stubs for plotting exercise

likelihood/lecture/likelihood.tex

@@ -211,7 +211,7 @@ cost function, e.g. the gradient descent \matlabfun{lsqcurvefit()}.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Fits von Wahrscheinlichkeitsverteilungen}
+\section{Fitting probability distributions}
 Finally let's consider the case in which we want to fit the parameters
 of a probability density function (e.g. the shape parameter of a
 \enterm{Gamma-distribution}) to a dataset.

plotting/exercises/exercises.tex

@@ -0,0 +1,213 @@
+\documentclass[12pt,a4paper,pdftex]{exam}
+
+\usepackage[german]{babel}
+\usepackage{pslatex}
+\usepackage[mediumspace,mediumqspace,Gray]{SIunits}      % \ohm, \micro
+\usepackage{xcolor}
+\usepackage{graphicx}
+\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
+
+%%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
+\pagestyle{headandfoot}
+\ifprintanswers
+\newcommand{\stitle}{: Solutions}
+\else
+\newcommand{\stitle}{}
+\fi
+\header{{\bfseries\large Exercise 12\stitle}}{{\bfseries\large Maximum likelihood}}{{\bfseries\large January 7th, 2019}}
+\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
+jan.benda@uni-tuebingen.de}
+\runningfooter{}{\thepage}{}
+
+\setlength{\baselineskip}{15pt}
+\setlength{\parindent}{0.0cm}
+\setlength{\parskip}{0.3cm}
+\renewcommand{\baselinestretch}{1.15}
+
+%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{listings}
+\lstset{
+  language=Matlab,
+  basicstyle=\ttfamily\footnotesize,
+  numbers=left,
+  numberstyle=\tiny,
+  title=\lstname,
+  showstringspaces=false,
+  commentstyle=\itshape\color{darkgray},
+  breaklines=true,
+  breakautoindent=true,
+  columns=flexible,
+  frame=single,
+  xleftmargin=1em,
+  xrightmargin=1em,
+  aboveskip=10pt
+}
+
+%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{amsmath}
+\usepackage{amssymb}
+\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}}
+
+%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\continue}{\ifprintanswers%
+\else
+\vfill\hspace*{\fill}$\rightarrow$\newpage%
+\fi}
+\newcommand{\continuepage}{\ifprintanswers%
+\newpage
+\else
+\vfill\hspace*{\fill}$\rightarrow$\newpage%
+\fi}
+\newcommand{\newsolutionpage}{\ifprintanswers%
+\newpage%
+\else
+\fi}
+
+%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\qt}[1]{\textbf{#1}\\}
+\newcommand{\pref}[1]{(\ref{#1})}
+\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
+\newcommand{\code}[1]{\texttt{#1}}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{document}
+
+\input{instructions}
+
+
+\begin{questions}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\question \qt{Maximum likelihood of the standard deviation}
+Let's compute the likelihood and the log-likelihood for the estimation
+of the standard deviation.
+\begin{parts}
+  \part Draw $n=50$ random numbers from a normal distribution with
+  mean $\mu=3$ and standard deviation $\sigma=2$.
+
+  \part Plot the likelihood (computed as the product of probabilities)
+  and the log-likelihood (sum of the logarithms of the probabilities)
+  as a function of the standard deviation. Compare the position of the
+  maxima with the standard deviation that you compute directly from
+  the data.
+
+  \part Increase $n$ to 1000. What happens to the likelihood, what
+  happens to the log-likelihood? Why?
+\end{parts}
+\begin{solution}
+  \lstinputlisting{mlestd.m}
+  \includegraphics[width=1\textwidth]{mlestd}\\
+
+  The more data the smaller the product of the probabilities ($\approx
+  p^n$ with $0 \le p < 1$) and the smaller the sum of the logarithms
+  of the probabilities ($\approx n\log p$, note that $\log p < 0$).
+
+  The product eventually gets smaller than the precision of the
+  floating point numbers support. Therefore for $n=1000$ the products
+  becomes zero. Using the logarithm avoids this numerical problem.
+\end{solution}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\question \qt{Maximum-likelihood estimator of a line through the origin} 
+In the lecture we derived the following equation for an
+maximum-likelihood estimate of the slope $\theta$ of a straight line
+through the origin fitted to $n$ pairs of data values $(x_i|y_i)$ with
+standard deviation $\sigma_i$:
+\[\theta = \frac{\sum_{i=1}^n \frac{x_i y_i}{\sigma_i^2}}{ \sum_{i=1}^n
+  \frac{x_i^2}{\sigma_i^2}} \]
+\begin{parts}
+  \part \label{mleslopefunc} Write a function that takes two vectors
+  $x$ and $y$ containing the data pairs and returns the slope,
+  computed according to this equation. For simplicity we assume
+  $\sigma_i=\sigma$ for all $1 \le i \le n$. How does this simplify
+  the equation for the slope?
