406 lines
14 KiB
TeX
406 lines
14 KiB
TeX
\documentclass[12pt]{report}
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%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
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\author{Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
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\date{WS 15/16}
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%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% \newcommand{\tr}[2]{#1} % en
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% \usepackage[english]{babel}
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\newcommand{\tr}[2]{#2} % de
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\usepackage[german]{babel}
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%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{pslatex} % nice font for pdf file
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\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
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%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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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\usepackage{graphicx}
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\usepackage{xcolor}
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\newcommand{\texpicture}[1]{{\sffamily\small\input{#1.tex}}}
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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\lstset{
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inputpath=../code,
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basicstyle=\ttfamily\footnotesize,
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numbers=left,
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showstringspaces=false,
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language=Matlab,
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commentstyle=\itshape\color{darkgray},
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keywordstyle=\color{blue},
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stringstyle=\color{green},
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backgroundcolor=\color{blue!10},
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breaklines=true,
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breakautoindent=true,
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columns=flexible,
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frame=single,
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caption={\protect\filename@parse{\lstname}\protect\filename@base},
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captionpos=t,
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xleftmargin=1em,
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xrightmargin=1em,
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aboveskip=10pt
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}
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%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{amsmath}
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\usepackage{bm}
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\usepackage{dsfont}
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\newcommand{\naZ}{\mathds{N}}
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\newcommand{\gaZ}{\mathds{Z}}
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\newcommand{\raZ}{\mathds{Q}}
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\newcommand{\reZ}{\mathds{R}}
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\newcommand{\reZp}{\mathds{R^+}}
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\newcommand{\reZpN}{\mathds{R^+_0}}
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\newcommand{\koZ}{\mathds{C}}
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%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{ifthen}
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\newcommand{\code}[1]{\texttt{#1}}
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\newcommand{\source}[1]{
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\begin{flushright}
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\color{gray}\scriptsize \url{#1}
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\end{flushright}
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}
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\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
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{\medskip}
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\newcommand{\showlisting}{yes}
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%\newcommand{\showlisting}{no}
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\newcounter{theexercise}
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\setcounter{theexercise}{1}
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\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
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\arabic{theexercise}:} \stepcounter{theexercise}\newline \newcommand{\exercisesource}{#1}}%
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{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\equal{\showlisting}{yes}}{\medskip\lstinputlisting{\exercisesource}}{}}\medskip}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\maketitle
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%\tableofcontents
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\chapter{\tr{Descriptive statistics}{Deskriptive Statistik}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Statistics of real-valued data}
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\begin{itemize}
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\item Location, central tendency
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\begin{itemize}
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\item arithmetic mean
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\item median
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\item mode
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\end{itemize}
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\item Spread, dispersion
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\begin{itemize}
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\item variance
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\item standard deviation
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\item interquartile range
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\item coefficient of variation
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\item minimum, maximum
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\end{itemize}
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\item Shape
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\begin{itemize}
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\item skewnees
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\item kurtosis
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\end{itemize}
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\item Dependence
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\begin{itemize}
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\item Pearson correlation coefficient
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\item Spearman's rank correlation coefficient
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\end{itemize}
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Median, Quartile, Percentile}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{median}
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\caption{\label{medianfig} Median.}
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\end{figure}
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\begin{definition}[\tr{median}{Median}]
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\tr{Half of the observations $X=(x_1, x_2, \ldots, x_n)$ are
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larger than the median and half of them are smaller than the
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median.} {Der Median teilt eine Liste von Messwerten so in zwei
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H\"alften, dass die eine H\"alfte der Daten nicht gr\"o{\ss}er
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und die andere H\"alfte nicht kleiner als der Median ist.}
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\end{definition}
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\begin{exercise}[mymedian.m]
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\tr{Write a function that computes the median of a vector.}
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{Schreibe eine Funktion, die den Median eines Vektors zur\"uckgibt.}
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\end{exercise}
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\code{matlab} stellt die Funktion \code{median()} zur Berechnung des Medians bereit.
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\begin{exercise}[checkmymedian.m]
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\tr{Write a script that tests whether your median function really
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returns a median above which are the same number of data than
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below. In particular the script should test data vectors of
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different length.} {Schreibe ein Skript, das testet ob die
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\code{mymedian} Funktion wirklich die Zahl zur\"uckgibt, \"uber
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der genauso viele Datenwerte liegen wie darunter. Das Skript sollte
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insbesondere verschieden lange Datenvektoren testen.}
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\end{exercise}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{quartile}
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\caption{\label{quartilefig} Median und Quartile.}
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\end{figure}
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\begin{definition}[\tr{quartile}{Quartile}]
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Die Quartile Q1, Q2 und Q3 unterteilen die Daten in vier gleich
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gro{\ss}e Gruppen, die jeweils ein Viertel der Daten enthalten.
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Das mittlere Quartil entspricht dem Median.
