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\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}

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\begin{document} 

\maketitle

%\tableofcontents

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\chapter{\tr{Descriptive statistics}{Deskriptive Statistik}}

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\section{Statistics of real-valued data}

  \begin{itemize}
  \item Location, central tendency
    \begin{itemize}
    \item arithmetic mean
    \item median
    \item mode
    \end{itemize}
  \item Spread, dispersion
    \begin{itemize}
    \item variance
    \item standard deviation
    \item interquartile range
    \item coefficient of variation
    \item minimum, maximum
    \end{itemize}
  \item Shape
    \begin{itemize}
    \item skewnees
    \item kurtosis
    \end{itemize}
  \item Dependence
    \begin{itemize}
    \item Pearson correlation coefficient
    \item Spearman's rank correlation coefficient
    \end{itemize}
  \end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Median, Quartile, Percentile}

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{median}
  \caption{\label{medianfig} Median.}
\end{figure}

\begin{definition}[\tr{median}{Median}]
  \tr{Half of the observations $X=(x_1, x_2, \ldots, x_n)$ are
    larger than the median and half of them are smaller than the
    median.}  {Der Median teilt eine Liste von Messwerten so in zwei
    H\"alften, dass die eine H\"alfte der Daten nicht gr\"o{\ss}er
    und die andere H\"alfte nicht kleiner als der Median ist.}
\end{definition}

\begin{exercise}[mymedian.m]
  \tr{Write a function that computes the median of a vector.}
  {Schreibe eine Funktion, die den Median eines Vektors zur\"uckgibt.}
\end{exercise}

\code{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
    returns a median above which are the same number of data than
    below. In particular the script should test data vectors of
    different length.}  {Schreibe ein Skript, das testet ob die
    \code{mymedian} Funktion wirklich die Zahl zur\"uckgibt, \"uber
    der genauso viele Datenwerte liegen wie darunter. Das Skript sollte
    insbesondere verschieden lange Datenvektoren testen.}
\end{exercise}

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{quartile}
  \caption{\label{quartilefig} Median und Quartile.}
\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{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}

\subsection{Histogram}

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.

\begin{exercise}[rollthedie.m]
  \tr{Write a function that simulates rolling a die $n$ times.}
  {Schreibe eine Funktion, die das $n$-malige W\"urfeln mit einem W\"urfel simuliert.}
\end{exercise}

\begin{exercise}[diehistograms.m]
  \tr{Plot histograms from rolling the die 20, 100, 1000 times.  Use
    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
    anschliessend das Histogram auf geeignete Weise.}
\end{exercise}

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{diehistograms}
  \caption{\label{diehistogramsfig} \tr{Histograms of rolling a die
      100 or 500 times.  Left: plain histograms counting the frequency
      of the six possible outcomes.  Right: the same data normalized
      to their sum.}{Histogramme des Ergebnisses von 100 oder 500 mal
      W\"urfeln. Links: das absolute Histogramm z\"ahlt die Anzahl des
      Auftretens jeder Augenzahl. Rechts: Normiert auf die Summe des
      Histogramms werden die beiden Messungen vergleichbar.}}
\end{figure}

Bei ganzzahligen Messdaten (z.B. die Augenzahl eines W\"urfels) 
kann f\"ur jede auftretende Zahl eine Klasse definiert werden.
Damit die H\"ohe der Histogrammbalken unabh\"angig von der Anzahl der Messwerte wird,
normiert man das Histogram auf die Anzahl der Messwerte.
Die H\"ohe der Histogrammbalken gibt dann die Wahrscheinlichkeit $P(x_i)$
des Auftretens der Gr\"o{\ss}e $x_i$ in der $i$-ten Klasse an
\[ P_i = \frac{n_i}{N} = \frac{n_i}{\sum_{i=1}^M n_i} \; . \]


\subsection{Probability density function}

Meistens haben wir es jedoch mit reellen Messgr\"o{\ss}en zu tun.

\begin{exercise}[gaussianbins.m]
  \tr{Draw 100 random data from a Gaussian distribution and plot
    histograms with different bin sizes of the data.}  {Ziehe 100
    normalverteilte Zufallszahlen und erzeuge Histogramme mit
    unterschiedlichen Klassenbreiten. Was f\"allt auf?}
\end{exercise}

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{pdfhistogram}
  \caption{\label{pdfhistogramfig} \tr{Histograms of normally
      distributed data with different bin sizes.}{Histogramme mit
      verschiednenen Klassenbreiten eines Datensatzes von
      normalverteilten Messwerten. Links: Die H\"ohe des absoluten
      Histogramms h\"angt von der Klassenbreite ab. Rechts: Bei auf
      das Integral normierten Histogrammen werden auch
      unterschiedliche Klassenbreiten vergleichbar.}}
\end{figure}

Histogramme von reellen Messwerten m\"ussen auf das Integral 1 normiert werden, so dass
das Integral (nicht die Summe) \"uber das Histogramm eins ergibt. Das Integral
ist die Fl\"ache des Histograms. 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
\[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i \]
und das normierte Histogramm hat die H\"ohe
\[ p(x_i) = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} \]
Es muss also nicht nur durch die Summe, sondern auch durch die Breite $\Delta x$ der Klassen
geteilt werden.