+  \begin{solution}
+    \lstinputlisting{mleslope.m}
+  \end{solution}
+
+  \part Write a script that generates data pairs that scatter around a
+  line through the origin with a given slope. Use the function from
+  \pref{mleslopefunc} to compute the slope from the generated data.
+  Compare the computed slope with the true slope that has been used to
+  generate the data. Plot the data togehther with the line from which
+  the data were generated and the maximum-likelihood fit.
+  \begin{solution}
+    \lstinputlisting{mlepropfit.m}
+    \includegraphics[width=1\textwidth]{mlepropfit}
+  \end{solution}
+
+  \part \label{mleslopecomp} Vary the number of data pairs, the slope,
+  as well as the variance of the data points around the true
+  line. Under which conditions is the maximum-likelihood estimation of
+  the slope closer to the true slope?
+
+  \part To answer \pref{mleslopecomp} more precisely, generate for
+  each condition let's say 1000 data sets and plot a histogram of the
+  estimated slopes. How does the histogram, its mean and standard
+  deviation relate to the true slope?
+\end{parts}
+\begin{solution}
+  \lstinputlisting{mlepropest.m}
+  \includegraphics[width=1\textwidth]{mlepropest}\\
+  The estimated slopes are centered around the true slope. The
+  standard deviation of the estimated slopes gets smaller for larger
+  $n$ and less noise in the data.
+\end{solution}
+
+\continue
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\question \qt{Maximum-likelihood-estimation of a probability-density function}
+Many probability-density functions have parameters that cannot be
+computed directly from the data, like, for example, the mean of
+normally-distributed data. Such parameter need to be estimated by
+means of the maximum-likelihood from the data.
+
+Let us demonstrate this approach by means of data that are drawn from a
+gamma distribution,
+\begin{parts}
+  \part Find out which \code{matlab} function computes the
+  probability-density function of the gamma distribution.
+
+  \part \label{gammaplot} Use this function to plot the
+  probability-density function of the gamma distribution for various
+  values of the (positive) ``shape'' parameter. Wet set the ``scale''
+  parameter to one.
+
+  \part Find out which \code{matlab} function generates random numbers
+  that are distributed according to a gamma distribution. Generate
+  with this function 50 random numbers using one of the values of the
+  ``shape'' parameter used in \pref{gammaplot}.
+
+  \part Compute and plot a properly normalized histogram of these
+  random numbers.
+
+  \part Find out which \code{matlab} function fit a distribution to a
+  vector of random numbers according to the maximum-likelihood method.
+  How do you need to use this function in order to fit a gamma
+  distribution to the data?
+
+  \part Estimate with this function the parameter of the gamma
+  distribution used to generate the data.
+
+  \part Finally, plot the fitted gamma distribution on top of the
+  normalized histogram of the data.
+\end{parts}
+\begin{solution}
+  \lstinputlisting{mlepdffit.m}
+  \includegraphics[width=1\textwidth]{mlepdffit}
+\end{solution}
+
+\end{questions}
+
+\end{document}

plotting/exercises/instructions.tex

@@ -0,0 +1,41 @@
+\vspace*{-8ex}
+\begin{center}
+\textbf{\Large Introduction to scientific computing}\\[1ex]
+{\large Jan Grewe, Jan Benda}\\[-3ex]
+Neuroethology lab \hfill --- \hfill Institute for Neurobiology \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
+\end{center}
+
+% \ifprintanswers%
+% \else
+
+% % Die folgenden Aufgaben dienen der Wiederholung, \"Ubung und
+% % Selbstkontrolle und sollten eigenst\"andig bearbeitet und gel\"ost
+% % werden. Die L\"osung soll in Form eines einzelnen Skriptes (m-files)
+% % im ILIAS hochgeladen werden. Jede Aufgabe sollte in einer eigenen
+% % ``Zelle'' gel\"ost sein. Die Zellen \textbf{m\"ussen} unabh\"angig
+% % voneinander ausf\"uhrbar sein. Das Skript sollte nach dem Muster:
+% % ``variablen\_datentypen\_\{nachname\}.m'' benannt werden
+% % (z.B. variablen\_datentypen\_mueller.m).
+
+% \begin{itemize}
+% \item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass
+%   sie auch wirklich das macht, was sie machen soll! Teste dies mit
+%   kleinen Beispielen direkt in der Kommandozeile.
