetzt betrachten wir noch den Fall, bei dem wir die Parameter einer
|
||||
Wahrscheinlichkeitsdichtefunktion (z.B. den shape-Parameter einer
|
||||
Gamma-Verteilung) an ein Datenset fitten wollen.
|
||||
|
||||
Ein erster Gedanke k\"onnte sein, die
|
||||
Wahrscheinlichkeitsdichtefunktion durch Minimierung des quadratischen
|
||||
@ -196,16 +223,7 @@ aufintegriert. Die beiden Annahmen normalverteilte und unabh\"angige
|
||||
Daten, die die Minimierung des quadratischen Abstands
|
||||
\eqnref{chisqmin} zu einem Maximum-Likelihood Sch\"atzer machen, sind
|
||||
also verletzt. (iii) Das Histogramm h\"angt von der Wahl der
|
||||
Klassenbreite ab.
|
||||
|
||||
Den direkten Weg, eine Wahrscheinlichkeitsdichtefunktion an ein
|
||||
Datenset zu fitten, haben wir oben schon bei dem Beispiel zur
|
||||
Absch\"atzung des Mittelwertes einer Normalverteilung gesehen ---
|
||||
Maximum Likelihood! Wir suchen einfach die Parameter $\theta$ der
|
||||
gesuchten Wahrscheinlichkeitsdichtefunktion bei der die Log-Likelihood
|
||||
\eqnref{loglikelihood} maximal wird. Das ist im allgemeinen ein
|
||||
nichtlinieares Optimierungsproblem, das mit numerischen Verfahren, wie
|
||||
z.B. dem Gradientenabstieg, gel\"ost wird.
|
||||
Klassenbreite ab (\figref{mlepdffig}).
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=1\textwidth]{mlepdf}
|
||||
@ -216,3 +234,69 @@ z.B. dem Gradientenabstieg, gel\"ost wird.
|
||||
normierte Histogramm der Daten zusammen mit dem \"uber Minimierung
|
||||
des quadratischen Abstands zum Histogramm berechneten Fit.}
|
||||
\end{figure}
|
||||
|
||||
Den direkten Weg, eine Wahrscheinlichkeitsdichtefunktion an ein
|
||||
Datenset zu fitten, haben wir oben schon bei dem Beispiel zur
|
||||
Absch\"atzung des Mittelwertes einer Normalverteilung gesehen ---
|
||||
Maximum Likelihood! Wir suchen einfach die Parameter $\theta$ der
|
||||
gesuchten Wahrscheinlichkeitsdichtefunktion bei der die Log-Likelihood
|
||||
\eqnref{loglikelihood} maximal wird. Das ist im allgemeinen ein
|
||||
nichtlinieares Optimierungsproblem, das mit numerischen Verfahren, wie
|
||||
z.B. dem Gradientenabstieg, gel\"ost wird \matlabfun{mle}.
|
||||
|
||||
\begin{exercise}[mlegammafit.m]
|
||||
Erzeuge Gammaverteilte Zufallszahlen und benutze Maximum-Likelihood,
|
||||
um die Parameter der Gammafunktion aus den Daten zu bestimmen.
|
||||
\newpage
|
||||
\end{exercise}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Neuronale Kodierung}
|
||||
In sensorischen Systemen kodieren Populationen von Neuronen mit ihrer
|
||||
Aktivit\"at Eigenschaften von sensorischen Stimuli. z.B. im visuellen
|
||||
Kortex V1 die Orientierung eines Balkens. Traditionell wird die
|
||||
Antwort der Neurone f\"ur verschiedene Stimuli (z.B. verschiedene
|
||||
Orientierungen des Balkens) gemessen. Die mittlere Antwort der Neurone
|
||||
als Funktion eines Stimulusparameters ist dann die ``Tuning-curve''
|
||||
(z.B. Feuerrate als Funktion des Orientierungswinkels).
|
||||
|
||||
\begin{figure}[tp]
|
||||
\includegraphics[width=1\textwidth]{mlecoding}
|
||||
\caption{\label{mlecodingfig} Maximum Likelihood Sch\"atzung eines
|
||||
Stimulusparameters aus der Aktivit\"at einer Population von
|
||||
Neuronen. Oben: Die Tuning-Kurve eines einzelnen Neurons in
|
||||
Abh\"angigkeit von der Orientierung eines Balkens. Der Stimulus
|
||||
der die st\"akste Aktivit\"at in diesem Neuron hervorruft ist ein
|
||||
senkrechter Balken (Pfeil, $\phi_i=90$\,\degree. Die rote Fl\"ache
|
||||
deutet die Variabilit\"at $p(r)$ der Aktivit\"at $r$ um die
|
||||
Tuning-Kurve herum an. Mitte: Jedes Neuron in der Population hat
|
||||
eine andere bevorzugte Orientierung des Stimulus (farbige Linien).
|
||||
Ein Stimulus einer bestimmten Orientierung aktiviert die Neurone
|
||||
in spezifischer Weise (Punkte). Unten: Die Log-Likelihood dieser
|
||||
Aktivit\"aten wir maximal in der N\"ahe der wahren Orientierung
|
||||
des Stimulus.}
|
||||
\end{figure}
|
||||
|
||||
Das Gehirn ist aber mit dem umgekehrten Problem konfrontiert: gegeben
|
||||
eine bestimmte Aktivit\"at der Neurone in der Population, was war der
|
||||
Stimulus (die Orientierung des Balkens)? Eine m\"ogliche Antwort ist
|
||||
im Sinne von Maximum-Likelihood: es war der Stimulus f\"ur den das
|
||||
Aktivit\"atsmuster am wahrscheinlichsten ist.
