Added correlation plots

statistics/code/diehistograms.m

@@ -3,16 +3,16 @@ nrolls = [ 20, 100, 1000 ];
 for i = [1:length(nrolls)]
   d = rollthedie( nrolls(i) );
   % plain hist:
-  % hist( d )
+  %hist( d )
 
   % check bin counts of plain hist:
   % h = hist( d )
 
   % force 6 bins:
-  % hist( d, 6 )
+  %hist( d, 6 )
 
   % set the right bin centers:
-  bins = 1:6;
+  %bins = 1:6;
   %hist( d, bins )
 
   % normalize histogram and compare to expectation:

statistics/lecture/correlation.py

@@ -0,0 +1,34 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+plt.xkcd()
+fig = plt.figure( figsize=(6,5) )
+n = 200
+for k, r  in enumerate( [ 1.0, 0.6, 0.0, -0.9 ] ) :
+    print r
+    x = np.random.randn( n )
+    y = r*x + np.sqrt(1.0-r*r)*np.random.randn( n )
+    ax = fig.add_subplot( 2, 2, k+1 )
+    ax.spines['right'].set_visible(False)
+    ax.spines['top'].set_visible(False)
+    ax.yaxis.set_ticks_position('left')
+    ax.xaxis.set_ticks_position('bottom')
+    ax.text( -2, 2.5, 'r=%.1f' % r )
+    if k == 0 :
+        ax.text( 2.8, -2, 'positively\ncorrelated', ha='right' )
+    elif k == 1 :
+        ax.text( 2.8, -2.5, 'weakly\ncorrelated', ha='right' )
+    elif k == 2 :
+        ax.text( 2.8, -2.5, 'not\ncorrelated', ha='right' )
+    elif k == 3 :
+        ax.text( -2.5, -2, 'negatively\ncorrelated', ha='left' )
+    ax.set_xlabel('x')
+    ax.set_ylabel('y')
+    ax.set_xlim( -3.0, 3.0)
+    ax.set_ylim( -3.0, 3.0)
+    ax.scatter( x, y )
+
+plt.tight_layout()
+plt.savefig('correlation.pdf')
+plt.show()
+

statistics/lecture/descriptivestatistics.tex

@@ -19,10 +19,141 @@
 \usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
 \setcounter{tocdepth}{1}
 
-%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%% 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}
-\newcommand{\texpicture}[1]{{\sffamily\small\input{#1.tex}}}
+\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}{Eq.}
+\newcommand{\Eqn}{Eq.}
+\newcommand{\eqns}{Eqs.}
+\newcommand{\Eqns}{Eqs.}
+\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}
@@ -290,6 +421,36 @@ spricht von einer Wahrscheinlichkeitsdichte.
 \end{figure}
 
 
+\subsection{Korrelation}
+
+\begin{figure}[t]
+  \includegraphics[width=1\textwidth]{correlation}
+  \caption{\label{correlationfig} Korrelationen zwischen zwei Datens\"atzen $x$ und $y$.}
+\end{figure}
+
+Bisher haben wir Eigenschaften einer einzelnen Me{\ss}gr\"o{\ss}e angeschaut.
+Bei mehreren Me{\ss}gr\"o{\ss}en, kann nach Abh\"angigkeiten gefragt werden.
+Der Korrelationskoeffizient 
+\[ r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle
+  (x-\langle x \rangle)(y-\langle y \rangle) \rangle}{\sqrt{\langle
+    (x-\langle x \rangle)^2} \rangle \sqrt{\langle (y-\langle y
+    \rangle)^2} \rangle} \] quantifiziert einfache lineare
+Zusammenh\"ange. Perfekt korrelierte Variablen ergeben einen
+Korrelationskoeffizienten von $+1$, antikorrelierte Daten einen
+Korrelationskoeffizienten von $-1$ und nicht korrelierte Daten einen
+Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
+
+\begin{figure}[t]
+  \includegraphics[width=1\textwidth]{nonlincorrelation}
+  \caption{\label{nonlincorrelationfig} Nichtlineare Zusammenh\"ange
+    werden durch den Korrelationskoeffizienten nicht erfasst! Sowohl
+    die quadratische Abh\"angigkeit (links) als auch eine
+    Rauschkorrelation (rechts), bei der die Streuung der $y$-Werte von
+    $x$ abh\"angen, ergeben Korrelationskeffizienten nahe Null.
+    $\xi$ sind normalverteilte Zufallszahlen.}
+\end{figure}
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Data types}
 

statistics/lecture/nonlincorrelation.py

@@ -0,0 +1,42 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+plt.xkcd()
+fig = plt.figure( figsize=(6,3) )
+n = 200
+x = np.random.randn( n )
+y = np.random.randn( n )
+
+z = x*x+0.2*y
+r =np.corrcoef(x,z)[0,1]
+ax = fig.add_subplot( 1, 2, 1 )
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.yaxis.set_ticks_position('left')
+ax.xaxis.set_ticks_position('bottom')
+ax.text( 0, 4.0, 'r=%.1f' % r, ha='center' )
+ax.text( 0, 5.5, r'$y = x^2+\xi/5$', ha='center' )
+ax.set_xlabel('x')
+ax.set_ylabel('y')
+ax.set_xlim( -3.0, 3.0)
+ax.set_ylim( -0.5, 6.0)
+ax.scatter( x, z )
+
+z = 0.5*x*y
+r =np.corrcoef(x,z)[0,1]
+ax = fig.add_subplot( 1, 2, 2 )
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.yaxis.set_ticks_position('left')
+ax.xaxis.set_ticks_position('bottom')
+ax.text( 0, 1.5, 'r=%.1f' % r, ha='center' )
+ax.text( 0, 2.5, r'$y = x \cdot \xi/2$', ha='center' )
+ax.set_xlabel('x')
+ax.set_ylabel('y')
+ax.set_xlim( -3.0, 3.0)
+ax.set_ylim( -3.0, 3.0)
+ax.scatter( x, z )
+
+plt.tight_layout()
+plt.savefig('nonlincorrelation.pdf')
+plt.show()