+% \item Versuche die L\"osungen der Aufgaben m\"oglichst in
+%   sinnvolle kleine Funktionen herunterzubrechen.
+%   Sobald etwas \"ahnliches mehr als einmal berechnet werden soll,
+%   lohnt es sich eine Funktion daraus zu schreiben!
+% \item Teste rechenintensive \code{for} Schleifen, Vektoren, Matrizen
+%   zuerst mit einer kleinen Anzahl von Wiederholungen oder kleiner
+%   Gr\"o{\ss}e, und benutze erst am Ende, wenn alles \"uberpr\"uft
+%   ist, eine gro{\ss}e Anzahl von Wiederholungen oder Elementen, um eine gute
+%   Statistik zu bekommen.
+% \item Benutze die Hilfsfunktion von \code{matlab} (\code{help
+%     commando} oder \code{doc commando}) und das Internet, um
+%   herauszufinden, wie bestimmte \code{matlab} Funktionen zu verwenden
+%   sind und was f\"ur M\"oglichkeiten sie bieten.
+%   Auch zu inhaltlichen Konzepten bietet das Internet oft viele
+%   Antworten!
+% \end{itemize}
+
+% \fi

plotting/exercises/plotting_exercise.py

@@ -0,0 +1,74 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import scipy.io as scio
+from IPython import embed
+
+def boltzmann(x, y_max, slope, inflection):
+    """
+    The underlying Boltzmann function.
+    .. math::
+        f(x) = y_max / \exp{-slope*(x-inflection}
+
+    :param x: The x values.
+    :param y_max: The maximum value.
+    :param slope: The slope parameter k
+    :param inflection: the position of the inflection point.
+    :return: the y values.
+    """
+    y = y_max / (1 + np.exp(-slope * (x - inflection)))
+    return y
+
+
+class Animal(object):
+
+    def __init__(self, delay, learning_rate, volatility, responsiveness):
+        """
+        :param delay:
+        :param learning_rate: delta percent_correct per session
+        :param volatility: 0 -> 1 the noise in the decision
+        :param responsiveness: 0 -> 1 probability of actually conducting a trial
+        """
+        self.__delay = delay
+        self.__learning_rate = learning_rate
+        self.__volatility = volatility
+        self.__responsiveness = responsiveness
+
+    def simulate(self, session_count=10, trials=20, task_difficulties=[]):
+        tasks = 1 if len(task_difficulties) == 0 else len(task_difficulties)
+        if len(task_difficulties) == 0:
+            task_difficulties = [0]
+        avg_perf = np.zeros((session_count, tasks))
+        err_perf = np.zeros((session_count, tasks))
+        trials_performed = np.zeros(session_count)
+        for i in range(session_count):
+            for j in range(tasks):
+                base_performance = boltzmann(i, 1.0, self.__learning_rate/20, self.__delay)
+                penalty = base_performance * task_difficulties[j] * 0.5
+                base_perf = 50 + 50 * (base_performance - penalty)
+                trials_completed = np.random.rand(trials) < self.__responsiveness
+                performances = np.random.randn(trials) * self.__volatility * 100 + base_perf
+                avg_perf[i, j] = np.mean(performances[trials_completed])
+                err_perf[i, j] = np.std(performances[trials_completed])
+                trials_performed = np.sum(trials_completed)
+        return avg_perf, err_perf, trials_performed
+
+
+if __name__ == "__main__":
+
+    session_count = 30
+    task_difficulties = [0, 0.3, 1.]