|
||||
|
||||
Bleiben wir mit einem Beispiel bei den orientierungssensitiven Zellen
|
||||
des V1. Das Tuning $\Omega_i(\phi)$ der Zellen $i$ auf ihre bevorzugte
|
||||
Orientierung $\phi_i$ l\"asst sich gut mit einer van-Mises Funktion
|
||||
(entspricht der Gaussfunktion auf einer zyklischen x-Achse)
|
||||
beschreiben (\figref{mlecodingfig}):
|
||||
\[ \Omega_i(\phi) = c \cdot e^{\cos(2(\phi-\phi_i))} \quad , \quad c
|
||||
\in \reZ \]
|
||||
Die Aktivit\"at der Neurone approximieren wir hier mit einer
|
||||
Normalverteilung um die Tuning-Kurve mit Standardabweichung
|
||||
$\sigma=\Omega/4$ proportional zu $\Omega$, so dass die
|
||||
Wahrscheinlichkeit $p_i(r|\phi)$ des $i$-ten Neurons die Aktivit\"at $r$ zu
|
||||
haben, wenn ein Stimulus mit Orientierung $\phi$ anliegt, gegeben ist durch
|
||||
\[ p_i(r|\phi) = \frac{1}{\sqrt{2\pi}\Omega_i(\phi)/4} e^{-\frac{1}{2}\left(\frac{r-\Omega_i(\phi)}{\Omega_i(\phi)/4}\right)^2} \; . \]
|
||||
Die Log-Likelihood der Stimulusorientierung $\phi$ gegeben die
|
||||
Aktivit\"aten $r_1$, $r_2$, ... $r_n$ ist damit
|
||||
\[ {\cal L}(\phi|r_1, r_2, \ldots r_n) = \sum_{i=1}^n \log p_i(r_i|\phi) \]
|
||||
|
||||
101
likelihood/lecture/mlecoding.py
Normal file
101
likelihood/lecture/mlecoding.py
Normal file
@ -0,0 +1,101 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
import matplotlib.cm as cm
|
||||
|
||||
plt.xkcd()
|
||||
fig = plt.figure( figsize=(6,6.8) )
|
||||
rng = np.random.RandomState(4637281)
|
||||
lmarg=0.1
|
||||
rmarg=0.1
|
||||
|
||||
ax = fig.add_axes([lmarg, 0.75, 1.0-rmarg, 0.25])
|
||||
ax.spines['bottom'].set_position('zero')
|
||||
ax.spines['left'].set_visible(False)
|
||||
ax.spines['right'].set_visible(False)
|
||||
ax.spines['top'].set_visible(False)
|
||||
ax.xaxis.set_ticks_position('bottom')
|
||||
ax.get_yaxis().set_visible(False)
|
||||
ax.set_xlim(0.0, np.pi)
|
||||
ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
|
||||
ax.set_xticklabels([])
|
||||
ax.set_ylim(0.0, 3.5)
|
||||
ax.text(-0.2, 0.5*3.5, 'Activity', rotation='vertical', va='center')
|
||||
ax.annotate('Tuning curve',
|
||||
xy=(0.42*np.pi, 2.5), xycoords='data',
|
||||
xytext=(0.3*np.pi, 3.2), textcoords='data', ha='right',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5),
|
||||
connectionstyle="angle3,angleA=-10,angleB=110") )
|
||||
ax.annotate('',
|
||||
xy=(0.5*np.pi, 0.1), xycoords='data',
|
||||
xytext=(0.5*np.pi, 2.6), textcoords='data',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
|
||||
connectionstyle="angle3,angleA=80,angleB=90") )
|
||||
ax.text(0.52*np.pi, 0.7, 'preferred\norientation')
|
||||
ax.plot([0, 0], [0.0, 3.5], 'k', zorder=10, clip_on=False)
|
||||
xx = np.arange(0.0, 2.0*np.pi, 0.01)
|
||||
pp = 0.5*np.pi
|
||||
yy = np.exp(np.cos(2.0*(xx+pp)))
|
||||
ax.fill_between(xx, yy+0.25*yy, yy-0.25*yy, color=cm.autumn(0.3, 1), alpha=0.5)
|
||||
ax.plot(xx, yy, color=cm.autumn(0.0, 1))
|
||||
|
||||
ax = fig.add_axes([lmarg, 0.34, 1.0-rmarg, 0.38])
|
||||
ax.spines['bottom'].set_position('zero')
|
||||
ax.spines['left'].set_visible(False)
|
||||
ax.spines['right'].set_visible(False)
|
||||
ax.spines['top'].set_visible(False)
|
||||
ax.xaxis.set_ticks_position('bottom')
|
||||
ax.get_yaxis().set_visible(False)
|
||||
ax.set_xlim(0.0, np.pi)
|
||||
ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
|
||||
ax.set_xticklabels([])
|
||||
ax.set_ylim(-1.5, 3.0)
|
||||
ax.text(0.5*np.pi, -1.8, 'Orientation', ha='center')
|
||||
ax.text(-0.2, 0.5*3.5, 'Activity', rotation='vertical', va='center')
|
||||
ax.plot([0, 0], [0.0, 3.0], 'k', zorder=10, clip_on=False)
|
||||
xx = np.arange(0.0, 1.0*np.pi, 0.01)
|
||||
prefphases = np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi)
|
||||
responses = []
|
||||
xresponse = 0.475*np.pi
|
||||
for pp in prefphases :
|
||||
yy = np.exp(np.cos(2.0*(xx+pp)))
|
||||
ax.plot(xx, yy, color=cm.autumn(2.0*np.abs(pp/np.pi-0.5), 1))
|
||||
y = np.exp(np.cos(2.0*(xresponse+pp)))
|
||||
responses.append(y + rng.randn()*0.25*y)
|
||||
ax.plot(xresponse, y, '.', markersize=20, color=cm.autumn(2.0*np.abs(pp/np.pi-0.5), 1))
|
||||
r=0.3
|
||||
y=-0.8
|
||||
ax.plot([pp-0.5*r*np.cos(pp), pp+0.5*r*np.cos(pp)], [y-r*np.sin(pp), y+r*np.sin(pp)], 'k', lw=6)
|
||||
responses = np.array(responses)
|
||||
|
||||
ax = fig.add_axes([lmarg, 0.05, 1.0-rmarg, 0.22])
|
||||
ax.spines['left'].set_visible(False)
|
||||
ax.spines['right'].set_visible(False)
|
||||
ax.spines['top'].set_visible(False)
|
||||
ax.xaxis.set_ticks_position('bottom')
|
||||
ax.get_yaxis().set_visible(False)
|
||||
ax.set_xlim(0.0, np.pi)
|
||||
ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
|
||||
ax.set_xticklabels([])
|
||||
ax.set_ylim(-1600, 0)
|
||||
ax.set_xlabel('Orientiation')
|
||||
ax.text(-0.2, -800, 'Log-Likelihood', rotation='vertical', va='center')
|
||||
ax.plot([0, 0], [-1600, 0], 'k', zorder=10, clip_on=False)
|
||||
phases = np.linspace(0.0, 1.1*np.pi, 100)
|
||||
probs = np.zeros((len(responses), len(phases)))
|
||||
for k, (pp, r) in enumerate(zip(prefphases, responses)) :
|
||||
