New design pattern chapter.
Next exercises for point processes.
This commit is contained in:
parent
0be73cce5c
commit
4bb0c77ac4
29
designpattern/lecture/Makefile
Normal file
29
designpattern/lecture/Makefile
Normal file
@ -0,0 +1,29 @@
|
||||
BASENAME=designpattern
|
||||
|
||||
PYFILES=$(wildcard *.py)
|
||||
PYPDFFILES=$(PYFILES:.py=.pdf)
|
||||
|
||||
all : pdf
|
||||
|
||||
# script:
|
||||
pdf : $(BASENAME)-chapter.pdf
|
||||
|
||||
$(BASENAME)-chapter.pdf : $(BASENAME)-chapter.tex $(BASENAME).tex $(PYPDFFILES)
|
||||
pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
|
||||
|
||||
$(PYPDFFILES) : %.pdf : %.py
|
||||
python $<
|
||||
|
||||
clean :
|
||||
rm -f *~
|
||||
rm -f $(BASENAME).aux $(BASENAME).log
|
||||
rm -f $(BASENAME)-chapter.aux $(BASENAME)-chapter.log $(BASENAME)-chapter.out
|
||||
rm -f $(PYPDFFILES) $(GPTTEXFILES)
|
||||
|
||||
cleanall : clean
|
||||
rm -f $(BASENAME)-chapter.pdf
|
||||
|
||||
watchpdf :
|
||||
while true; do ! make -q pdf && make pdf; sleep 0.5; done
|
||||
|
||||
|
||||
225
designpattern/lecture/designpattern-chapter.tex
Normal file
225
designpattern/lecture/designpattern-chapter.tex
Normal file
@ -0,0 +1,225 @@
|
||||
\documentclass[12pt]{report}
|
||||
|
||||
%%%%% title %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\title{\tr{Introduction to Scientific Computing}{Einf\"uhrung in die wissenschaftliche Datenverarbeitung}}
|
||||
\author{Jan Grewe \& Jan Benda\\Abteilung Neuroethologie\\[2ex]\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}}
|
||||
\date{WS 15/16}
|
||||
|
||||
%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% \newcommand{\tr}[2]{#1} % en
|
||||
% \usepackage[english]{babel}
|
||||
\newcommand{\tr}[2]{#2} % de
|
||||
\usepackage[german]{babel}
|
||||
|
||||
%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{pslatex} % nice font for pdf file
|
||||
\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
|
||||
|
||||
%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage[left=25mm,right=25mm,top=20mm,bottom=30mm]{geometry}
|
||||
\setcounter{tocdepth}{1}
|
||||
|
||||
%%%%% 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}
|
||||
\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}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\Eqn}{\tr{Eq}{Gl}.}
|
||||
\newcommand{\eqns}{\tr{Eqs}{Gln}.}
|
||||
\newcommand{\Eqns}{\tr{Eqs}{Gln}.}
|
||||
\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}
|
||||
\lstset{
|
||||
inputpath=../code,
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
numbers=left,
|
||||
showstringspaces=false,
|
||||
language=Matlab,
|
||||
commentstyle=\itshape\color{darkgray},
|
||||
keywordstyle=\color{blue},
|
||||
stringstyle=\color{green},
|
||||
backgroundcolor=\color{blue!10},
|
||||
breaklines=true,
|
||||
breakautoindent=true,
|
||||
columns=flexible,
|
||||
frame=single,
|
||||
caption={\protect\filename@parse{\lstname}\protect\filename@base},
|
||||
captionpos=t,
|
||||
xleftmargin=1em,
|
||||
