Merge branch 'master' of https://whale.am28.uni-tuebingen.de/git/teaching/scientificComputing

likelihood/exercises/exercises01-de.tex

@@ -0,0 +1,192 @@
+\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}{: L\"osungen}
+\else
+\newcommand{\stitle}{}
+\fi
+\header{{\bfseries\large \"Ubung 4\stitle}}{{\bfseries\large Statistik}}{{\bfseries\large 26. Oktober, 2015}}
+\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 der Standardabweichung} 
+Wir wollen uns die Likelihood und die Log-Likelihood am Beispiel der
+Absch\"atzung der Standardabweichung verdeutlichen.
+\begin{parts}
+  \part Ziehe $n=50$ normalverteilte Zufallsvariablen mit Mittelwert $\mu=3$
+  und einer Standardabweichung $\sigma=2$.
+
+  \part
+  Plotte die Likelihood (aus dem Produkt der Wahrscheinlichkeiten) und
+  die Log-Likelihood (aus der Summe der logarithmierten
+  Wahrscheinlichkeiten) f\"ur die Standardabweichung als Parameter. Vergleiche die
+  Position der Maxima mit der aus den Daten berechneten Standardabweichung.
+
+  \part
+  Erh\"ohe $n$ auf 1000. Was passiert mit der Likelihood, was mit der Log-Likelihood? Warum?
+\end{parts}
+\begin{solution}
+  \lstinputlisting{mlestd.m}
+  \includegraphics[width=1\textwidth]{mlestd}
+\end{solution}
+
+\continue
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\question \qt{Maximum-Likelihood-Sch\"atzer einer Ursprungsgeraden} 
+In der Vorlesung haben wir folgende Formel f\"ur die Maximum-Likelihood
+Absch\"atzung der Steigung $\theta$ einer Ursprungsgeraden durch $n$ Datenpunkte $(x_i|y_i)$ mit Standardabweichung $\sigma_i$ hergeleitet:
+\[\theta = \frac{\sum_{i=1}^n \frac{x_iy_i}{\sigma_i^2}}{ \sum_{i=1}^n
+  \frac{x_i^2}{\sigma_i^2}} \]
+\begin{parts}
+  \part \label{mleslopefunc} Schreibe eine Funktion, die in einem $x$ und einem
+  $y$ Vektor die Datenpaare \"uberreicht bekommt und die Steigung der
+  Ursprungsgeraden, die die Likelihood maximiert, zur\"uckgibt
+  ($\sigma=\text{const}$).
+
+  \part
+  Schreibe ein Skript, das Datenpaare erzeugt, die um eine
+  Ursprungsgerade mit vorgegebener Steigung streuen. Berechne mit der
+  Funktion aus \pref{mleslopefunc} die Steigung aus den Daten,
+  vergleiche mit der wahren Steigung, und plotte die urspr\"ungliche
+  sowie die gefittete Gerade zusammen mit den Daten.
+
+  \part
+  Ver\"andere die Anzahl der Datenpunkte, die Steigung, sowie die
+  Streuung der Daten um die Gerade.
+\end{parts}
+\begin{solution}
+  \lstinputlisting{mleslope.m}
+  \lstinputlisting{mlepropfit.m}
+  \includegraphics[width=1\textwidth]{mlepropfit}
+\end{solution}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\question \qt{Maximum-Likelihood-Sch\"atzer einer Wahrscheinlichkeitsdichtefunktion} 
+Verschiedene Wahrscheinlichkeitsdichtefunktionen haben Parameter, die
+nicht so einfach wie der Mittelwert und die Standardabweichung einer
+Normalverteilung direkt aus den Daten berechnet werden k\"onnen. Solche Parameter
+m\"ussen dann aus den Daten mit der Maximum-Likelihood-Methode gefittet werden.
+
+Um dies zu veranschaulichen ziehen wir uns diesmal nicht normalverteilte Zufallszahlen, sondern Zufallszahlen aus der Gamma-Verteilung.
+\begin{parts}
+  \part
+  Finde heraus welche \code{matlab} Funktion die
+  Wahrscheinlichkeitsdichtefunktion (probability density function) der
+  Gamma-Verteilung berechnet.
+
+  \part
+  Plotte mit Hilfe dieser Funktion die  Wahrscheinlichkeitsdichtefunktion
+  der Gamma-Verteilung f\"ur verschiedene Werte des (positiven) ``shape'' Parameters.
+  Den ``scale'' Parameter setzen wir auf Eins.
+
+  \part
+  Finde heraus mit welcher Funktion Gammaverteilte Zufallszahlen in
+  \code{matlab} gezogen werden k\"onnen. Erzeuge mit dieser Funktion
+  50 Zufallszahlen mit einem der oben geplotteten ``shape'' Parameter.
+
+  \part
+  Berechne und plotte ein normiertes Histogramm dieser Zufallszahlen.
+
+  \part
+  Finde heraus mit welcher \code{matlab}-Funktion eine beliebige
+  Verteilung (``distribution'') an die Zufallszahlen nach der
+  Maximum-Likelihood Methode gefittet werden kann. Wie wird diese
+  Funktion benutzt, um die Gammaverteilung an die Daten zu fitten?
+
+  \part
+  Bestimme mit dieser Funktion die Parameter der Gammaverteilung aus
+  den Zufallszahlen.
+
+  \part
+  Plotte anschlie{\ss}end die Gammaverteilung mit den gefitteten
+  Parametern.
+\end{parts}
+\begin{solution}
+  \lstinputlisting{mlepdffit.m}
+  \includegraphics[width=1\textwidth]{mlepdffit}
+\end{solution}
+
+\end{questions}
+
+\end{document}