y = np.exp(np.cos(2.0*(phases+pp)))
|
||||
sigma = 0.1*y
|
||||
probs[k,:] = np.exp(-0.5*((r-y)/sigma)**2.0)/np.sqrt(2.0*np.pi)/sigma
|
||||
loglikelihood = np.sum(np.log(probs), 0)
|
||||
maxl = np.max(loglikelihood)
|
||||
maxp = phases[np.argmax(loglikelihood)]
|
||||
ax.annotate('',
|
||||
xy=(maxp, -1600), xycoords='data',
|
||||
xytext=(maxp, -30), textcoords='data',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
|
||||
connectionstyle="angle3,angleA=80,angleB=90") )
|
||||
ax.text(maxp+0.05, -1300, 'most\nlikely\norientation')
|
||||
ax.plot(phases, loglikelihood, '-b')
|
||||
|
||||
plt.savefig('mlecoding.pdf')
|
||||
#plt.show();
|
||||
@ -1,217 +1,8 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
\author{Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
|
||||
\date{WS 15/16}
|
||||
\input{../../header}
|
||||
|
||||
%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \newcommand{\tr}[2]{#1} % en
|
||||
% \usepackage[english]{babel}
|
||||
\newcommand{\tr}[2]{#2} % de
|
||||
\usepackage[german]{babel}
|
||||
|
||||
%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{pslatex} % nice font for pdf file
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
|
||||
\setcounter{tocdepth}{1}
|
||||
|
||||
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[sf,bf,it,big,clearempty]{titlesec}
|
||||
\setcounter{secnumdepth}{1}
|
||||
|
||||
|
||||
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
|
||||
|
||||
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{graphicx}
|
||||
\usepackage{xcolor}
|
||||
\pagecolor{white}
|
||||
|
||||
\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
|
||||
\put(0,4){\line(1,0){170}}%
|
||||
\multiput(0,2)(10,0){18}{\line(0,1){4}}%
|
||||
\multiput(0,3)(1,0){170}{\line(0,1){2}}%
|
||||
\put(0,0){\makebox(0,0){{\tiny 0}}}%
|
||||
\put(10,0){\makebox(0,0){{\tiny 1}}}%
|
||||
\put(20,0){\makebox(0,0){{\tiny 2}}}%
|
||||
\put(30,0){\makebox(0,0){{\tiny 3}}}%
|
||||
\put(40,0){\makebox(0,0){{\tiny 4}}}%
|
||||
\put(50,0){\makebox(0,0){{\tiny 5}}}%
|
||||
\put(60,0){\makebox(0,0){{\tiny 6}}}%
|
||||
\put(70,0){\makebox(0,0){{\tiny 7}}}%
|
||||
\put(80,0){\makebox(0,0){{\tiny 8}}}%
|
||||
\put(90,0){\makebox(0,0){{\tiny 9}}}%
|
||||
\put(100,0){\makebox(0,0){{\tiny 10}}}%
|
||||
\put(110,0){\makebox(0,0){{\tiny 11}}}%
|
||||
\put(120,0){\makebox(0,0){{\tiny 12}}}%
|
||||
\put(130,0){\makebox(0,0){{\tiny 13}}}%
|
||||
\put(140,0){\makebox(0,0){{\tiny 14}}}%
|
||||
\put(150,0){\makebox(0,0){{\tiny 15}}}%
|
||||
\put(160,0){\makebox(0,0){{\tiny 16}}}%
|
||||
\put(170,0){\makebox(0,0){{\tiny 17}}}%
|
||||
\end{picture}\par}
|
||||
|
||||
% figures:
|
||||
\setlength{\fboxsep}{0pt}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
|
||||
%\newcommand{\texpicture}[1]{}
|
||||
\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
|
||||
|
||||
% maximum number of floats:
|
||||
\setcounter{topnumber}{2}
|
||||
\setcounter{bottomnumber}{0}
|
||||
\setcounter{totalnumber}{2}
|
||||
|
||||
% float placement fractions:
|
||||
\renewcommand{\textfraction}{0.2}
|
||||
\renewcommand{\topfraction}{0.8}
|
||||
\renewcommand{\bottomfraction}{0.0}
|
||||
\renewcommand{\floatpagefraction}{0.5}
|
||||
|
||||
% spacing for floats:
|
||||
\setlength{\floatsep}{12pt plus 2pt minus 2pt}
|
||||
\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
|
||||
\setlength{\intextsep}{12pt plus 2pt minus 2pt}
|
||||
|
||||
% spacing for a floating page:
|
||||
\makeatletter
|
||||
\setlength{\@fptop}{0pt}
|
||||
\setlength{\@fpsep}{8pt plus 2.0fil}
|
||||
\setlength{\@fpbot}{0pt plus 1.0fil}
|
||||
\makeatother
|
||||
|
||||
% rules for floats:
|
||||
\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
|
||||
\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
|
||||
|
||||
% captions:
|
||||
\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
|
||||
|
||||
% put caption on separate float:
|
||||
\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
|
||||
|
||||
% references to panels of a figure within the caption:
|
||||
\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
|
||||
% references to figures:
|
||||
\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
|
||||
\newcommand{\fref}[1]{\textup{\ref{#1}}}
|
||||
\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
|
||||
% references to figures in normal text:
|
||||
\newcommand{\fig}{Fig.}
|
||||
\newcommand{\Fig}{Figure}
|
||||
\newcommand{\figs}{Figs.}
|
||||
\newcommand{\Figs}{Figures}
|
||||
\newcommand{\figref}[1]{\fig~\fref{#1}}
|
||||
\newcommand{\Figref}[1]{\Fig~\fref{#1}}
|
||||
\newcommand{\figsref}[1]{\figs~\fref{#1}}
|
||||
\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
|
||||
\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
|
||||
% references to figures within bracketed text:
|
||||
\newcommand{\figb}{Fig.}
|
||||
\newcommand{\figsb}{Figs.}
|
||||
\newcommand{\figrefb}[1]{\figb~\fref{#1}}
|
||||
\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
|
||||
\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
|
||||
|
||||
% references to tables:
|
||||
