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Jan Benda 2015-10-22 08:42:53 +02:00
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statistics/lecture/descriptivestatistics.tex
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@ -21,7 +21,7 @@
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[sf,bf,it,big,clearempty]{titlesec} \usepackage[sf,bf,it,big,clearempty]{titlesec}
\setcounter{secnumdepth}{-1} \setcounter{secnumdepth}{1}
%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -144,7 +144,7 @@
%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\eqref}[1]{(\ref{#1})} %\newcommand{\eqref}[1]{(\ref{#1})}
\newcommand{\eqn}{Eq.} \newcommand{\eqn}{Eq.}
\newcommand{\Eqn}{Eq.} \newcommand{\Eqn}{Eq.}
\newcommand{\eqns}{Eqs.} \newcommand{\eqns}{Eqs.}
@ -229,7 +229,7 @@
\chapter{\tr{Descriptive statistics}{Deskriptive Statistik}} \chapter{\tr{Descriptive statistics}{Deskriptive Statistik}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Statistics of real-valued data} %\section{Statistics of real-valued data}
\begin{itemize} \begin{itemize}
\item Location, central tendency \item Location, central tendency
@ -259,7 +259,7 @@
\end{itemize} \end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Median, Quartile, Percentile} \section{\tr{Median, quartile, etc.}{Median, Quartil, etc.}}
\begin{figure}[t] \begin{figure}[t]
\includegraphics[width=1\textwidth]{median} \includegraphics[width=1\textwidth]{median}
@ -307,7 +307,7 @@
{Schreibe eine Funktion, die das erste, zweite und dritte Quartil als Vektor zur\"uckgibt.} {Schreibe eine Funktion, die das erste, zweite und dritte Quartil als Vektor zur\"uckgibt.}
\end{exercise} \end{exercise}
\subsection{Histogram} \section{\tr{Histogram}{Histogramm}}
Histogramme z\"ahlen die H\"aufigkeit $n_i$ des Auftretens von Histogramme z\"ahlen die H\"aufigkeit $n_i$ des Auftretens von
$N=\sum_{i=1}^M n_i$ Messwerten in $M$ Messbereichsklassen $i$ (Bins). $N=\sum_{i=1}^M n_i$ Messwerten in $M$ Messbereichsklassen $i$ (Bins).
@ -349,7 +349,7 @@ des Auftretens der Gr\"o{\ss}e $x_i$ in der $i$-ten Klasse an
\[ P_i = \frac{n_i}{N} = \frac{n_i}{\sum_{i=1}^M n_i} \; . \] \[ P_i = \frac{n_i}{N} = \frac{n_i}{\sum_{i=1}^M n_i} \; . \]
\subsection{Probability density function} \section{\tr{Probability density function}{Wahrscheinlichkeitsdichte}}
Meistens haben wir es jedoch mit reellen Messgr\"o{\ss}en zu tun. Meistens haben wir es jedoch mit reellen Messgr\"o{\ss}en zu tun.
@ -371,11 +371,14 @@ Meistens haben wir es jedoch mit reellen Messgr\"o{\ss}en zu tun.
unterschiedliche Klassenbreiten vergleichbar.}} unterschiedliche Klassenbreiten vergleichbar.}}
\end{figure} \end{figure}
Histogramme von reellen Messwerten m\"ussen auf das Integral 1 normiert werden, so dass Histogramme von reellen Messwerten m\"ussen auf das Integral 1
das Integral (nicht die Summe) \"uber das Histogramm eins ergibt. Das Integral normiert werden, so dass das Integral (nicht die Summe) \"uber das
ist die Fl\"ache des Histograms. Diese setzt sich zusammen aus der Fl\"ache der einzelnen Histogramm eins ergibt --- denn die Wahrscheinlichkeit, dass
Histogrammbalken. Diese haben die H\"ohe $n_i$ und die Breite $\Delta x$. Die Gesamtfl\"ache irgendeiner der Messwerte auftritt mu{\ss} Eins sein. Das Integral ist
$A$ des Histogramms ist also die Fl\"ache des Histogramms. Diese setzt sich zusammen aus der
Fl\"ache der einzelnen Histogrammbalken. Diese haben die H\"ohe $n_i$
und die Breite $\Delta x$. Die Gesamtfl\"ache $A$ des Histogramms ist
also
\[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i \] \[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i \]
und das normierte Histogramm hat die H\"ohe und das normierte Histogramm hat die H\"ohe
\[ p(x_i) = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} \] \[ p(x_i) = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} \]
@ -421,16 +424,17 @@ spricht von einer Wahrscheinlichkeitsdichte.