likelihood/exercises/exercises01.tex

@@ -11,11 +11,11 @@
 \usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
 \pagestyle{headandfoot}
 \ifprintanswers
-\newcommand{\stitle}{: L\"osungen}
+\newcommand{\stitle}{: Solutions}
 \else
 \newcommand{\stitle}{}
 \fi
-\header{{\bfseries\large \"Ubung 4\stitle}}{{\bfseries\large Statistik}}{{\bfseries\large 26. Oktober, 2015}}
+\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}{}
@@ -89,98 +89,119 @@ jan.benda@uni-tuebingen.de}
 \begin{questions}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\question \qt{Maximum Likelihood der Standardabweichung} 
+\question \qt{Maximum likelihood of the standard deviation}
-Wir wollen uns die Likelihood und die Log-Likelihood am Beispiel der
+Let's compute the likelihood and the log-likelihood for the estimation
-Absch\"atzung der Standardabweichung verdeutlichen.
+of the standard deviation.
 \begin{parts}
-  \part Ziehe $n=50$ normalverteilte Zufallsvariablen mit Mittelwert $\mu=3$
+  \part Draw $n=50$ random numbers from a normal distribution with
-  und einer Standardabweichung $\sigma=2$.
+  mean $\mu=3$ and standard deviation $\sigma=2$.
 
-  \part
+  \part Plot the likelihood (computed as the product of probabilities)
-  Plotte die Likelihood (aus dem Produkt der Wahrscheinlichkeiten) und
+  and the log-likelihood (sum of the logarithms of the probabilities)
-  die Log-Likelihood (aus der Summe der logarithmierten
+  as a function of the standard deviation. Compare the position of the
-  Wahrscheinlichkeiten) f\"ur die Standardabweichung als Parameter. Vergleiche die
+  maxima with the standard deviation that you compute directly from
-  Position der Maxima mit der aus den Daten berechneten Standardabweichung.
+  the data.
 
-  \part
+  \part Increase $n$ to 1000. What happens to the likelihood, what
-  Erh\"ohe $n$ auf 1000. Was passiert mit der Likelihood, was mit der Log-Likelihood? Warum?
+  happens to the log-likelihood? Why?
 \end{parts}
 \begin{solution}
   \lstinputlisting{mlestd.m}
-  \includegraphics[width=1\textwidth]{mlestd}
+  \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}
 
-\continue
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\question \qt{Maximum-Likelihood-Sch\"atzer einer Ursprungsgeraden} 
+\question \qt{Maximum-likelihood estimator of a line through the origin} 
-In der Vorlesung haben wir folgende Formel f\"ur die Maximum-Likelihood
+In the lecture we derived the following equation for an
-Absch\"atzung der Steigung $\theta$ einer Ursprungsgeraden durch $n$ Datenpunkte $(x_i|y_i)$ mit Standardabweichung $\sigma_i$ hergeleitet:
+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} Schreibe eine Funktion, die in einem $x$ und einem
+  \part \label{mleslopefunc} Write a function that takes two vectors
-  $y$ Vektor die Datenpaare \"uberreicht bekommt und die Steigung der
+  $x$ and $y$ containing the data pairs and returns the slope,
-  Ursprungsgeraden, die die Likelihood maximiert, zur\"uckgibt
+  computed according to this equation. For simplicity we assume
-  ($\sigma=\text{const}$).
+  $\sigma_i=\sigma$ for all $1 \le i \le n$. How does this simplify
-
+  the equation for the slope?
-  \part
-  Schreibe ein Skript, das Datenpaare erzeugt, die um eine
-  Ursprungsgerade mit vorgegebener Steigung streuen. Berechne mit der
-  Funktion aus \pref{mleslopefunc} die Steigung aus den Daten,
-  vergleiche mit der wahren Steigung, und plotte die urspr\"ungliche
-  sowie die gefittete Gerade zusammen mit den Daten.
-
-  \part
-  Ver\"andere die Anzahl der Datenpunkte, die Steigung, sowie die
-  Streuung der Daten um die Gerade.
-\end{parts}
   \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
-\question \qt{Maximum-Likelihood-Sch\"atzer einer Wahrscheinlichkeitsdichtefunktion} 
+  each condition let's say 1000 data sets and plot a histogram of the
-Verschiedene Wahrscheinlichkeitsdichtefunktionen haben Parameter, die
+  estimated slopes. How does the histogram, its mean and standard
-nicht so einfach wie der Mittelwert und die Standardabweichung einer
+  deviation relate to the true slope?
-Normalverteilung direkt aus den Daten berechnet werden k\"onnen. Solche Parameter
+\end{parts}
-m\"ussen dann aus den Daten mit der Maximum-Likelihood-Methode gefittet werden.
+\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}
 