\newcommand{\tref}[1]{\textup{\ref{#1}}}
|
||||
% references to tables in normal text:
|
||||
\newcommand{\tab}{Tab.}
|
||||
\newcommand{\Tab}{Table}
|
||||
\newcommand{\tabs}{Tabs.}
|
||||
\newcommand{\Tabs}{Tables}
|
||||
\newcommand{\tabref}[1]{\tab~\tref{#1}}
|
||||
\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
|
||||
\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
|
||||
\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
|
||||
% references to tables within bracketed text:
|
||||
\newcommand{\tabb}{Tab.}
|
||||
\newcommand{\tabsb}{Tab.}
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\newcommand{\eqref}[1]{(\ref{#1})}
|
||||
\newcommand{\eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\Eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\Eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
|
||||
\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
|
||||
\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
|
||||
\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
|
||||
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
inputpath=../code,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{red!60!black},
|
||||
keywordstyle=\color{blue!50!black},
|
||||
stringstyle=\color{green!50!black},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{9} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
\lstset{inputpath=../code}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
@ -1,219 +1,8 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
\author{Jan Grewe \& Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
|
||||
\date{WS 15/16}
|
||||
\input{header}
|
||||
|
||||
%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \newcommand{\tr}[2]{#1} % en
|
||||
% \usepackage[english]{babel}
|
||||
\newcommand{\tr}[2]{#2} % de
|
||||
\usepackage[german]{babel}
|
||||
|
||||
%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{pslatex} % nice font for pdf file
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
|
||||
\setcounter{tocdepth}{1}
|
||||
|
||||
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[sf,bf,it,big,clearempty]{titlesec}
|
||||
\setcounter{secnumdepth}{1}
|
||||
|
||||
|
||||
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
|
||||
|
||||
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{graphicx}
|
||||
\usepackage{xcolor}
|
||||
\pagecolor{white}
|
||||
|
||||
\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
|
||||
\put(0,4){\line(1,0){170}}%
|
||||
\multiput(0,2)(10,0){18}{\line(0,1){4}}%
|
||||
\multiput(0,3)(1,0){170}{\line(0,1){2}}%
|
||||
\put(0,0){\makebox(0,0){{\tiny 0}}}%
|
||||
\put(10,0){\makebox(0,0){{\tiny 1}}}%
|
||||
\put(20,0){\makebox(0,0){{\tiny 2}}}%
|
||||
\put(30,0){\makebox(0,0){{\tiny 3}}}%
|
||||
\put(40,0){\makebox(0,0){{\tiny 4}}}%
|
||||
\put(50,0){\makebox(0,0){{\tiny 5}}}%
|
||||
\put(60,0){\makebox(0,0){{\tiny 6}}}%
|
||||
\put(70,0){\makebox(0,0){{\tiny 7}}}%
|
||||
\put(80,0){\makebox(0,0){{\tiny 8}}}%
|
||||
\put(90,0){\makebox(0,0){{\tiny 9}}}%
|
||||
\put(100,0){\makebox(0,0){{\tiny 10}}}%
|
||||
\put(110,0){\makebox(0,0){{\tiny 11}}}%
|
||||
\put(120,0){\makebox(0,0){{\tiny 12}}}%
|
||||
\put(130,0){\makebox(0,0){{\tiny 13}}}%
|
||||
\put(140,0){\makebox(0,0){{\tiny 14}}}%
|
||||
\put(150,0){\makebox(0,0){{\tiny 15}}}%
|
||||
\put(160,0){\makebox(0,0){{\tiny 16}}}%
|
||||
\put(170,0){\makebox(0,0){{\tiny 17}}}%
|
||||
\end{picture}\par}
|
||||
|
||||
% figures:
|
||||
\setlength{\fboxsep}{0pt}
|
||||
\newcommand{\texinputpath}{}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{\texinputpath#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
|
||||
%\newcommand{\texpicture}[1]{}
|
||||
\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
|
||||
|
||||
% maximum number of floats:
|
||||
\setcounter{topnumber}{2}
|
||||
\setcounter{bottomnumber}{0}
|
||||
\setcounter{totalnumber}{2}
|
||||
|
||||
% float placement fractions:
|
||||
\renewcommand{\textfraction}{0.2}
|
||||
\renewcommand{\topfraction}{0.8}
|
||||
\renewcommand{\bottomfraction}{0.0}
|
||||
\renewcommand{\floatpagefraction}{0.5}
|
||||
|
||||
% spacing for floats:
|
||||
\setlength{\floatsep}{12pt plus 2pt minus 2pt}
|
||||
\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
|
||||
\setlength{\intextsep}{12pt plus 2pt minus 2pt}
|
||||
|
||||
% spacing for a floating page:
|
||||
\makeatletter
|
||||
\setlength{\@fptop}{0pt}
|
||||
\setlength{\@fpsep}{8pt plus 2.0fil}
|
||||
\setlength{\@fpbot}{0pt plus 1.0fil}
|
||||
\makeatother
|
||||
|
||||
% rules for floats:
|
||||
\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
|
||||
\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
|
||||
|
||||
% captions:
|
||||
\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
|
||||
|
||||
% put caption on separate float:
|
||||
\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
|
||||
|
||||
% references to panels of a figure within the caption:
|
||||
\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
|
||||
% references to figures:
|
||||
\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