$p(x_i)$ kann keine Wahrscheinlichkeit sein, da $p(x_i)$ nun eine
Einheit hat --- das Inverse der Einheit der Messgr\"osse $x$. Man
spricht von einer Wahrscheinlichkeitsdichte.

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{pdfprobabilities}
  \caption{\label{pdfprobabilitiesfig} Wahrscheinlichkeiten bei
  einer Wahrscheinlichkeitsdichtefunktion.}
\end{figure}

\begin{exercise}[gaussianpdf.m]
  \tr{Plot the Gaussian probability density}{Plotte die Gauss'sche Wahrscheinlichkeitsdichte }
  \[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]
  \tr{What does it mean?}{Was bedeutet die folgende Wahrscheinlichkeit?}
  \[ P(x_1 < x < x2) = \int\limits_{x_1}^{x_2} p(x) \, dx \]
  \tr{How large is}{Wie gro{\ss} ist}
  \[ \int\limits_{-\infty}^{+\infty} p(x) \, dx \; ?\]
  \tr{Why?}{Warum?}
\end{exercise}


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\subsection{Data types}

\subsubsection{Nominal scale}
\begin{itemize}
\item Binary
  \begin{itemize}
  \item ``yes/no'',
  \item ``true/false'',
  \item ``success/failure'', etc.
  \end{itemize}
\item Categorial
  \begin{itemize}
  \item cell type (``rod/cone/horizontal cell/bipolar cell/ganglion cell''),
  \item blood type (``A/B/AB/0''),
  \item parts of speech (``noun/veerb/preposition/article/...''),
  \item taxonomic groups (``Coleoptera/Lepidoptera/Diptera/Hymenoptera''), etc.
  \end{itemize}
\item Each observation/measurement/sample is put into one category
\item There is no reasonable order among the categories.\\
  example: [rods, cones] vs. [cones, rods]
\item Statistics: mode, i.e. the most common item
\end{itemize}

\subsubsection{Ordinal scale}
\begin{itemize}
\item Like nominal scale, but with an order
\item Examples: ranks, ratings
  \begin{itemize}
  \item ``bad/ok/good'',
  \item ``cold/warm/hot'',
  \item ``young/old'', etc.
  \end{itemize}
\item {\bf But:} there is no reasonable measure of {\em distance}
  between the classes
\item Statistics: mode, median
\end{itemize}

\subsubsection{Interval scale}
\begin{itemize}
\item Quantitative/metric values
\item Reasonable measure of distance between values, but no absolute zero
\item Examples: 
  \begin{itemize}
  \item Temperature in $^\circ$C ($20^\circ$C is not twice as hot as $10^\circ$C)
  \item Direction measured in degrees from magnetic or true north
  \end{itemize}
\item Statistics:
  \begin{itemize}
  \item Central tendency: mode, median, arithmetic mean
  \item Dispersion: range, standard deviation
  \end{itemize}
\end{itemize}

\subsubsection{Absolute/ratio scale}
\begin{itemize}
\item Like interval scale, but with absolute origin/zero
\item Examples: 
  \begin{itemize}
  \item Temperature in $^\circ$K
  \item Length, mass, duration, electric charge, ...
  \item Plane angle, etc.
  \item Count (e.g. number of spikes in response to a stimulus)
  \end{itemize}
\item Statistics:
  \begin{itemize}
  \item Central tendency: mode, median, arithmetic, geometric, harmonic mean
  \item Dispersion: range, standard deviation
  \item Coefficient of variation (ratio standard deviation/mean)
  \item All other statistical measures
  \end{itemize}
\end{itemize}

\subsubsection{Data types}
\begin{itemize}
\item Data type selects
  \begin{itemize}
  \item statistics 
  \item type of plots (bar graph versus x-y plot)
  \item correct tests
  \end{itemize}
\item Scales exhibit increasing information content from nominal
  to absolute.\\
  Conversion  ,,downwards'' is always possible
\item For example: size measured in meter (ratio scale) $\rightarrow$
  categories ``small/medium/large'' (ordinal scale)
\end{itemize}

\subsubsection{Examples from neuroscience}
\begin{itemize}
\item {\bf absolute:}
  \begin{itemize}
  \item size of neuron/brain
  \item length of axon
  \item ion concentration
  \item membrane potential
  \item firing rate
  \end{itemize}

\item {\bf interval:}
  \begin{itemize}
  \item edge orientation
  \end{itemize}

\item {\bf ordinal:}
  \begin{itemize}
  \item stages of a disease
  \item ratings
  \end{itemize}

\item {\bf nominal:}
  \begin{itemize}
  \item cell type
  \item odor
  \item states of an ion channel
  \end{itemize}

\end{itemize}


\end{document}

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\subsection{Statistics}
What is "a statistic"? % dt. Sch\"atzfunktion
\begin{definition}[statistic]
  A statistic (singular) is a single measure of some attribute of a
  sample (e.g., its arithmetic mean value). It is calculated by
  applying a function (statistical algorithm) to the values of the
  items of the sample, which are known together as a set of data.
  
  \source{http://en.wikipedia.org/wiki/Statistic}
\end{definition}