Added correlation plots
This commit is contained in:
parent
2787e55094
commit
69b9afbc80
@ -3,16 +3,16 @@ nrolls = [ 20, 100, 1000 ];
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for i = [1:length(nrolls)]
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d = rollthedie( nrolls(i) );
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% plain hist:
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% hist( d )
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%hist( d )
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% check bin counts of plain hist:
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% h = hist( d )
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% force 6 bins:
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% hist( d, 6 )
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%hist( d, 6 )
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% set the right bin centers:
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bins = 1:6;
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%bins = 1:6;
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%hist( d, bins )
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% normalize histogram and compare to expectation:
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34
statistics/lecture/correlation.py
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34
statistics/lecture/correlation.py
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@ -0,0 +1,34 @@
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import numpy as np
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import matplotlib.pyplot as plt
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plt.xkcd()
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fig = plt.figure( figsize=(6,5) )
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n = 200
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for k, r in enumerate( [ 1.0, 0.6, 0.0, -0.9 ] ) :
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print r
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x = np.random.randn( n )
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y = r*x + np.sqrt(1.0-r*r)*np.random.randn( n )
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ax = fig.add_subplot( 2, 2, k+1 )
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.yaxis.set_ticks_position('left')
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ax.xaxis.set_ticks_position('bottom')
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ax.text( -2, 2.5, 'r=%.1f' % r )
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if k == 0 :
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ax.text( 2.8, -2, 'positively\ncorrelated', ha='right' )
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elif k == 1 :
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ax.text( 2.8, -2.5, 'weakly\ncorrelated', ha='right' )
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elif k == 2 :
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ax.text( 2.8, -2.5, 'not\ncorrelated', ha='right' )
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elif k == 3 :
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ax.text( -2.5, -2, 'negatively\ncorrelated', ha='left' )
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ax.set_xlabel('x')
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ax.set_ylabel('y')
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ax.set_xlim( -3.0, 3.0)
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ax.set_ylim( -3.0, 3.0)
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ax.scatter( x, y )
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plt.tight_layout()
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plt.savefig('correlation.pdf')
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plt.show()
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@ -19,10 +19,141 @@
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\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
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\setcounter{tocdepth}{1}
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%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[sf,bf,it,big,clearempty]{titlesec}
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\setcounter{secnumdepth}{-1}
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%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
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%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{graphicx}
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\usepackage{xcolor}
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\newcommand{\texpicture}[1]{{\sffamily\small\input{#1.tex}}}
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\pagecolor{white}
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\newcommand{\ruler}{\par\noindent\setlength{\unitlength}{1mm}\begin{picture}(0,6)%
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\put(0,4){\line(1,0){170}}%
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\multiput(0,2)(10,0){18}{\line(0,1){4}}%
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\multiput(0,3)(1,0){170}{\line(0,1){2}}%
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\put(0,0){\makebox(0,0){{\tiny 0}}}%
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\put(10,0){\makebox(0,0){{\tiny 1}}}%
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\put(20,0){\makebox(0,0){{\tiny 2}}}%
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\put(30,0){\makebox(0,0){{\tiny 3}}}%
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\put(40,0){\makebox(0,0){{\tiny 4}}}%
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\put(50,0){\makebox(0,0){{\tiny 5}}}%
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\put(60,0){\makebox(0,0){{\tiny 6}}}%
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\put(70,0){\makebox(0,0){{\tiny 7}}}%
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\put(80,0){\makebox(0,0){{\tiny 8}}}%
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\put(90,0){\makebox(0,0){{\tiny 9}}}%
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\put(100,0){\makebox(0,0){{\tiny 10}}}%
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\put(110,0){\makebox(0,0){{\tiny 11}}}%
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\put(120,0){\makebox(0,0){{\tiny 12}}}%
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\put(130,0){\makebox(0,0){{\tiny 13}}}%
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\put(140,0){\makebox(0,0){{\tiny 14}}}%
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\put(150,0){\makebox(0,0){{\tiny 15}}}%
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\put(160,0){\makebox(0,0){{\tiny 16}}}%
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\put(170,0){\makebox(0,0){{\tiny 17}}}%
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\end{picture}\par}
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% figures:
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\setlength{\fboxsep}{0pt}
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\newcommand{\texpicture}[1]{{\sffamily\footnotesize\input{#1.tex}}}
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%\newcommand{\texpicture}[1]{\fbox{\sffamily\footnotesize\input{#1.tex}}}
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%\newcommand{\texpicture}[1]{\setlength{\fboxsep}{2mm}\fbox{#1}}