+
+    delays = [5, 10, 12, 20]
+    learning_rates = np.array([5, 10, 2, 20])
+    volatilities = np.random.rand(4) * 0.5
+    responsivness = np.random.rand(4) * 0.5 + 0.5
+
+    for i in range(len(delays)):
+        d = delays[i]
+        lr = learning_rates[i]
+        v = volatilities[i],
+        r = responsivness[i]
+        a = Animal(d, lr, v, r)
+        ap, ep, tp = a.simulate(session_count=session_count, task_difficulties=[0, 0.3, 0.6])
+        plt.plot(ap)
+        embed()

pointprocesses/lecture/pointprocessscetchA.eps

@@ -1,7 +1,7 @@
 %!PS-Adobe-2.0 EPSF-2.0
 %%Title: pointprocessscetchA.tex
-%%Creator: gnuplot 4.6 patchlevel 4
+%%Creator: gnuplot 5.0 patchlevel 6
-%%CreationDate: Mon Nov 26 17:57:49 2018
+%%CreationDate: Thu Oct 17 14:13:46 2019
 %%DocumentFonts: 
 %%BoundingBox: 50 50 373 135
 %%EndComments
@@ -18,6 +18,7 @@ gnudict begin
 /Dashlength 1 def
 /Landscape false def
 /Level1 false def
+/Level3 false def
 /Rounded true def
 /ClipToBoundingBox false def
 /SuppressPDFMark false def
@@ -29,11 +30,11 @@ gnudict begin
 %
 /vshift -73 def
 /dl1 {
-  10.0 Dashlength mul mul
+  10.0 Dashlength userlinewidth gnulinewidth div mul mul mul
   Rounded { currentlinewidth 0.75 mul sub dup 0 le { pop 0.01 } if } if
 } def
 /dl2 {
-  10.0 Dashlength mul mul
+  10.0 Dashlength userlinewidth gnulinewidth div mul mul mul
   Rounded { currentlinewidth 0.75 mul add } if
 } def
 /hpt_ 31.5 def
@@ -47,7 +48,7 @@ gnudict begin
   } if
 } def
 %
-% Gnuplot Prolog Version 4.6 (September 2012)
+% Gnuplot Prolog Version 5.1 (Oct 2015)
 %
 %/SuppressPDFMark true def
 %
@@ -64,11 +65,11 @@ gnudict begin
 /vpt2 vpt 2 mul def
 /hpt2 hpt 2 mul def
 /Lshow {currentpoint stroke M 0 vshift R 
-	Blacktext {gsave 0 setgray show grestore} {show} ifelse} def
+	Blacktext {gsave 0 setgray textshow grestore} {textshow} ifelse} def
 /Rshow {currentpoint stroke M dup stringwidth pop neg vshift R
-	Blacktext {gsave 0 setgray show grestore} {show} ifelse} def
+	Blacktext {gsave 0 setgray textshow grestore} {textshow} ifelse} def
 /Cshow {currentpoint stroke M dup stringwidth pop -2 div vshift R 
-	Blacktext {gsave 0 setgray show grestore} {show} ifelse} def
+	Blacktext {gsave 0 setgray textshow grestore} {textshow} ifelse} def
 /UP {dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
   /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def} def
 /DL {Color {setrgbcolor Solid {pop []} if 0 setdash}
@@ -82,7 +83,7 @@ gnudict begin
 /PL {stroke userlinewidth setlinewidth
 	Rounded {1 setlinejoin 1 setlinecap} if} def
 3.8 setmiterlimit
-% Default Line colors
+% Classic Line colors (version 5.0)
 /LCw {1 1 1} def
 /LCb {0 0 0} def
 /LCa {0 0 0} def
@@ -95,19 +96,21 @@ gnudict begin
 /LC6 {0 0 0} def
 /LC7 {1 0.3 0} def
 /LC8 {0.5 0.5 0.5} def
-% Default Line Types
+% Default dash patterns (version 5.0)
+/LTB {BL [] LCb DL} def
 /LTw {PL [] 1 setgray} def
-/LTb {BL [] LCb DL} def
+/LTb {PL [] LCb DL} def
 /LTa {AL [1 udl mul 2 udl mul] 0 setdash LCa setrgbcolor} def
 /LT0 {PL [] LC0 DL} def
-/LT1 {PL [4 dl1 2 dl2] LC1 DL} def
+/LT1 {PL [2 dl1 3 dl2] LC1 DL} def
-/LT2 {PL [2 dl1 3 dl2] LC2 DL} def
+/LT2 {PL [1 dl1 1.5 dl2] LC2 DL} def
-/LT3 {PL [1 dl1 1.5 dl2] LC3 DL} def
+/LT3 {PL [6 dl1 2 dl2 1 dl1 2 dl2] LC3 DL} def
-/LT4 {PL [6 dl1 2 dl2 1 dl1 2 dl2] LC4 DL} def
+/LT4 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC4 DL} def
-/LT5 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC5 DL} def
+/LT5 {PL [4 dl1 2 dl2] LC5 DL} def
-/LT6 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC6 DL} def