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\end{definition}
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\begin{exercise}[quartiles.m]
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\tr{Write a function that computes the first, second, and third quartile of a vector.}
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{Schreibe eine Funktion, die das erste, zweite und dritte Quartil als Vektor zur\"uckgibt.}
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\end{exercise}
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\subsection{Histogram}
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Histogramme z\"ahlen die H\"aufigkeit $n_i$ des Auftretens von
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$N=\sum_{i=1}^M n_i$ Messwerten in $M$ Messbereichsklassen $i$ (Bins).
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Die Klassen unterteilen den Wertebereich meist in angrenzende und
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gleich gro{\ss}e Intervalle. Histogramme sch\"atzen die
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Wahrscheinlichkeitsverteilung der Messwerte ab.
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\begin{exercise}[rollthedie.m]
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\tr{Write a function that simulates rolling a die $n$ times.}
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{Schreibe eine Funktion, die das $n$-malige W\"urfeln mit einem W\"urfel simuliert.}
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\end{exercise}
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\begin{exercise}[diehistograms.m]
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\tr{Plot histograms from rolling the die 20, 100, 1000 times. Use
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the plain hist(x) function, force 6 bins via hist( x, 6 ), and set
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meaningfull bins positions.} {Plotte Histogramme von 20, 100, und
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1000-mal w\"urfeln. Benutze \code{hist(x)}, erzwinge sechs Bins
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mit \code{hist(x,6)}, und setze selbst sinnvolle Bins. Normiere
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anschliessend das Histogram auf geeignete Weise.}
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\end{exercise}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{diehistograms}
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\caption{\label{diehistogramsfig} \tr{Histograms of rolling a die
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100 or 500 times. Left: plain histograms counting the frequency
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of the six possible outcomes. Right: the same data normalized
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to their sum.}{Histogramme des Ergebnisses von 100 oder 500 mal
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W\"urfeln. Links: das absolute Histogramm z\"ahlt die Anzahl des
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Auftretens jeder Augenzahl. Rechts: Normiert auf die Summe des
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Histogramms werden die beiden Messungen vergleichbar.}}
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\end{figure}
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Bei ganzzahligen Messdaten (z.B. die Augenzahl eines W\"urfels)
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kann f\"ur jede auftretende Zahl eine Klasse definiert werden.
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Damit die H\"ohe der Histogrammbalken unabh\"angig von der Anzahl der Messwerte wird,
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normiert man das Histogram auf die Anzahl der Messwerte.
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Die H\"ohe der Histogrammbalken gibt dann die Wahrscheinlichkeit $P(x_i)$
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des Auftretens der Gr\"o{\ss}e $x_i$ in der $i$-ten Klasse an
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\[ P_i = \frac{n_i}{N} = \frac{n_i}{\sum_{i=1}^M n_i} \; . \]
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\subsection{Probability density function}
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Meistens haben wir es jedoch mit reellen Messgr\"o{\ss}en zu tun.
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\begin{exercise}[gaussianbins.m]
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\tr{Draw 100 random data from a Gaussian distribution and plot
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histograms with different bin sizes of the data.} {Ziehe 100
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normalverteilte Zufallszahlen und erzeuge Histogramme mit
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unterschiedlichen Klassenbreiten. Was f\"allt auf?}
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\end{exercise}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{pdfhistogram}
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\caption{\label{pdfhistogramfig} \tr{Histograms of normally
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distributed data with different bin sizes.}{Histogramme mit
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verschiednenen Klassenbreiten eines Datensatzes von
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normalverteilten Messwerten. Links: Die H\"ohe des absoluten
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Histogramms h\"angt von der Klassenbreite ab. Rechts: Bei auf
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das Integral normierten Histogrammen werden auch
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unterschiedliche Klassenbreiten vergleichbar.}}
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\end{figure}
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Histogramme von reellen Messwerten m\"ussen auf das Integral 1 normiert werden, so dass
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das Integral (nicht die Summe) \"uber das Histogramm eins ergibt. Das Integral
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ist die Fl\"ache des Histograms. Diese setzt sich zusammen aus der Fl\"ache der einzelnen
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Histogrammbalken. Diese haben die H\"ohe $n_i$ und die Breite $\Delta x$. Die Gesamtfl\"ache
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$A$ des Histogramms ist also
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\[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i \]
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und das normierte Histogramm hat die H\"ohe
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\[ p(x_i) = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} \]
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Es muss also nicht nur durch die Summe, sondern auch durch die Breite $\Delta x$ der Klassen
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geteilt werden.
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$p(x_i)$ kann keine Wahrscheinlichkeit sein, da $p(x_i)$ nun eine
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Einheit hat --- das Inverse der Einheit der Messgr\"osse $x$. Man
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spricht von einer Wahrscheinlichkeitsdichte.