xrightmargin=1em,
|
||||
aboveskip=10pt
|
||||
}
|
||||
|
||||
%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{amsmath}
|
||||
\usepackage{bm}
|
||||
\usepackage{dsfont}
|
||||
\newcommand{\naZ}{\mathds{N}}
|
||||
\newcommand{\gaZ}{\mathds{Z}}
|
||||
\newcommand{\raZ}{\mathds{Q}}
|
||||
\newcommand{\reZ}{\mathds{R}}
|
||||
\newcommand{\reZp}{\mathds{R^+}}
|
||||
\newcommand{\reZpN}{\mathds{R^+_0}}
|
||||
\newcommand{\koZ}{\mathds{C}}
|
||||
|
||||
|
||||
%%%%% structure: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\usepackage{ifthen}
|
||||
|
||||
\newcommand{\code}[1]{\texttt{#1}}
|
||||
|
||||
\newcommand{\source}[1]{
|
||||
\begin{flushright}
|
||||
\color{gray}\scriptsize \url{#1}
|
||||
\end{flushright}
|
||||
}
|
||||
|
||||
\newenvironment{definition}[1][]{\medskip\noindent\textbf{Definition}\ifthenelse{\equal{#1}{}}{}{ #1}:\newline}%
|
||||
{\medskip}
|
||||
|
||||
\newcounter{maxexercise}
|
||||
\setcounter{maxexercise}{9} % show listings up to exercise maxexercise
|
||||
\newcounter{theexercise}
|
||||
\setcounter{theexercise}{1}
|
||||
\newenvironment{exercise}[1][]{\medskip\noindent\textbf{\tr{Exercise}{\"Ubung}
|
||||
\arabic{theexercise}:}\newline \newcommand{\exercisesource}{#1}}%
|
||||
{\ifthenelse{\equal{\exercisesource}{}}{}{\ifthenelse{\value{theexercise}>\value{maxexercise}}{}{\medskip\lstinputlisting{\exercisesource}}}\medskip\stepcounter{theexercise}}
|
||||
|
||||
\graphicspath{{figures/}}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{document}
|
||||
|
||||
\include{designpattern}
|
||||
|
||||
\end{document}
|
||||
166
designpattern/lecture/designpattern.tex
Normal file
166
designpattern/lecture/designpattern.tex
Normal file
@ -0,0 +1,166 @@
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\chapter{Design Pattern}
|
||||
|
||||
Beim Programmieren sind sich viel Codes in ihrer Grundstruktur sehr
|
||||
\"ahnlich. Viele Konstrukte kommen in den verschiedensten Kontexten
|
||||
immer wieder in \"ahnlicher Weise vor. In diesem Kapitel stellen wir
|
||||
einige dieser ``Design pattern'' zusammen.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Plotten einer mathematischen Funktion}
|
||||
Eine mathematische Funktion ordnet einem beliebigen $x$-Wert einen
|
||||
$y$-Wert zu. Um eine solche Funktion zeichnen zu k\"onnen, m\"ussen
|
||||
wir uns eine Wertetabelle aus vielen $x$-Werten und den
|
||||
dazugeh\"origen Funktionswerten $y=f(x)$ erstellen.
|
||||
|
||||
Wir erstellen uns dazu einen Vektor mit geeigneten $x$-Werten, die von
|
||||
dem kleinsten bis zu dem gr\"o{\ss}ten $x$-Wert laufen, den wir
|
||||
plotten wollen. Die Schrittweite f\"ur die $x$-Werte w\"ahlen wir
|
||||
klein genug, um eine sch\"one glatte Kurve zu bekommen. F\"ur jeden
|
||||
Wert $x_i$ dieses Vektors berechnen wir den entsprechenden
|
||||
Funktionswert und erzeugen damit einen Vektor mit den $y$-Werten. Die
|
||||
Werte des $y$-Vektors k\"onnen dann gegen die Werte des $x$-Vektors
|
||||
geplottet werden.