-Um dies zu veranschaulichen ziehen wir uns diesmal nicht normalverteilte Zufallszahlen, sondern Zufallszahlen aus der Gamma-Verteilung.
+\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
+  \part Find out which \code{matlab} function computes the
-  Finde heraus welche \code{matlab} Funktion die
+  probability-density function of the gamma distribution.
-  Wahrscheinlichkeitsdichtefunktion (probability density function) der
+
-  Gamma-Verteilung berechnet.
+  \part \label{gammaplot} Use this function to plot the
-
+  probability-density function of the gamma distribution for various
-  \part
+  values of the (positive) ``shape'' parameter. Wet set the ``scale''
-  Plotte mit Hilfe dieser Funktion die  Wahrscheinlichkeitsdichtefunktion
+  parameter to one.
-  der Gamma-Verteilung f\"ur verschiedene Werte des (positiven) ``shape'' Parameters.
+
-  Den ``scale'' Parameter setzen wir auf Eins.
+  \part Find out which \code{matlab} function generates random numbers
-
+  that are distributed according to a gamma distribution. Generate
-  \part
+  with this function 50 random numbers using one of the values of the
-  Finde heraus mit welcher Funktion Gammaverteilte Zufallszahlen in
+  ``shape'' parameter used in \pref{gammaplot}.
-  \code{matlab} gezogen werden k\"onnen. Erzeuge mit dieser Funktion
+
-  50 Zufallszahlen mit einem der oben geplotteten ``shape'' Parameter.
+  \part Compute and plot a properly normalized histogram of these
-
+  random numbers.
-  \part
+
-  Berechne und plotte ein normiertes Histogramm dieser Zufallszahlen.
+  \part Find out which \code{matlab} function fit a distribution to a
-
+  vector of random numbers according to the maximum-likelihood method.
-  \part
+  How do you need to use this function in order to fit a gamma
-  Finde heraus mit welcher \code{matlab}-Funktion eine beliebige
+  distribution to the data?
-  Verteilung (``distribution'') an die Zufallszahlen nach der
+
-  Maximum-Likelihood Methode gefittet werden kann. Wie wird diese
+  \part Estimate with this function the parameter of the gamma
-  Funktion benutzt, um die Gammaverteilung an die Daten zu fitten?
+  distribution used to generate the data.
-
+
-  \part
+  \part Finally, plot the fitted gamma distribution on top of the
-  Bestimme mit dieser Funktion die Parameter der Gammaverteilung aus
+  normalized histogram of the data.
-  den Zufallszahlen.
-
-  \part
-  Plotte anschlie{\ss}end die Gammaverteilung mit den gefitteten
-  Parametern.
 \end{parts}
 \begin{solution}
   \lstinputlisting{mlepdffit.m}

likelihood/exercises/instructions.tex

@@ -1,41 +1,41 @@
-\vspace*{-6.5ex}
+\vspace*{-8ex}
 \begin{center}
-\textbf{\Large Einf\"uhrung in die wissenschaftliche Datenverarbeitung}\\[1ex]
+\textbf{\Large Introduction to scientific computing}\\[1ex]
 {\large Jan Grewe, Jan Benda}\\[-3ex]
-Abteilung Neuroethologie \hfill --- \hfill Institut f\"ur Neurobiologie \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
+Neuroethology lab \hfill --- \hfill Institute for Neurobiology \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
 \end{center}
 