|
||||
\newcommand{\fref}[1]{\textup{\ref{#1}}}
|
||||
\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
|
||||
% references to figures in normal text:
|
||||
\newcommand{\fig}{Fig.}
|
||||
\newcommand{\Fig}{Figure}
|
||||
\newcommand{\figs}{Figs.}
|
||||
\newcommand{\Figs}{Figures}
|
||||
\newcommand{\figref}[1]{\fig~\fref{#1}}
|
||||
\newcommand{\Figref}[1]{\Fig~\fref{#1}}
|
||||
\newcommand{\figsref}[1]{\figs~\fref{#1}}
|
||||
\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
|
||||
\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
|
||||
% references to figures within bracketed text:
|
||||
\newcommand{\figb}{Fig.}
|
||||
\newcommand{\figsb}{Figs.}
|
||||
\newcommand{\figrefb}[1]{\figb~\fref{#1}}
|
||||
\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
|
||||
\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
|
||||
|
||||
% references to tables:
|
||||
\newcommand{\tref}[1]{\textup{\ref{#1}}}
|
||||
% references to tables in normal text:
|
||||
\newcommand{\tab}{Tab.}
|
||||
\newcommand{\Tab}{Table}
|
||||
\newcommand{\tabs}{Tabs.}
|
||||
\newcommand{\Tabs}{Tables}
|
||||
\newcommand{\tabref}[1]{\tab~\tref{#1}}
|
||||
\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
|
||||
\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
|
||||
\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
|
||||
% references to tables within bracketed text:
|
||||
\newcommand{\tabb}{Tab.}
|
||||
\newcommand{\tabsb}{Tab.}
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\newcommand{\eqref}[1]{(\ref{#1})}
|
||||
\newcommand{\eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\Eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\Eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
|
||||
\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
|
||||
\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
|
||||
\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
|
||||
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{red!60!black},
|
||||
keywordstyle=\color{blue!50!black},
|
||||
stringstyle=\color{green!50!black},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
%%%%% exercises: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{10} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newcommand{\codepath}{}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\codepath\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
\setcounter{maxexercise}{10000} % show listings up to exercise maxexercise
|
||||
|
||||
\graphicspath{{statistics/lecture/}{statistics/lecture/figures/}{bootstrap/lecture/}{bootstrap/lecture/figures/}{likelihood/lecture/}{likelihood/lecture/figures/}{pointprocesses/lecture/}{pointprocesses/lecture/figures/}}
|
||||
|
||||
@ -226,20 +15,20 @@
|
||||
|
||||
\tableofcontents
|
||||
|
||||
\renewcommand{\codepath}{statistics/code/}
|
||||
\lstset{inputpath=statistics/code}
|
||||
\include{statistics/lecture/statistics}
|
||||
|
||||
\renewcommand{\codepath}{bootstrap/code/}
|
||||
\lstset{inputpath=bootstrap/code}
|
||||
\include{bootstrap/lecture/bootstrap}
|
||||
|
||||
\renewcommand{\codepath}{likelihood/code/}
|
||||
\lstset{inputpath=likelihood/code}
|
||||
\include{likelihood/lecture/likelihood}
|
||||
|
||||
\renewcommand{\codepath}{pointprocesses/code/}
|
||||
\lstset{inputpath=pointprocesses/code/}
|
||||
\renewcommand{\texinputpath}{pointprocesses/lecture/}
|
||||
\include{pointprocesses/lecture/pointprocesses}
|
||||
|
||||
\renewcommand{\codepath}{designpattern/code/}
|
||||
\lstset{inputpath=designpattern/code/}
|
||||
\include{designpattern/lecture/designpattern}
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -1,20 +1,70 @@
|
||||
TEXFILES=$(wildcard *.tex)
|
||||
TEXFILES=psth_sta.tex
|
||||
BASENAME=psth_sta
|
||||
|
||||
PDFFILES=$(TEXFILES:.tex=.pdf)
|
||||
PYFILES=$(wildcard *.py)
|
||||
PYPDFFILES=$(PYFILES:.py=.pdf)
|
||||
|
||||
pdf : $(PDFFILES)
|
||||
GPTFILES=$(wildcard *.gpt)
|
||||
GPTTEXFILES=$(GPTFILES:.gpt=.tex)
|
||||
|
||||
$(PDFFILES) : %.pdf : %.tex
|
||||
|
||||
all: pdf slides thumbs
|
||||
|
||||
# script:
|
||||
pdf : $(BASENAME)-chapter.pdf
|
||||
$(BASENAME)-chapter.pdf : $(BASENAME)-chapter.tex $(BASENAME).tex $(GPTTEXFILES) $(PYPDFFILES)
|
||||
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
|
||||
|
||||
clean :
|
||||
rm -f *~ $(TEXFILES:.tex=.aux) $(TEXFILES:.tex=.log) $(TEXFILES:.tex=.out) $(TEXFILES:.tex=.nav) $(TEXFILES:.tex=.snm) $(TEXFILES:.tex=.toc) $(TEXFILES:.tex=.vrb)
|
||||
|
||||
cleanall : clean
|
||||
rm -f $(PDFFILES)
|
||||
# slides:
|
||||
slides: $(BASENAME)-slides.pdf
|
||||
$(BASENAME)-slides.pdf : $(BASENAME)-slides.tex $(GPTTEXFILES) $(PYPDFFILES)
|
||||
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
|
||||
|
||||
# thumbnails:
|
||||
thumbs: $(BASENAME)-handout.pdf
|
||||
$(BASENAME)-handout.pdf: $(BASENAME)-slides.tex $(GPTTEXFILES)
|
||||
sed -e 's/setboolean{presentation}{true}/setboolean{presentation}{false}/; s/usepackage{crop}/usepackage[frame]{crop}/' $< > thumbsfoils.tex
|
||||
pdflatex thumbsfoils | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex thumbsfoils || true