|
||||
|
||||
Folgende Programme berechnen und plotten die Funktion $f(x)=e^{-x^2}$:
|
||||
\begin{lstlisting}
|
||||
xmin = -1.0;
|
||||
xmax = 2.0;
|
||||
dx = 0.01; % Schrittweite
|
||||
x = xmin:dx:xmax; % Vektor mit x-Werten
|
||||
y = exp(-x.*x); % keine for Schleife! '.*' fuer elementweises multiplizieren
|
||||
plot(x, y);
|
||||
\end{lstlisting}
|
||||
|
||||
\begin{lstlisting}
|
||||
x = -1:0.01:2; % Vektor mit x-Werten
|
||||
y = exp(-x.*x); % keine for Schleife! '.*' fuer elementweises multiplizieren
|
||||
plot(x, y);
|
||||
\end{lstlisting}
|
||||
|
||||
\begin{lstlisting}
|
||||
x = -1:0.01:2; % Vektor mit x-Werten
|
||||
plot(x, exp(-x.*x));
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Skalieren und Verschieben nicht nur von Zufallszahlen}
|
||||
Zufallsgeneratoren geben oft nur Zufallszahlen mit festen Mittelwerten
|
||||
und Standardabweichungen (auch Skalierungen) zur\"uck. Multiplikation
|
||||
mit einem Faktor skaliert die Standardabweichung und Addition einer Zahl
|
||||
verschiebt den Mittelwert.
|
||||
|
||||
\begin{lstlisting}
|
||||
% 100 random numbers draw from a Gaussian distribution with mean 0 and standard deviation 1.
|
||||
x = randn(100, 1);
|
||||
|
||||
% 100 random numbers drawn from a Gaussian distribution with mean 4.8 and standard deviation 2.3.
|
||||
mu = 4.8;
|
||||
sigma = 2.3;
|
||||
y = randn(100, 1)*sigma + mu;
|
||||
\end{lstlisting}
|
||||
|
||||
Das ist manchmal auch sinnvoll f\"ur \code{zeros} oder \code{ones}:
|
||||
\begin{lstlisting}
|
||||
x = -1:0.01:2; % Vektor mit x-Werten
|
||||
plot(x, exp(-x.*x));
|
||||
% Plotte f\"ur die gleichen x-Werte eine Linie mit y=0.8:
|
||||
plot(x, zeros(size(x))+0.8);
|
||||
% ... Linie mit y=0.5:
|
||||
plot(x, ones(size(x))*0.5);
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{for Schleifen \"uber Vektoren}
|
||||
Manchmal m\"ochte man doch mit einer for-Schleife \"uber einen Vektor iterieren:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
for i=1:length(x)
|
||||
% Benutze den Wert des Vektors x an der Stelle des Indexes i:
|
||||
do_something( x(i) );
|
||||
end
|
||||
\end{lstlisting}
|
||||
|
||||
Wenn in der Schleife das Ergebnis in einen Vektor gespeichert werden soll,
|
||||
sollten wir uns vor der Schleife schon einen Vektor f\"ur die Ergebnisse
|
||||
erstellen:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
y = zeros(size(x)); % Platz fuer die Ergebnisse
|
||||
for i=1:length(x)
|
||||
% Schreibe den Rueckgabewert der Funktion get_something an die i-te
|
||||
% Stelle von y:
|
||||
y(i) = get_something( x(i) );
|
||||
end
|
||||
% jetzt koennen wir den Ergebnisvektor weiter bearbeiten:
|
||||
mean(y)
|
||||
\end{lstlisting}
|
||||
|
||||
Die Berechnungen in der Schleife k\"onnen statt einer Zahl auch einen Vektor
|
||||
zur\"uckgeben. Wenn die L\"ange diese Vektors bekannt ist, dann kann vorher
|
||||
eine entsprechend gro{\ss}e Matrix angelegt werden:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
y = zeros(length(x),10); % Platz fuer die Ergebnisse
|
||||
for i=1:length(x)
|
||||
% Schreibe den Rueckgabewert der Funktion get_something - jetzt ein
|
||||
% Vektor mit 10 Elementen - in die i-te
|
||||
% Zeile von y:
|
||||
y(i,:) = get_something( x(i) );
|
||||
end
|
||||
% jetzt koennen wir die Ergebnismatrix weiter bearbeiten:
|
||||
mean(y, 1)
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
Alternativ k\"onnen die in der Schleife erzeugten Vektoren zu einem
|
||||
einzigen, durchgehenden Vektor zusammengestellt werden:
|
||||
\begin{lstlisting}
|
||||
x = [2:3:20]; % irgendein Vektor
|
||||
y = []; % Leerer Vektor fuer die Ergebnisse
|
||||
for i=1:length(x)
|
||||
% Die Funktion get_something gibt uns einen Vektor zurueck:
|
||||
z = get_something( x(i) );
|
||||
% dessen Inhalt h\"angen wir an unseren Ergebnissvektor an:
|
||||
y = [y z(:)];
|
||||
end
|
||||
% jetzt koennen wir dem Ergebnisvektor weiter bearbeiten:
|
||||
mean(y)
|
||||
\end{lstlisting}
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Normierung von Histogrammen}
|
||||
Meistens sollten Histogramme normiert werden, damit sie vergleichbar
|
||||
mit anderen Histogrammen oder mit theoretischen
|
||||
Wahrscheinlichkeitsverteilungen werden.