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%\newcommand{\texpicture}[1]{}
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\newcommand{\figlabel}[1]{\textsf{\textbf{\large \uppercase{#1}}}}
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% maximum number of floats:
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\setcounter{topnumber}{2}
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\setcounter{bottomnumber}{0}
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\setcounter{totalnumber}{2}
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% float placement fractions:
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\renewcommand{\textfraction}{0.2}
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\renewcommand{\topfraction}{0.8}
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\renewcommand{\bottomfraction}{0.0}
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\renewcommand{\floatpagefraction}{0.5}
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% spacing for floats:
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\setlength{\floatsep}{12pt plus 2pt minus 2pt}
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\setlength{\textfloatsep}{20pt plus 4pt minus 2pt}
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\setlength{\intextsep}{12pt plus 2pt minus 2pt}
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% spacing for a floating page:
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\makeatletter
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\setlength{\@fptop}{0pt}
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\setlength{\@fpsep}{8pt plus 2.0fil}
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\setlength{\@fpbot}{0pt plus 1.0fil}
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\makeatother
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% rules for floats:
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\newcommand{\topfigrule}{\vspace*{10pt}{\hrule height0.4pt}\vspace*{-10.4pt}}
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\newcommand{\bottomfigrule}{\vspace*{-10.4pt}{\hrule height0.4pt}\vspace*{10pt}}
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% captions:
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\usepackage[format=plain,singlelinecheck=off,labelfont=bf,font={small,sf}]{caption}
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% put caption on separate float:
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\newcommand{\breakfloat}{\end{figure}\begin{figure}[t]}
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% references to panels of a figure within the caption:
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\newcommand{\figitem}[1]{\textsf{\bfseries\uppercase{#1}}}
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% references to figures:
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\newcommand{\panel}[1]{\textsf{\uppercase{#1}}}
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\newcommand{\fref}[1]{\textup{\ref{#1}}}
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\newcommand{\subfref}[2]{\textup{\ref{#1}}\,\panel{#2}}
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% references to figures in normal text:
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\newcommand{\fig}{Fig.}
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\newcommand{\Fig}{Figure}
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\newcommand{\figs}{Figs.}
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\newcommand{\Figs}{Figures}
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\newcommand{\figref}[1]{\fig~\fref{#1}}
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\newcommand{\Figref}[1]{\Fig~\fref{#1}}
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\newcommand{\figsref}[1]{\figs~\fref{#1}}
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\newcommand{\Figsref}[1]{\Figs~\fref{#1}}
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\newcommand{\subfigref}[2]{\fig~\subfref{#1}{#2}}
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\newcommand{\Subfigref}[2]{\Fig~\subfref{#1}{#2}}
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\newcommand{\subfigsref}[2]{\figs~\subfref{#1}{#2}}
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\newcommand{\Subfigsref}[2]{\Figs~\subfref{#1}{#2}}
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% references to figures within bracketed text:
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\newcommand{\figb}{Fig.}
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\newcommand{\figsb}{Figs.}
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\newcommand{\figrefb}[1]{\figb~\fref{#1}}
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\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
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\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
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\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
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% references to tables:
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\newcommand{\tref}[1]{\textup{\ref{#1}}}
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% references to tables in normal text:
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\newcommand{\tab}{Tab.}
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\newcommand{\Tab}{Table}
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\newcommand{\tabs}{Tabs.}
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\newcommand{\Tabs}{Tables}
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\newcommand{\tabref}[1]{\tab~\tref{#1}}
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\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
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\newcommand{\tabsref}[1]{\tabs~\tref{#1}}
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\newcommand{\Tabsref}[1]{\Tabs~\tref{#1}}
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% references to tables within bracketed text:
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\newcommand{\tabb}{Tab.}
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\newcommand{\tabsb}{Tab.}
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\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
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\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
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%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\eqref}[1]{(\ref{#1})}
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\newcommand{\eqn}{Eq.}
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\newcommand{\Eqn}{Eq.}
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\newcommand{\eqns}{Eqs.}
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\newcommand{\Eqns}{Eqs.}
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\newcommand{\eqnref}[1]{\eqn~\eqref{#1}}
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\newcommand{\Eqnref}[1]{\Eqn~\eqref{#1}}
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\newcommand{\eqnsref}[1]{\eqns~\eqref{#1}}
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\newcommand{\Eqnsref}[1]{\Eqns~\eqref{#1}}
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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@ -290,6 +421,36 @@ spricht von einer Wahrscheinlichkeitsdichte.