+/LT6 {PL [1.5 dl1 1.5 dl2 1.5 dl1 1.5 dl2 1.5 dl1 6 dl2] LC6 DL} def
-/LT7 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC7 DL} def
+/LT7 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC7 DL} def
-/LT8 {PL [2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 4 dl2] LC8 DL} def
+/LT8 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC8 DL} def
+/SL {[] 0 setdash} def
 /Pnt {stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore} def
 /Dia {stroke [] 0 setdash 2 copy vpt add M
   hpt neg vpt neg V hpt vpt neg V
@@ -329,9 +332,14 @@ gnudict begin
 %
 /languagelevel where
  {pop languagelevel} {1} ifelse
- 2 lt
+dup 2 lt
-	{/InterpretLevel1 true def}
+	{/InterpretLevel1 true def
-	{/InterpretLevel1 Level1 def}
+	 /InterpretLevel3 false def}
+	{/InterpretLevel1 Level1 def
+	 2 gt
+	    {/InterpretLevel3 Level3 def}
+	    {/InterpretLevel3 false def}
+	 ifelse }
  ifelse
 %
 % PostScript level 2 pattern fill definitions
@@ -420,6 +428,7 @@ Level1 {Level1PatternFill} {Level2PatternFill} ifelse
 /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
 dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
 currentdict end definefont pop
+%
 Level1 SuppressPDFMark or 
 {} {
 /SDict 10 dict def
@@ -429,14 +438,42 @@ systemdict /pdfmark known not {
 SDict begin [
   /Title (pointprocessscetchA.tex)
   /Subject (gnuplot plot)
-  /Creator (gnuplot 4.6 patchlevel 4)
+  /Creator (gnuplot 5.0 patchlevel 6)
-  /Author (benda)
+  /Author (grewe)
 %  /Producer (gnuplot)
 %  /Keywords ()
-  /CreationDate (Mon Nov 26 17:57:49 2018)
+  /CreationDate (Thu Oct 17 14:13:46 2019)
   /DOCINFO pdfmark
 end
 } ifelse
+%
+% Support for boxed text - Ethan A Merritt May 2005
+%
+/InitTextBox { userdict /TBy2 3 -1 roll put userdict /TBx2 3 -1 roll put
+           userdict /TBy1 3 -1 roll put userdict /TBx1 3 -1 roll put
+	   /Boxing true def } def
+/ExtendTextBox { Boxing
+    { gsave dup false charpath pathbbox
+      dup TBy2 gt {userdict /TBy2 3 -1 roll put} {pop} ifelse
+      dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse
+      dup TBy1 lt {userdict /TBy1 3 -1 roll put} {pop} ifelse
+      dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse
+      grestore } if } def
+/PopTextBox { newpath TBx1 TBxmargin sub TBy1 TBymargin sub M
+               TBx1 TBxmargin sub TBy2 TBymargin add L
+	       TBx2 TBxmargin add TBy2 TBymargin add L
+	       TBx2 TBxmargin add TBy1 TBymargin sub L closepath } def
+/DrawTextBox { PopTextBox stroke /Boxing false def} def
+/FillTextBox { gsave PopTextBox 1 1 1 setrgbcolor fill grestore /Boxing false def} def
+0 0 0 0 InitTextBox
+/TBxmargin 20 def
+/TBymargin 20 def
+/Boxing false def
+/textshow { ExtendTextBox Gshow } def
+%
+% redundant definitions for compatibility with prologue.ps older than 5.0.2
+/LTB {BL [] LCb DL} def
+/LTb {PL [] LCb DL} def
 end
 %%EndProlog
 %%Page: 1 1
@@ -450,10 +487,9 @@ newpath
 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 1.000 UP
-LTb
 LCb setrgbcolor
-LTb
 1.000 UL
 LTb
 gsave 6208 824 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
@@ -467,9 +503,11 @@ gsave 6208 824 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
 stroke
 2.000 UL
 LTb
-0.00 0.00 0.00 C 10.000 UL
+0.00 0.00 0.00 C % Begin plot #1
-LT0
+10.000 UL
-LC0 setrgbcolor
+LTb
+LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 910 573 M
 0 503 V
 1412 573 M
@@ -488,11 +526,16 @@ LC0 setrgbcolor
 0 503 V
 5685 573 M
 0 503 V
-1.000 UP
 stroke
+LTw
+% End plot #1
 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
+1.000 UP
+2.000 UL
+LTb