|
||||
|
||||
Die \code{histogram} Funktion macht das mit den entsprechenden Parametern automatisch:
|
||||
\begin{lstlisting}
|
||||
x = randn(100, 1); % irgendwelche reellwertige Daten
|
||||
histogram(x, 'Normalization', 'pdf');
|
||||
\end{lstlisting}
|
||||
|
||||
\begin{lstlisting}
|
||||
x = randi(6, 100, 1); % irgendwelche integer Daten
|
||||
histogram(x, 'Normalization', 'probability');
|
||||
\end{lstlisting}
|
||||
|
||||
So geht es aber auch:
|
||||
\begin{lstlisting}
|
||||
x = randn(100, 1); % irgendwelche reellwertige Daten
|
||||
[h, b] = hist(x); % Histogram berechnen
|
||||
h = h/sum(h)/(b(2)-b(1)); % normieren zu einer Wahrscheinlichkeitsdichte
|
||||
bar(b, h); % und plotten.
|
||||
\end{lstlisting}
|
||||
|
||||
\begin{lstlisting}
|
||||
x = randi(6, 100, 1); % irgendwelche integer Daten
|
||||
[h, b] = hist(x); % Histogram berechnen
|
||||
h = h/sum(h); % normieren zu Wahrscheinlichkeiten
|
||||
bar(b, h); % und plotten.
|
||||
\end{lstlisting}
|
||||
@ -14,6 +14,6 @@ function spikes = hompoissonspikes( trials, rate, tmax )
|
||||
x = rand( trials, ceil(tmax/dt) );
|
||||
spikes = cell( trials, 1 );
|
||||
for k=1:trials
|
||||
spikes{k} = find( x(k,:) >= 1.0-p ) * dt;
|
||||
spikes{k} = find( x(k,:) < p ) * dt;
|
||||
end
|
||||
end
|
||||
|
||||
@ -93,7 +93,7 @@ jan.benda@uni-tuebingen.de}
|
||||
Wir wollen den homogenen Poisson Prozess benutzen um Spikes zu generieren,
|
||||
mit denen wir die Analysfunktionen des vorherigen \"Ubungszettel \"uberpr\"ufen k\"onnen.
|
||||
|
||||
Ein homogener Poisson Prozess mit der Rate $\lambda$ (measured in Hertz) ist ein Punktprozess,
|
||||
Ein homogener Poisson Prozess mit der Rate $\lambda$ (gemessen in Hertz) ist ein Punktprozess,
|
||||
bei dem die Wahrschienlichkeit eines Ereignisses unabh\"angig von der Zeit $t$ und
|
||||
unabh\"angig von vorherigen Ereignissen ist.
|
||||
Die Wahrscheinlichkeit $P$ eines Ereignisses innerhalb eines Bins der Breite $\Delta t$ ist
|
||||
@ -102,7 +102,7 @@ f\"ur gen\"ugend kleine $\Delta t$.
|
||||
\begin{parts}
|
||||
|
||||
\part Schreibe eine Funktion die $n$ homogene Poisson Spiketrains
|
||||
einer gegebenen Dauer $T_{max}$ mit rate $\lambda$ erzeugt.
|
||||
einer gegebenen Dauer $T_{max}$ mit Rate $\lambda$ erzeugt.