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\end{figure}
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\subsection{Korrelation}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{correlation}
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\caption{\label{correlationfig} Korrelationen zwischen zwei Datens\"atzen $x$ und $y$.}
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\end{figure}
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Bisher haben wir Eigenschaften einer einzelnen Me{\ss}gr\"o{\ss}e angeschaut.
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Bei mehreren Me{\ss}gr\"o{\ss}en, kann nach Abh\"angigkeiten gefragt werden.
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Der Korrelationskoeffizient
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\[ r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle
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(x-\langle x \rangle)(y-\langle y \rangle) \rangle}{\sqrt{\langle
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(x-\langle x \rangle)^2} \rangle \sqrt{\langle (y-\langle y
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\rangle)^2} \rangle} \] quantifiziert einfache lineare
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Zusammenh\"ange. Perfekt korrelierte Variablen ergeben einen
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Korrelationskoeffizienten von $+1$, antikorrelierte Daten einen
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Korrelationskoeffizienten von $-1$ und nicht korrelierte Daten einen
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Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{nonlincorrelation}
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\caption{\label{nonlincorrelationfig} Nichtlineare Zusammenh\"ange
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werden durch den Korrelationskoeffizienten nicht erfasst! Sowohl
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die quadratische Abh\"angigkeit (links) als auch eine
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Rauschkorrelation (rechts), bei der die Streuung der $y$-Werte von
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$x$ abh\"angen, ergeben Korrelationskeffizienten nahe Null.
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$\xi$ sind normalverteilte Zufallszahlen.}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Data types}
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42
statistics/lecture/nonlincorrelation.py
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42
statistics/lecture/nonlincorrelation.py
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import numpy as np
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import matplotlib.pyplot as plt
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plt.xkcd()
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fig = plt.figure( figsize=(6,3) )
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n = 200
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x = np.random.randn( n )
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y = np.random.randn( n )
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z = x*x+0.2*y
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r =np.corrcoef(x,z)[0,1]
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ax = fig.add_subplot( 1, 2, 1 )
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.yaxis.set_ticks_position('left')
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ax.xaxis.set_ticks_position('bottom')
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ax.text( 0, 4.0, 'r=%.1f' % r, ha='center' )
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ax.text( 0, 5.5, r'$y = x^2+\xi/5$', ha='center' )
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ax.set_xlabel('x')
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ax.set_ylabel('y')
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ax.set_xlim( -3.0, 3.0)
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ax.set_ylim( -0.5, 6.0)
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ax.scatter( x, z )
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z = 0.5*x*y
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r =np.corrcoef(x,z)[0,1]
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ax = fig.add_subplot( 1, 2, 2 )
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.yaxis.set_ticks_position('left')
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ax.xaxis.set_ticks_position('bottom')
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ax.text( 0, 1.5, 'r=%.1f' % r, ha='center' )
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ax.text( 0, 2.5, r'$y = x \cdot \xi/2$', ha='center' )
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ax.set_xlabel('x')
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ax.set_ylabel('y')
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ax.set_xlim( -3.0, 3.0)
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ax.set_ylim( -3.0, 3.0)
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ax.scatter( x, z )
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plt.tight_layout()
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plt.savefig('nonlincorrelation.pdf')
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plt.show()
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