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{pdfprobabilities}
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\caption{\label{pdfprobabilitiesfig} Wahrscheinlichkeiten bei
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einer Wahrscheinlichkeitsdichtefunktion.}
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\end{figure}
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\begin{exercise}[gaussianpdf.m]
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\tr{Plot the Gaussian probability density}{Plotte die Gauss'sche Wahrscheinlichkeitsdichte }
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\[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]
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\tr{What does it mean?}{Was bedeutet die folgende Wahrscheinlichkeit?}
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\[ P(x_1 < x < x2) = \int\limits_{x_1}^{x_2} p(x) \, dx \]
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\tr{How large is}{Wie gro{\ss} ist}
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\[ \int\limits_{-\infty}^{+\infty} p(x) \, dx \; ?\]
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\tr{Why?}{Warum?}
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\end{exercise}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Data types}
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\subsubsection{Nominal scale}
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\begin{itemize}
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\item Binary
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\begin{itemize}
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\item ``yes/no'',
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\item ``true/false'',
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\item ``success/failure'', etc.
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\end{itemize}
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\item Categorial
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\begin{itemize}
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\item cell type (``rod/cone/horizontal cell/bipolar cell/ganglion cell''),
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\item blood type (``A/B/AB/0''),
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\item parts of speech (``noun/veerb/preposition/article/...''),
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\item taxonomic groups (``Coleoptera/Lepidoptera/Diptera/Hymenoptera''), etc.
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\end{itemize}
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\item Each observation/measurement/sample is put into one category
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\item There is no reasonable order among the categories.\\
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example: [rods, cones] vs. [cones, rods]
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\item Statistics: mode, i.e. the most common item
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\end{itemize}
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\subsubsection{Ordinal scale}
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\begin{itemize}
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\item Like nominal scale, but with an order
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\item Examples: ranks, ratings
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\begin{itemize}
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\item ``bad/ok/good'',
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\item ``cold/warm/hot'',
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\item ``young/old'', etc.
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\end{itemize}
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\item {\bf But:} there is no reasonable measure of {\em distance}
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between the classes
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\item Statistics: mode, median
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\end{itemize}
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\subsubsection{Interval scale}
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\begin{itemize}
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\item Quantitative/metric values
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\item Reasonable measure of distance between values, but no absolute zero
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\item Examples:
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\begin{itemize}
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\item Temperature in $^\circ$C ($20^\circ$C is not twice as hot as $10^\circ$C)
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\item Direction measured in degrees from magnetic or true north
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\end{itemize}
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\item Statistics:
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\begin{itemize}
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\item Central tendency: mode, median, arithmetic mean
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\item Dispersion: range, standard deviation
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\end{itemize}
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\end{itemize}
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\subsubsection{Absolute/ratio scale}
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\begin{itemize}
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\item Like interval scale, but with absolute origin/zero
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\item Examples:
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\begin{itemize}
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\item Temperature in $^\circ$K
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\item Length, mass, duration, electric charge, ...
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\item Plane angle, etc.
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\item Count (e.g. number of spikes in response to a stimulus)
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\end{itemize}
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\item Statistics:
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\begin{itemize}
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\item Central tendency: mode, median, arithmetic, geometric, harmonic mean
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\item Dispersion: range, standard deviation
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\item Coefficient of variation (ratio standard deviation/mean)
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\item All other statistical measures
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\end{itemize}
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\end{itemize}
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\subsubsection{Data types}
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\begin{itemize}
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\item Data type selects
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\begin{itemize}
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\item statistics
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\item type of plots (bar graph versus x-y plot)
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\item correct tests
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\end{itemize}
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\item Scales exhibit increasing information content from nominal
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to absolute.\\
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Conversion ,,downwards'' is always possible
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\item For example: size measured in meter (ratio scale) $\rightarrow$
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categories ``small/medium/large'' (ordinal scale)
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\end{itemize}
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\subsubsection{Examples from neuroscience}
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\begin{itemize}
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\item {\bf absolute:}
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\begin{itemize}
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\item size of neuron/brain
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\item length of axon
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\item ion concentration
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\item membrane potential
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\item firing rate
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\end{itemize}
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\item {\bf interval:}
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\begin{itemize}
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\item edge orientation
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\end{itemize}
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\item {\bf ordinal:}
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\begin{itemize}
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\item stages of a disease
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\item ratings
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\end{itemize}
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\item {\bf nominal:}
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\begin{itemize}
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\item cell type
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\item odor
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\item states of an ion channel
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\end{itemize}
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\end{itemize}
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\end{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Statistics}
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What is "a statistic"? % dt. Sch\"atzfunktion
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\begin{definition}[statistic]
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A statistic (singular) is a single measure of some attribute of a
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sample (e.g., its arithmetic mean value). It is calculated by
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applying a function (statistical algorithm) to the values of the
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items of the sample, which are known together as a set of data.
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\source{http://en.wikipedia.org/wiki/Statistic}
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\end{definition}
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