|
||||
pdfnup --nup 2x4 --no-landscape --paper a4paper --trim "-1cm -1cm -1cm -1cm" --outfile $@ thumbsfoils.pdf # 1-19
|
||||
rm thumbsfoils.*
|
||||
|
||||
watch :
|
||||
watchpdf :
|
||||
while true; do ! make -q pdf && make pdf; sleep 0.5; done
|
||||
|
||||
watchslides :
|
||||
while true; do ! make -q slides && make slides; sleep 0.5; done
|
||||
|
||||
# python plots:
|
||||
$(PYPDFFILES) : %.pdf: %.py
|
||||
python $<
|
||||
|
||||
# gnuplot plots:
|
||||
$(GPTTEXFILES) : %.tex: %.gpt whitestyles.gp
|
||||
gnuplot whitestyles.gp $<
|
||||
epstopdf $*.eps
|
||||
|
||||
|
||||
clean :
|
||||
rm -f *~
|
||||
rm -f $(BASENAME).aux $(BASENAME).log
|
||||
rm -f $(BASENAME)-chapter.aux $(BASENAME)-chapter.log $(BASENAME)-chapter.out
|
||||
rm -f $(BASENAME)-slides.aux $(BASENAME)-slides.log $(BASENAME)-slides.out $(BASENAME)-slides.toc $(BASENAME)-slides.nav $(BASENAME)-slides.snm $(BASENAME)-slides.vrb
|
||||
rm -f $(PYPDFFILES) $(GPTTEXFILES)
|
||||
|
||||
cleanall : clean
|
||||
rm -f $(BASENAME)-chapter.pdf $(BASENAME)-slides.pdf $(BASENAME)-handout.pdf
|
||||
|
||||
|
||||
help :
|
||||
@echo -e \
|
||||
"make pdf: make the pdf file of the script.\n"\
|
||||
"make slides: make the pdf file of the slides.\n"\
|
||||
"make thumbs: make color thumbnails of the talk.\n"\
|
||||
"make watchpdf: make the pdf file of the script\n"\
|
||||
" whenever the tex file is modified.\n"\
|
||||
"make watchpdf: make the pdf file of the slides\n"\
|
||||
" whenever the tex file is modified.\n"\
|
||||
"make clean: remove all intermediate files,\n"\
|
||||
" just leave the source files and the final .pdf files.\n"\
|
||||
"make cleanup: remove all intermediate files as well as\n"\
|
||||
" the final .pdf files.\n"\
|
||||
|
||||
17
spike_trains/lecture/psth_sta-chapter.tex
Normal file
17
spike_trains/lecture/psth_sta-chapter.tex
Normal file
@ -0,0 +1,17 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
\input{../../header}
|
||||
|
||||
\lstset{inputpath=../code}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\include{psth_sta}
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -3,7 +3,7 @@
|
||||
for i = 1:140 % loop over different length
|
||||
for k = 1:10 % try several times
|
||||
a = randn( i, 1 ); % generate some data
|
||||
m = mymedian( a ) % compute median
|
||||
m = mymedian( a ); % compute median
|
||||
if length( a(a>m) ) ~= length( a(a<m) ) % check
|
||||
disp( 'error!' )
|
||||
end
|
||||
|
||||
@ -1,24 +1,27 @@
|
||||
% dependence of histogram on number of rolls:
|
||||
nrolls = [ 20, 100, 1000 ];
|
||||
nrolls = [20, 100, 1000];
|
||||
for i = [1:length(nrolls)]
|
||||
d = rollthedie( nrolls(i) );
|
||||
d = rollthedie(nrolls(i));
|
||||
% plain hist:
|
||||
%hist( d )
|
||||
%hist(d)
|
||||
|
||||
% check bin counts of plain hist:
|
||||
% h = hist( d )
|
||||
% h = hist(d)
|
||||
|
||||
% force 6 bins:
|
||||
%hist( d, 6 )
|
||||
%hist(d, 6)
|
||||
|
||||
% set the right bin centers:
|
||||
%bins = 1:6;
|
||||
%hist( d, bins )
|
||||
bins = 1:6;
|
||||
%hist(d, bins)
|
||||
|
||||
% normalize histogram and compare to expectation:
|
||||
plot([0 7], [1/6 1/6], '-r', 'linewidth', 10)
|
||||
[h, b] = hist(d, bins);
|
||||
h = h/sum(h) % normalization
|
||||
hold on
|
||||
plot( [0 7], [1/6 1/6], '-r', 'linewidth', 10 )
|
||||
hist( d, bins, 1.0, 'facecolor', 'b' )
|
||||
bar(b, h, 'facecolor', 'b')
|
||||
hold off
|
||||
pause
|
||||
title(sprintf('N=%d', length(d)))
|
||||
pause( 2.0 )
|
||||
end
|
||||
|
||||
@ -7,9 +7,9 @@ bins2 = -4:db2:4; % small bins
|
||||
[h2,b2] = hist(x,bins2);
|
||||
|
||||
subplot( 1, 2, 1 );
|
||||
bar(b1,hn1)
|
||||
bar(b1,h1)
|
||||
hold on
|
||||
bar(b2,hn2, 'facecolor', 'r' )
|
||||
bar(b2,h2, 'facecolor', 'r' )
|
||||
xlabel('x')
|
||||
ylabel('Frequency')
|
||||
hold off
|
||||
|
||||
@ -1,30 +1,27 @@
|
||||
% plot Gaussian pdf:
|
||||
dx=0.1;
|
||||
dx=0.01;
|
||||
x = [-4.0:dx:4.0];
|
||||
p = exp(-0.5*x.^2)/sqrt(2.0*pi);
|
||||
plot(x, p, 'linewidth', 4)
|
||||
hold on
|
||||
plot(x, p, 'linewidth', 10)
|
||||
% show area of integral:
|
||||
x1=1.0;
|
||||
x2=2.0;
|
||||
area(x((x>=x1)&(x<=x2)), p((x>=x1)&(x<=x2)), 'FaceColor', 'r' )
|
||||
hold off
|
||||
|
||||
% compute integral between x1 and x2:
|
||||
x1=1.0;
|
||||
x2=2.0;
|
||||
P = sum(p((x>=x1)&(x<x2)))*dx;
|
||||
disp( [ 'The integral between ', num2str(x1, 1), ' and ', num2str(x2, 1), ' is ', num2str(P, 3) ] );
|
||||
fprintf( 'The integral between %.2g and %.2g is %.3g\n', x1, x2, P );
|
||||
|
||||
% draw random numbers:
|
||||
%r = randn( 10000, 1 );
|
||||
%hist(r,x,1.0/dx)
|
||||
r = randn( 10000, 1 );
|
||||
|
||||
% check P:
|
||||
Pr = sum((r>=x1)&(r<x2))/length(r);
|
||||
disp( [ 'The probability of getting a number between ', num2str(x1, 1), ' and ', num2str(x2, 1), ' is ', num2str(Pr, 3) ] );
|
||||
fprintf( 'The probability of getting a number between %.2g and %.2g is %.3g\n', x1, x2, Pr );
|
||||
|
||||
% infinite integral:
|
||||
P = sum(p)*dx;
|
||||
disp( [ 'The integral between -infinity and +infinity is ', num2str(P, 3) ] );