|
||||
\begin{solution}
|
||||
\lstinputlisting{hompoissonspikes.m}
|
||||
\end{solution}
|
||||
@ -122,8 +122,8 @@ f\"ur gen\"ugend kleine $\Delta t$.
|
||||
Poisson Spiketrains mit der Rate $\lambda=100$\,Hz. Ver\"andere
|
||||
\"uber die Dauer $T_{max}$ der Spiketrains und die Anzahl $n$ der
|
||||
Trials die Anzahl der Intervalle und ver\"andere auch die Binbreite
|
||||
des Histograms (fange mit 1\,ms an). Wieviele Interspikeintervalle
|
||||
werden ben\"otigt um ein ``sch\"ones'' Histogramm zu erhalten? Wie
|
||||
des Histograms (fang mit 1\,ms an). Wieviele Interspikeintervalle
|
||||
werden ben\"otigt, um ein ``sch\"ones'' Histogramm zu erhalten? Wie
|
||||
lange m\"usste man also von dem Neuron ableiten?
|
||||
\begin{solution}
|
||||
About 5000 intervals for 25 bins. This corresponds to a $5000 /
|
||||
@ -131,56 +131,62 @@ f\"ur gen\"ugend kleine $\Delta t$.
|
||||
100\,\hertz.
|
||||
\end{solution}
|
||||
|
||||
\part Vergleiche das Histogramm mit der zu erwartenden Verteilung
|
||||
\part Vergleiche Interspike-Intervall Histogramme von Poisson-Spikes
|
||||
verschiedener Raten $\lambda$ mit der theoretisch zu erwartenden Verteilung
|
||||
der Intervalle $T$ des Poisson Prozesses
|
||||
\[ p(T) = \lambda e^{-\lambda T} \]
|
||||
mit rate $\lambda$.
|
||||
\[ p(T) = \lambda e^{-\lambda T} \; .\]
|
||||
\begin{solution}
|
||||
\lstinputlisting{hompoissonisih.m}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih100hz}}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih20hz}}
|
||||
\end{solution}
|
||||
|
||||
\part \extra Was pasiert mit den Histogrammen, wenn die Binbreite der Histogramme kleiner
|
||||
als das bei der Erzeugung der $\Delta t$ der
|
||||
used for generating the Poisson spikes?
|
||||
\part \extra Was pasiert mit den Histogrammen, wenn die Binbreite
|
||||
der Histogramme kleiner als das bei der Erzeugung der Poisson
|
||||
Spiketrains verwendete $\Delta t$ ist?
|
||||
\begin{solution}
|
||||
The bins between the discretization have zero entries. Therefore
|
||||
the other ones become higher than they should be.
|
||||
Die Bins zwischen der durch $\Delta t$ vorgegebenen
|
||||
Diskretisierung haben den Wert 0. Dadurch werden aber die anderen
|
||||
durch die Normierung h\"oher als sie sein sollten.
|
||||
\end{solution}
|
||||
|
||||
\part Plot the mean interspike interval, the corresponding standard deviation, and the CV
|
||||
as a function of the rate $\lambda$ of the Poisson process.
|
||||
Compare the ../code with the theoretical expectations for the dependence on $\lambda$.
|
||||
\part Plotte den Mittelwert der Interspikeintervalle, die
|
||||
dazugeh\"orige Standardabweichung und den Variationskoeffizienten
|
||||
als Funktion der Rate $\lambda$ des Poisson Prozesses. Vergleiche
|
||||
die Ergebnisse mit den theoretischen Erwartungen (siehe Vorlesungsskript).
|
||||
\begin{solution}
|
||||
\lstinputlisting{hompoissonisistats.m}
|
||||
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonisistats}}
|
||||
\end{solution}
|
||||
|
||||
\part Write a function that computes serial correlations for the interspike intervals
|
||||
for a range of lags.
|
||||
The serial correlations $\rho_k$ at lag $k$ are defined as
|
||||
\[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \]
|
||||
Use this function to show that interspike intervals of Poisson spikes are independent.