 0.00 0.00 0.00 C stroke
 grestore
 end

pointprocesses/lecture/pointprocessscetchB.eps

@@ -1,7 +1,7 @@
 %!PS-Adobe-2.0 EPSF-2.0
 %%Title: pointprocessscetchB.tex
-%%Creator: gnuplot 4.6 patchlevel 4
+%%Creator: gnuplot 5.0 patchlevel 6
-%%CreationDate: Mon Nov 26 17:57:49 2018
+%%CreationDate: Thu Oct 17 14:13:46 2019
 %%DocumentFonts: 
 %%BoundingBox: 50 50 373 237
 %%EndComments
@@ -18,6 +18,7 @@ gnudict begin
 /Dashlength 1 def
 /Landscape false def
 /Level1 false def
+/Level3 false def
 /Rounded true def
 /ClipToBoundingBox false def
 /SuppressPDFMark false def
@@ -29,11 +30,11 @@ gnudict begin
 %
 /vshift -73 def
 /dl1 {
-  10.0 Dashlength mul mul
+  10.0 Dashlength userlinewidth gnulinewidth div mul mul mul
   Rounded { currentlinewidth 0.75 mul sub dup 0 le { pop 0.01 } if } if
 } def
 /dl2 {
-  10.0 Dashlength mul mul
+  10.0 Dashlength userlinewidth gnulinewidth div mul mul mul
   Rounded { currentlinewidth 0.75 mul add } if
 } def
 /hpt_ 31.5 def
@@ -47,7 +48,7 @@ gnudict begin
   } if
 } def
 %
-% Gnuplot Prolog Version 4.6 (September 2012)
+% Gnuplot Prolog Version 5.1 (Oct 2015)
 %
 %/SuppressPDFMark true def
 %
@@ -64,11 +65,11 @@ gnudict begin
 /vpt2 vpt 2 mul def
 /hpt2 hpt 2 mul def
 /Lshow {currentpoint stroke M 0 vshift R 
-	Blacktext {gsave 0 setgray show grestore} {show} ifelse} def
+	Blacktext {gsave 0 setgray textshow grestore} {textshow} ifelse} def
 /Rshow {currentpoint stroke M dup stringwidth pop neg vshift R
-	Blacktext {gsave 0 setgray show grestore} {show} ifelse} def
+	Blacktext {gsave 0 setgray textshow grestore} {textshow} ifelse} def
 /Cshow {currentpoint stroke M dup stringwidth pop -2 div vshift R 
-	Blacktext {gsave 0 setgray show grestore} {show} ifelse} def
+	Blacktext {gsave 0 setgray textshow grestore} {textshow} ifelse} def
 /UP {dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
   /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def} def
 /DL {Color {setrgbcolor Solid {pop []} if 0 setdash}
@@ -82,7 +83,7 @@ gnudict begin
 /PL {stroke userlinewidth setlinewidth
 	Rounded {1 setlinejoin 1 setlinecap} if} def
 3.8 setmiterlimit
-% Default Line colors
+% Classic Line colors (version 5.0)
 /LCw {1 1 1} def
 /LCb {0 0 0} def
 /LCa {0 0 0} def
@@ -95,19 +96,21 @@ gnudict begin
 /LC6 {0 0 0} def
 /LC7 {1 0.3 0} def
 /LC8 {0.5 0.5 0.5} def
-% Default Line Types
+% Default dash patterns (version 5.0)
+/LTB {BL [] LCb DL} def
 /LTw {PL [] 1 setgray} def
-/LTb {BL [] LCb DL} def
+/LTb {PL [] LCb DL} def
 /LTa {AL [1 udl mul 2 udl mul] 0 setdash LCa setrgbcolor} def
 /LT0 {PL [] LC0 DL} def
-/LT1 {PL [4 dl1 2 dl2] LC1 DL} def
+/LT1 {PL [2 dl1 3 dl2] LC1 DL} def
-/LT2 {PL [2 dl1 3 dl2] LC2 DL} def
+/LT2 {PL [1 dl1 1.5 dl2] LC2 DL} def
-/LT3 {PL [1 dl1 1.5 dl2] LC3 DL} def
+/LT3 {PL [6 dl1 2 dl2 1 dl1 2 dl2] LC3 DL} def
-/LT4 {PL [6 dl1 2 dl2 1 dl1 2 dl2] LC4 DL} def
+/LT4 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC4 DL} def
-/LT5 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC5 DL} def
+/LT5 {PL [4 dl1 2 dl2] LC5 DL} def
-/LT6 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC6 DL} def
+/LT6 {PL [1.5 dl1 1.5 dl2 1.5 dl1 1.5 dl2 1.5 dl1 6 dl2] LC6 DL} def
-/LT7 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC7 DL} def
+/LT7 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC7 DL} def
-/LT8 {PL [2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 4 dl2] LC8 DL} def
+/LT8 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC8 DL} def
+/SL {[] 0 setdash} def
 /Pnt {stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore} def