|
||||
disp( [ 'I.e. the probability to get any number is ', num2str(P, 3) ] );
|
||||
|
||||
|
||||
fprintf( 'The integral between -infinity and +infinity is %.3g\n', P );
|
||||
fprintf( 'I.e. the probability to get any number is %.3g\n', P );
|
||||
|
||||
@ -8,7 +8,7 @@ x1=0.0
|
||||
x2=1.0
|
||||
|
||||
plt.xkcd()
|
||||
fig = plt.figure( figsize=(6,4) )
|
||||
fig = plt.figure( figsize=(6,3.8) )
|
||||
ax = fig.add_subplot( 1, 1, 1 )
|
||||
ax.spines['right'].set_visible(False)
|
||||
ax.spines['top'].set_visible(False)
|
||||
|
||||
@ -1,217 +1,8 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
\author{Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
|
||||
\date{WS 15/16}
|
||||
\input{../../header}
|
||||
|
||||
%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \newcommand{\tr}[2]{#1} % en
|
||||
% \usepackage[english]{babel}
|
||||
\newcommand{\tr}[2]{#2} % de
|
||||
\usepackage[german]{babel}
|
||||
|
||||
%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{pslatex} % nice font for pdf file
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
|
||||
\setcounter{tocdepth}{1}
|
||||
|
||||
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[sf,bf,it,big,clearempty]{titlesec}
|
||||
\setcounter{secnumdepth}{1}
|
||||
|
||||
|
||||
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
|
||||
|
||||
|
||||
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{graphicx}
|
||||
\usepackage{xcolor}
|
||||
\pagecolor{white}
|
||||
|
||||
\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
|
||||
\put(0,4){\line(1,0){170}}%
|
||||
\multiput(0,2)(10,0){18}{\line(0,1){4}}%
|
||||
\multiput(0,3)(1,0){170}{\line(0,1){2}}%
|
||||
\put(0,0){\makebox(0,0){{\tiny 0}}}%
|
||||
\put(10,0){\makebox(0,0){{\tiny 1}}}%
|
||||
\put(20,0){\makebox(0,0){{\tiny 2}}}%
|
||||
\put(30,0){\makebox(0,0){{\tiny 3}}}%
|
||||
\put(40,0){\makebox(0,0){{\tiny 4}}}%
|
||||
\put(50,0){\makebox(0,0){{\tiny 5}}}%
|
||||
\put(60,0){\makebox(0,0){{\tiny 6}}}%
|
||||
\put(70,0){\makebox(0,0){{\tiny 7}}}%
|
||||
\put(80,0){\makebox(0,0){{\tiny 8}}}%
|
||||
\put(90,0){\makebox(0,0){{\tiny 9}}}%
|
||||
\put(100,0){\makebox(0,0){{\tiny 10}}}%
|
||||
\put(110,0){\makebox(0,0){{\tiny 11}}}%
|
||||
\put(120,0){\makebox(0,0){{\tiny 12}}}%
|
||||
\put(130,0){\makebox(0,0){{\tiny 13}}}%
|
||||
\put(140,0){\makebox(0,0){{\tiny 14}}}%
|
||||
\put(150,0){\makebox(0,0){{\tiny 15}}}%
|
||||
\put(160,0){\makebox(0,0){{\tiny 16}}}%
|
||||
\put(170,0){\makebox(0,0){{\tiny 17}}}%
|
||||
\end{picture}\par}
|
||||
|
||||
% figures:
|
||||
\setlength{\fboxsep}{0pt}
|
||||
\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
|
||||
%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
|
||||
%\newcommand{\texpicture}[1]{}
|
||||
\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
|
||||
|
||||
% maximum number of floats:
|
||||
\setcounter{topnumber}{2}
|
||||
\setcounter{bottomnumber}{0}
|
||||
\setcounter{totalnumber}{2}
|
||||
|
||||
% float placement fractions:
|
||||
\renewcommand{\textfraction}{0.2}
|
||||
\renewcommand{\topfraction}{0.8}
|
||||
\renewcommand{\bottomfraction}{0.0}
|
||||
\renewcommand{\floatpagefraction}{0.5}
|
||||
|
||||
% spacing for floats:
|
||||
\setlength{\floatsep}{12pt plus 2pt minus 2pt}
|
||||
\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
|
||||
\setlength{\intextsep}{12pt plus 2pt minus 2pt}
|
||||
|
||||
% spacing for a floating page:
|
||||
\makeatletter
|
||||
\setlength{\@fptop}{0pt}
|
||||
\setlength{\@fpsep}{8pt plus 2.0fil}
|
||||
\setlength{\@fpbot}{0pt plus 1.0fil}
|
||||
\makeatother
|
||||
|
||||
% rules for floats:
|
||||
\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
|
||||
\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
|
||||
|
||||
% captions:
|
||||
\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
|
||||
|
||||
% put caption on separate float:
|
||||
\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
|
||||
|
||||
% references to panels of a figure within the caption:
|
||||
\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
|
||||
% references to figures:
|
||||
\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
|
||||
\newcommand{\fref}[1]{\textup{\ref{#1}}}
|
||||
\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
|
||||
% references to figures in normal text:
|
||||
\newcommand{\fig}{Fig.}
|
||||
\newcommand{\Fig}{Figure}
|
||||
\newcommand{\figs}{Figs.}
|
||||
\newcommand{\Figs}{Figures}
|
||||
\newcommand{\figref}[1]{\fig~\fref{#1}}
|
||||
\newcommand{\Figref}[1]{\Fig~\fref{#1}}
|
||||
\newcommand{\figsref}[1]{\figs~\fref{#1}}
|
||||
\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
|
||||
\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
|
||||
\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
|
||||
% references to figures within bracketed text:
|
||||
\newcommand{\figb}{Fig.}
|
||||
\newcommand{\figsb}{Figs.}
|
||||
\newcommand{\figrefb}[1]{\figb~\fref{#1}}
|
||||
\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
|
||||
\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