\end{figure} \end{figure}
\subsection{Korrelation} \section{\tr{Correlations}{Korrelationen}}
\begin{figure}[t] \begin{figure}[t]
\includegraphics[width=1\textwidth]{correlation} \includegraphics[width=1\textwidth]{correlation}
\caption{\label{correlationfig} Korrelationen zwischen zwei Datens\"atzen $x$ und $y$.} \caption{\label{correlationfig} Korrelationen zwischen zwei Datens\"atzen $x$ und $y$.}
\end{figure} \end{figure}
Bisher haben wir Eigenschaften einer einzelnen Me{\ss}gr\"o{\ss}e angeschaut. Bisher haben wir Eigenschaften einer einzelnen Me{\ss}gr\"o{\ss}e
Bei mehreren Me{\ss}gr\"o{\ss}en, kann nach Abh\"angigkeiten gefragt werden. angeschaut. Bei mehreren Me{\ss}gr\"o{\ss}en, kann nach
Der Korrelationskoeffizient Abh\"angigkeiten zwischen den beiden Gr\"o{\ss}en gefragt werden. Der
Korrelationskoeffizient
\[ r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle \[ r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle
(x-\langle x \rangle)(y-\langle y \rangle) \rangle}{\sqrt{\langle (x-\langle x \rangle)(y-\langle y \rangle) \rangle}{\sqrt{\langle
(x-\langle x \rangle)^2} \rangle \sqrt{\langle (y-\langle y (x-\langle x \rangle)^2} \rangle \sqrt{\langle (y-\langle y
@ -452,9 +456,9 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Data types} \section{Data types}
\subsubsection{Nominal scale} \subsection{Nominal scale}
\begin{itemize} \begin{itemize}
\item Binary \item Binary
\begin{itemize} \begin{itemize}
@ -475,7 +479,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
\item Statistics: mode, i.e. the most common item \item Statistics: mode, i.e. the most common item
\end{itemize} \end{itemize}
\subsubsection{Ordinal scale} \subsection{Ordinal scale}
\begin{itemize} \begin{itemize}
\item Like nominal scale, but with an order \item Like nominal scale, but with an order
\item Examples: ranks, ratings \item Examples: ranks, ratings
@ -489,7 +493,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
\item Statistics: mode, median \item Statistics: mode, median
\end{itemize} \end{itemize}
\subsubsection{Interval scale} \subsection{Interval scale}
\begin{itemize} \begin{itemize}
\item Quantitative/metric values \item Quantitative/metric values
\item Reasonable measure of distance between values, but no absolute zero \item Reasonable measure of distance between values, but no absolute zero
@ -505,7 +509,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\subsubsection{Absolute/ratio scale} \subsection{Absolute/ratio scale}
\begin{itemize} \begin{itemize}
\item Like interval scale, but with absolute origin/zero \item Like interval scale, but with absolute origin/zero
\item Examples: \item Examples:
@ -524,7 +528,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\subsubsection{Data types} \subsection{Data types}
\begin{itemize} \begin{itemize}
\item Data type selects \item Data type selects
\begin{itemize} \begin{itemize}
@ -539,7 +543,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
categories ``small/medium/large'' (ordinal scale) categories ``small/medium/large'' (ordinal scale)
\end{itemize} \end{itemize}
\subsubsection{Examples from neuroscience} \subsection{Examples from neuroscience}
\begin{itemize} \begin{itemize}
\item {\bf absolute:} \item {\bf absolute:}
\begin{itemize} \begin{itemize}
@ -615,7 +619,7 @@ aus der Stichprobe. Das hat mehrere Vorteile:
\section{Bootstrap des Standardfehlers} \section{Bootstrap des Standardfehlers}
Beim Bootstrap erzeugen wir durch resampling neue Stichproben und Beim Bootstrap erzeugen wir durch Resampling neue Stichproben und
benutzen diese um die Stichprobenverteilung einer Statistik zu benutzen diese um die Stichprobenverteilung einer Statistik zu
berechnen. Die Bootstrap Stichproben haben jeweils den gleichen Umfang berechnen. Die Bootstrap Stichproben haben jeweils den gleichen Umfang
wie die urspr\"unglich gemessene Stichprobe und werden durch Ziehen wie die urspr\"unglich gemessene Stichprobe und werden durch Ziehen
@ -639,7 +643,7 @@ Stichprobe vorkommen.
\end{document} \end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Statistics} \section{Statistics}
What is "a statistic"? % dt. Sch\"atzfunktion What is "a statistic"? % dt. Sch\"atzfunktion
\begin{definition}[statistic] \begin{definition}[statistic]
A statistic (singular) is a single measure of some attribute of a A statistic (singular) is a single measure of some attribute of a