 /Dia {stroke [] 0 setdash 2 copy vpt add M
   hpt neg vpt neg V hpt vpt neg V
@@ -329,9 +332,14 @@ gnudict begin
 %
 /languagelevel where
  {pop languagelevel} {1} ifelse
- 2 lt
+dup 2 lt
-	{/InterpretLevel1 true def}
+	{/InterpretLevel1 true def
-	{/InterpretLevel1 Level1 def}
+	 /InterpretLevel3 false def}
+	{/InterpretLevel1 Level1 def
+	 2 gt
+	    {/InterpretLevel3 Level3 def}
+	    {/InterpretLevel3 false def}
+	 ifelse }
  ifelse
 %
 % PostScript level 2 pattern fill definitions
@@ -420,6 +428,7 @@ Level1 {Level1PatternFill} {Level2PatternFill} ifelse
 /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont
 dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall
 currentdict end definefont pop
+%
 Level1 SuppressPDFMark or 
 {} {
 /SDict 10 dict def
@@ -429,14 +438,42 @@ systemdict /pdfmark known not {
 SDict begin [
   /Title (pointprocessscetchB.tex)
   /Subject (gnuplot plot)
-  /Creator (gnuplot 4.6 patchlevel 4)
+  /Creator (gnuplot 5.0 patchlevel 6)
-  /Author (benda)
+  /Author (grewe)
 %  /Producer (gnuplot)
 %  /Keywords ()
-  /CreationDate (Mon Nov 26 17:57:49 2018)
+  /CreationDate (Thu Oct 17 14:13:46 2019)
   /DOCINFO pdfmark
 end
 } ifelse
+%
+% Support for boxed text - Ethan A Merritt May 2005
+%
+/InitTextBox { userdict /TBy2 3 -1 roll put userdict /TBx2 3 -1 roll put
+           userdict /TBy1 3 -1 roll put userdict /TBx1 3 -1 roll put
+	   /Boxing true def } def
+/ExtendTextBox { Boxing
+    { gsave dup false charpath pathbbox
+      dup TBy2 gt {userdict /TBy2 3 -1 roll put} {pop} ifelse
+      dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse
+      dup TBy1 lt {userdict /TBy1 3 -1 roll put} {pop} ifelse
+      dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse
+      grestore } if } def
+/PopTextBox { newpath TBx1 TBxmargin sub TBy1 TBymargin sub M
+               TBx1 TBxmargin sub TBy2 TBymargin add L
+	       TBx2 TBxmargin add TBy2 TBymargin add L
+	       TBx2 TBxmargin add TBy1 TBymargin sub L closepath } def
+/DrawTextBox { PopTextBox stroke /Boxing false def} def
+/FillTextBox { gsave PopTextBox 1 1 1 setrgbcolor fill grestore /Boxing false def} def
+0 0 0 0 InitTextBox
+/TBxmargin 20 def
+/TBymargin 20 def
+/Boxing false def
+/textshow { ExtendTextBox Gshow } def
+%
+% redundant definitions for compatibility with prologue.ps older than 5.0.2
+/LTB {BL [] LCb DL} def
+/LTb {PL [] LCb DL} def
 end
 %%EndProlog
 %%Page: 1 1
@@ -450,10 +487,9 @@ newpath
 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 1.000 UP
-LTb
 LCb setrgbcolor
-LTb
 1.000 UL
 LTb
 gsave 6208 3165 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
@@ -467,9 +503,11 @@ gsave 6208 3165 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
 stroke
 2.000 UL
 LTb
-0.00 0.00 0.00 C 10.000 UL
+0.00 0.00 0.00 C % Begin plot #1
-LT0
+10.000 UL
-LC0 setrgbcolor
+LTb
+LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 910 3029 M
 0 272 V
 502 -272 R
@@ -488,18 +526,22 @@ LC0 setrgbcolor
 0 272 V
 5685 3029 M
 0 272 V
-1.000 UP
 stroke
+LTw
+% End plot #1
 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
+1.000 UP
+2.000 UL
+LTb
 0.00 0.00 0.00 C 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 1.000 UP
-LTb
 LCb setrgbcolor
-LTb
 1.000 UL
 LTb
 gsave 6208 2043 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
@@ -510,76 +552,86 @@ gsave 6208 2043 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
 0 -32 V
 -5680 0 R
 5680 0 V
+stroke
 1291 1929 M
 121 32 V
 -121 32 V
--260 -64 R
+-260 0 R
--121 32 V
+910 1961 L
-121 32 V
+121 -32 V
-910 1961 M
+-121 32 R
 502 0 V