|
||||
\part Plotte die seriellen Korrelationen von Poisson-Spiketrains und
|
||||
erkl\"are kurz das Ergebniss.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/isiserialcorr.m}
|
||||
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonserial100hz}}
|
||||
\mbox{}\\[-2ex]\hspace*{2cm}
|
||||
\colorbox{white}{\includegraphics[width=0.8\textwidth]{poissonserial100hz}}\\
|
||||
Alle Korrelationen zwischen Interspikeintervallen sind Null, da
|
||||
beim Poisson Prozess das Auftreten jedes Spikes unabh\"angig von
|
||||
den vorherigen Spikes ist.
|
||||
\end{solution}
|
||||
|
||||
\part Write a function that generates from spike times
|
||||
a histogram of spike counts in a count window of given duration $W$.
|
||||
The function should also plot the Poisson distribution
|
||||
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
|
||||
for the rate $\lambda$ determined from the spike trains.
|
||||
\part Vergleiche Histogramme von Spikecounts gez\"ahlt in Fenstern
|
||||
der Breite $W$ mit der Poisson Verteilung
|
||||
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \; , \]
|
||||
wobei die Rate $\lambda$ aus den Daten bestimmt werden
|
||||
soll. Hinweis: es gibt eine \code{matlab} Funktion, die die
|
||||
Fakult\"at $n!$ berechnet.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/counthist.m}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}}
|
||||
\colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}}
|
||||
\end{solution}
|
||||
|
||||
\part Write a function that computes mean count, variance of count and the corresponding Fano factor
|
||||
for a range of count window durations. The function should generate tow plots: one plotting
|
||||
the count variance against the mean, the other one the Fano factor as a function of the window duration.
|
||||
\part Schreibe eine Funktion, die die mittlere Anzahl, die Varianz
|
||||
und den Fano-Faktor der Anzahl der Spikes in einem Fenster der
|
||||
Breite $W$ bestimmt. Benutze die Funktion, um diese Parameter f\"ur
|
||||
verschiedene Fensterbreiten $W$ zu bestimmen. Zwei Plots sollen aus
|
||||
den Ergebnissen angefertigt werden: (i) Varianz gegen Mittelwert der counts.
|
||||
(ii) Fano Faktor als Funktion der Fensterbreite.
|
||||
\begin{solution}
|
||||
\lstinputlisting{../code/fano.m}
|
||||
\colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonfano100hz}}
|
||||
@ -191,11 +197,3 @@ f\"ur gen\"ugend kleine $\Delta t$.
|
||||
\end{questions}
|
||||
|
||||
\end{document}
|
||||
|
||||
|
||||
Zus\"atzlich soll die Funktion
|
||||
die Poisson-Verteilung
|
||||
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \] mit der Rate
|
||||
$\lambda$, die aus den Daten bestimmt werden kann, mit zu dem
|
||||
Histogramm hineinzeichen. Hinweis: es gibt eine \code{matlab} Funktion,
|
||||
die die Fakult\"at $n!$ berechnet.
|
||||
@ -224,48 +224,3 @@
|
||||
|
||||
\end{document}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{\tr{Homogeneous Poisson process}{Homogener Poisson Prozess}}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=1\textwidth]{poissonraster100hz}
|
||||
\caption{\label{hompoissonfig}Rasterplot von Poisson-Spikes.}
|
||||
\end{figure}
|
||||
|
||||
The probability $p(t)\delta t$ of an event occuring at time $t$
|
||||
is independent of $t$ and independent of any previous event
|
||||
(independent of event history).
|
||||
|
||||
The probability $P$ for an event occuring within a time bin of width $\Delta t$
|
||||
is
|
||||
\[ P=\lambda \cdot \Delta t \]
|
||||
for a Poisson process with rate $\lambda$.