|
||||
\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
|
||||
|
||||
% references to tables:
|
||||
\newcommand{\tref}[1]{\textup{\ref{#1}}}
|
||||
% references to tables in normal text:
|
||||
\newcommand{\tab}{Tab.}
|
||||
\newcommand{\Tab}{Table}
|
||||
\newcommand{\tabs}{Tabs.}
|
||||
\newcommand{\Tabs}{Tables}
|
||||
\newcommand{\tabref}[1]{\tab~\tref{#1}}
|
||||
\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
|
||||
\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
|
||||
\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
|
||||
% references to tables within bracketed text:
|
||||
\newcommand{\tabb}{Tab.}
|
||||
\newcommand{\tabsb}{Tab.}
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\newcommand{\eqref}[1]{(\ref{#1})}
|
||||
\newcommand{\eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\Eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\Eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
|
||||
\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
|
||||
\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
|
||||
\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
|
||||
|
||||
|
||||
%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{listings}
|
||||
\lstset{
|
||||
inputpath=../code,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{red!60!black},
|
||||
keywordstyle=\color{blue!50!black},
|
||||
stringstyle=\color{green!50!black},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{9} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
\lstset{inputpath=../code}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
@ -53,7 +53,7 @@
|
||||
{Schreibe eine Funktion, die den Median eines Vektors zur\"uckgibt.}
|
||||
\end{exercise}
|
||||
|
||||
\code{matlab} stellt die Funktion \code{median()} zur Berechnung des Medians bereit.
|
||||
\matlab{} stellt die Funktion \code{median()} zur Berechnung des Medians bereit.
|
||||
|
||||
\begin{exercise}[checkmymedian.m]
|
||||
\tr{Write a script that tests whether your median function really
|
||||
@ -70,24 +70,24 @@
|
||||
\caption{\label{quartilefig} Median und Quartile einer Normalverteilung.}
|
||||
\end{figure}
|
||||
|
||||
\begin{definition}[\tr{quartile}{Quartile}]
|
||||
Die Quartile Q1, Q2 und Q3 unterteilen die Daten in vier gleich
|
||||
gro{\ss}e Gruppen, die jeweils ein Viertel der Daten enthalten.
|
||||
Das mittlere Quartil entspricht dem Median.
|
||||
\end{definition}
|
||||
% \begin{definition}[\tr{quartile}{Quartile}]
|
||||
% Die Quartile Q1, Q2 und Q3 unterteilen die Daten in vier gleich
|
||||
% gro{\ss}e Gruppen, die jeweils ein Viertel der Daten enthalten.
|
||||
% Das mittlere Quartil entspricht dem Median.
|
||||
% \end{definition}
|
||||
|
||||
\begin{exercise}[quartiles.m]
|
||||
\tr{Write a function that computes the first, second, and third quartile of a vector.}
|
||||
{Schreibe eine Funktion, die das erste, zweite und dritte Quartil als Vektor zur\"uckgibt.}
|
||||
\end{exercise}
|
||||
% \begin{exercise}[quartiles.m]
|
||||
% \tr{Write a function that computes the first, second, and third quartile of a vector.}
|
||||
% {Schreibe eine Funktion, die das erste, zweite und dritte Quartil als Vektor zur\"uckgibt.}
|
||||
% \end{exercise}
|
||||
|
||||
\section{\tr{Histogram}{Histogramm}}
|
||||
|
||||
Histogramme z\"ahlen die H\"aufigkeit $n_i$ des Auftretens von
|
||||
$N=\sum_{i=1}^M n_i$ Messwerten in $M$ Messbereichsklassen $i$ (Bins).
|
||||
Die Klassen unterteilen den Wertebereich meist in angrenzende und
|
||||
gleich gro{\ss}e Intervalle. Histogramme sch\"atzen die
|
||||
Wahrscheinlichkeitsverteilung der Messwerte ab.
|
||||
gleich gro{\ss}e Intervalle. Histogramme k\"onnen verwendet werden, um die
|
||||
Wahrscheinlichkeitsverteilung der Messwerte abzusch\"atzen.
|
||||
|
||||
\begin{exercise}[rollthedie.m]
|
||||
\tr{Write a function that simulates rolling a die $n$ times.}
|
||||
@ -99,7 +99,7 @@ Wahrscheinlichkeitsverteilung der Messwerte ab.
|
||||
the plain hist(x) function, force 6 bins via hist( x, 6 ), and set
|
||||
meaningfull bins positions.} {Plotte Histogramme von 20, 100, und
|
||||
1000-mal w\"urfeln. Benutze \code{hist(x)}, erzwinge sechs Bins
|
||||
mit \code{hist(x,6)}, und setze selbst sinnvolle Bins. Normiere
|
||||
mit \code{hist(x,6)}, oder setze selbst sinnvolle Bins. Normiere
|
||||
anschliessend das Histogram auf geeignete Weise.}
|
||||
\end{exercise}
|
||||
|
||||
@ -216,7 +216,7 @@ Korrelationskoeffizient
|
||||
(x-\langle x \rangle)(y-\langle y \rangle) \rangle}{\sqrt{\langle
|
||||
(x-\langle x \rangle)^2} \rangle \sqrt{\langle (y-\langle y
|
||||
\rangle)^2} \rangle} \] quantifiziert einfache lineare
|
||||
Zusammenh\"ange. Perfekt korrelierte Variablen ergeben einen
|
||||
Zusammenh\"ange \matlabfun{corr}. Perfekt korrelierte Variablen ergeben einen
|
||||
Korrelationskoeffizienten von $+1$, antikorrelierte Daten einen
|
||||
Korrelationskoeffizienten von $-1$ und nicht korrelierte Daten einen
|
||||
Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
|
||||
|
||||
Reference in New Issue
Block a user