-590 -32 R
+stroke
+2002 1929 M
 121 32 V
 -121 32 V
--469 -64 R
+-469 0 R
--121 32 V
+-121 -32 V
-121 32 V
+121 -32 V
--121 -32 R
+-121 32 R
 711 0 V
-206 -22 R
+stroke
+2329 1939 M
 84 22 V
 -84 22 V
--122 -44 R
+-122 0 R
--84 22 V
+-84 -22 V
-84 22 V
+84 -22 V
--84 -22 R
+-84 22 R
 290 0 V
-216 -23 R
+stroke
+2629 1938 M
 88 23 V
 -88 23 V
--128 -46 R
+-128 0 R
--88 23 V
+-88 -23 V
-88 23 V
+88 -23 V
--88 -23 R
+-88 23 R
 304 0 V
-329 -32 R
+stroke
+3046 1929 M
 121 32 V
 -121 32 V
--208 -64 R
+-208 0 R
--121 32 V
+-121 -32 V
-121 32 V
+121 -32 V
--121 -32 R
+-121 32 R
 450 0 V
-745 -32 R
+stroke
+3912 1929 M
 121 32 V
 -121 32 V
--624 -64 R
+-624 0 R
--121 32 V
+-121 -32 V
-121 32 V
+121 -32 V
--121 -32 R
+-121 32 R
 866 0 V
-496 -32 R
+stroke
+4529 1929 M
 121 32 V
 -121 32 V
--375 -64 R
+-375 0 R
--121 32 V
+-121 -32 V
-121 32 V
+121 -32 V
--121 -32 R
+-121 32 R
 617 0 V
-914 -32 R
+stroke
+5564 1929 M
 121 32 V
 -121 32 V
--793 -64 R
+-793 0 R
--121 32 V
+-121 -32 V
-121 32 V
+121 -32 V
--121 -32 R
+-121 32 R
 1035 0 V
 stroke
 2.000 UL
 LTb
-0.00 0.00 0.00 C 10.000 UL
+0.00 0.00 0.00 C % Begin plot #1
-LT0
+10.000 UL
-LC0 setrgbcolor
+LTb
+LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 910 1907 M
 0 272 V
 502 -272 R
@@ -598,47 +650,60 @@ LC0 setrgbcolor
 0 272 V
 5685 1907 M
 0 272 V
-1.000 UP
 stroke
+LTw
+% End plot #1
 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
+1.000 UP
+2.000 UL
+LTb
 0.00 0.00 0.00 C 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 528 268 M
 -63 0 V
 stroke
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 528 460 M
 -63 0 V
 stroke
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 528 652 M
 -63 0 V
 stroke
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 528 844 M
 -63 0 V
 stroke
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 528 1036 M
 -63 0 V
 stroke
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 528 1228 M
 -63 0 V
 stroke
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 528 1276 M
 528 220 L
 5801 0 R
@@ -646,9 +711,7 @@ LCb setrgbcolor
 -5801 0 R
 1.000 UP
 stroke
-LTb
 LCb setrgbcolor
-LTb
 1.000 UL
 LTb
 gsave 6208 268 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
@@ -662,9 +725,11 @@ gsave 6208 268 N 0 -32 V 121 32 V -121 32 V 0 -32 V 1 PolyFill
 stroke
 2.000 UL
 LTb
-0.00 0.00 0.00 C 3.000 UL
+0.00 0.00 0.00 C % Begin plot #1
-LT0
+3.000 UL
-LC0 setrgbcolor
+LTb
+LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 528 268 M
 382 0 V
 0 96 R
@@ -685,11 +750,15 @@ LC0 setrgbcolor
 1035 0 V
 0 96 R
 533 0 V
-1.500 UP
 stroke
+LTw
+% End plot #1
+% Begin plot #2
+1.500 UP
 2.000 UL
-LT0
+LTb
-LC0 setrgbcolor
+LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 910 268 CircleF
 1412 364 CircleF
 2123 460 CircleF
@@ -699,10 +768,14 @@ LC0 setrgbcolor
 4033 844 CircleF
 4650 940 CircleF
 5685 1036 CircleF
+LTw
+% End plot #2
+% Begin plot #3
 1.000 UP
 2.000 UL
-LT0
+LTb
-LC0 setrgbcolor
+LCb setrgbcolor
+[] 0 setdash
 1.00 1.00 1.00 C 910 268 CircleF
 1412 364 CircleF
 2123 460 CircleF
@@ -712,10 +785,14 @@ LC0 setrgbcolor
 4033 844 CircleF
 4650 940 CircleF
 5685 1036 CircleF
+LTw
+% End plot #3
+% Begin plot #4
 1.500 UP
 2.000 UL
-LT0
+LTb
-LC0 setrgbcolor
+LCb setrgbcolor
+[] 0 setdash
 0.00 0.00 0.00 C 910 364 CircleF
 1412 460 CircleF
 2123 556 CircleF
@@ -725,10 +802,15 @@ LC0 setrgbcolor
 4033 940 CircleF
 4650 1036 CircleF
 5685 1132 CircleF
-1.000 UP
+LTw
+% End plot #4
 2.000 UL
 LTb
 LCb setrgbcolor
+[] 0 setdash
+1.000 UP
+2.000 UL
+LTb
 0.00 0.00 0.00 C stroke
 grestore
 end