|
||||
|
||||
\subsection{Statistics of homogeneous Poisson process}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
|
||||
\includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
|
||||
\caption{\label{hompoissonisihfig}Interspike interval histograms of poisson spike train.}
|
||||
\end{figure}
|
||||
|
||||
\begin{itemize}
|
||||
\item Exponential distribution of intervals $T$: $p(T) = \lambda e^{-\lambda T}$
|
||||
\item Mean interval $\mu_{ISI} = \frac{1}{\lambda}$
|
||||
\item Variance of intervals $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$
|
||||
\item Coefficient of variation $CV_{ISI} = 1$
|
||||
\item Serial correlation $\rho_k =0$ for $k>0$ (renewal process!)
|
||||
\item Fano factor $F=1$
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Count statistics of Poisson process}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
|
||||
\caption{\label{hompoissoncountfig}Count statistics of poisson spike train.}
|
||||
\end{figure}
|
||||
|
||||
Poisson distribution:
|
||||
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
|
||||
@ -113,3 +113,52 @@ Insbesondere ist die mittlere Rate der Ereignisse $r$ (``Spikes pro Zeit'', Feue
|
||||
% \end{figure}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Homogener Poisson Prozess}
|
||||
F\"ur kontinuierliche Me{\ss}gr\"o{\ss}en ist die Normalverteilung
|
||||
u.a. wegem dem Zentralen Grenzwertsatz die Standardverteilung. Eine
|
||||
\"ahnliche Rolle spilet bei Punktprozessen der ``Poisson Prozess''.
|
||||
|
||||
Beim homogenen Poisson Prozess treten Ereignisse mit einer festen Rate
|
||||
$\lambda=\text{const.}$ auf und sind unabh\"angig von der Zeit $t$ und
|
||||
unabh\"angig von den Zeitpunkten fr\"uherer Ereignisse. Die
|
||||
Wahrscheinlichkeit zu irgendeiner Zeit ein Ereigniss in einem kleinen
|
||||
Zeitfenster der Breite $\Delta t$ zu bekommen ist
|
||||
\[ P = \lambda \cdot \Delta t \; . \]
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=1\textwidth]{poissonraster100hz}
|
||||
\caption{\label{hompoissonfig}Rasterplot von Spikes eine homogenen
|
||||
Poisson Prozesse mit $\lambda=100$\,Hz.}
|
||||
\end{figure}
|
||||
|
||||
Beim inhomogenen Poisson Prozess h\"angt die Rate $\lambda$ von der
|
||||
Zeit ab: $\lambda = \lambda(t)$.
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
|
||||
\includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
|
||||
\caption{\label{hompoissonisihfig}Interspikeintervallverteilungen
|
||||
zweier Poissonprozesse.}
|
||||
\end{figure}
|
||||
|
||||
Der homogne Poissonprozess hat folgende Eigenschaften:
|
||||
\begin{itemize}
|
||||
\item Die Intervalle $T$ sind exponentiell verteilt: $p(T) = \lambda e^{-\lambda T}$ .
|
||||
\item Das mittlere Intervall ist $\mu_{ISI} = \frac{1}{\lambda}$ .
|
||||
\item Die Varianz der Intervalle ist $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$ .
|
||||
\item Der Variationskoeffizient ist also immer $CV_{ISI} = 1$ .
|
||||
\item Die seriellen Korrelationen $\rho_k =0$ for $k>0$, da das
|
||||
Auftreten der Ereignisse unabh\"angig von der Vorgeschichte ist. Ein
|
||||
solcher Prozess wird auch Erneuerungsprozess genannt (``renewal
|
||||
process'').
|
||||
\item Die Anzahl der Ereignisse $k$ innerhalb eines Fensters der L\"ange W ist Poissonverteilt:
|
||||
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
|
||||
\item Der Fano Faktor ist immer $F=1$ .
|
||||
\end{itemize}
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
|
||||
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
|
||||
\caption{\label{hompoissoncountfig}Z\"ahlstatistik von Poisson Spikes.}
|
||||
\end{figure}
|
||||
|
||||
@ -239,4 +239,7 @@
|
||||
\renewcommand{\texinputpath}{pointprocesses/lecture/}
|
||||
\include{pointprocesses/lecture/pointprocesses}
|
||||
|
||||
\renewcommand{\codepath}{designpattern/code/}
|
||||
\include{designpattern/lecture/designpattern}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user