-\ifprintanswers%
+% \ifprintanswers%
-\else
+% \else
 
-% Die folgenden Aufgaben dienen der Wiederholung, \"Ubung und
+% % Die folgenden Aufgaben dienen der Wiederholung, \"Ubung und
-% Selbstkontrolle und sollten eigenst\"andig bearbeitet und gel\"ost
+% % Selbstkontrolle und sollten eigenst\"andig bearbeitet und gel\"ost
-% werden. Die L\"osung soll in Form eines einzelnen Skriptes (m-files)
+% % werden. Die L\"osung soll in Form eines einzelnen Skriptes (m-files)
-% im ILIAS hochgeladen werden. Jede Aufgabe sollte in einer eigenen
+% % im ILIAS hochgeladen werden. Jede Aufgabe sollte in einer eigenen
-% ``Zelle'' gel\"ost sein. Die Zellen \textbf{m\"ussen} unabh\"angig
+% % ``Zelle'' gel\"ost sein. Die Zellen \textbf{m\"ussen} unabh\"angig
-% voneinander ausf\"uhrbar sein. Das Skript sollte nach dem Muster:
+% % voneinander ausf\"uhrbar sein. Das Skript sollte nach dem Muster:
-% ``variablen\_datentypen\_\{nachname\}.m'' benannt werden
+% % ``variablen\_datentypen\_\{nachname\}.m'' benannt werden
-% (z.B. variablen\_datentypen\_mueller.m).
+% % (z.B. variablen\_datentypen\_mueller.m).
 
-\begin{itemize}
+% \begin{itemize}
-\item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass
+% \item \"Uberzeuge dich von jeder einzelnen Zeile deines Codes, dass
-  sie auch wirklich das macht, was sie machen soll! Teste dies mit
+%   sie auch wirklich das macht, was sie machen soll! Teste dies mit
-  kleinen Beispielen direkt in der Kommandozeile.
+%   kleinen Beispielen direkt in der Kommandozeile.
-\item Versuche die L\"osungen der Aufgaben m\"oglichst in
+% \item Versuche die L\"osungen der Aufgaben m\"oglichst in
-  sinnvolle kleine Funktionen herunterzubrechen.
+%   sinnvolle kleine Funktionen herunterzubrechen.
-  Sobald etwas \"ahnliches mehr als einmal berechnet werden soll,
+%   Sobald etwas \"ahnliches mehr als einmal berechnet werden soll,
-  lohnt es sich eine Funktion daraus zu schreiben!
+%   lohnt es sich eine Funktion daraus zu schreiben!
-\item Teste rechenintensive \code{for} Schleifen, Vektoren, Matrizen
+% \item Teste rechenintensive \code{for} Schleifen, Vektoren, Matrizen
-  zuerst mit einer kleinen Anzahl von Wiederholungen oder kleiner
+%   zuerst mit einer kleinen Anzahl von Wiederholungen oder kleiner
-  Gr\"o{\ss}e, und benutze erst am Ende, wenn alles \"uberpr\"uft
+%   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
+%   ist, eine gro{\ss}e Anzahl von Wiederholungen oder Elementen, um eine gute
-  Statistik zu bekommen.
+%   Statistik zu bekommen.
-\item Benutze die Hilfsfunktion von \code{matlab} (\code{help
+% \item Benutze die Hilfsfunktion von \code{matlab} (\code{help
-    commando} oder \code{doc commando}) und das Internet, um
+%     commando} oder \code{doc commando}) und das Internet, um
-  herauszufinden, wie bestimmte \code{matlab} Funktionen zu verwenden
+%   herauszufinden, wie bestimmte \code{matlab} Funktionen zu verwenden
-  sind und was f\"ur M\"oglichkeiten sie bieten.
+%   sind und was f\"ur M\"oglichkeiten sie bieten.
-  Auch zu inhaltlichen Konzepten bietet das Internet oft viele
+%   Auch zu inhaltlichen Konzepten bietet das Internet oft viele
-  Antworten!
+%   Antworten!
-\end{itemize}
+% \end{itemize}
 
-\fi
+% \fi

likelihood/exercises/mlepropest.m

@@ -0,0 +1,26 @@
+m = 2.0;               % slope
+sigmas = [0.1, 1.0];   % standard deviations
+ns = [100, 1000];      % number of data pairs
+trials = 1000;         % number of data sets
+
+for i = 1:length(sigmas)
+  sigma = sigmas(i);
+  for j = 1:length(ns)
+    n = ns(j);
+    slopes = zeros(trials, 1);
+    for k=1:trials
+      % data pairs:
+      x = 5.0*rand(n, 1);
+      y = m*x + sigma*randn(n, 1);
+      % fit:
+      slopes(k) = mleslope(x, y);
+    end
+    subplot(2, 2, 2*(i-1)+j);
+    bins = [1.9:0.005:2.1];
+    hist(slopes, bins);
+    xlabel('estimated slope');
+    title(sprintf('sigma=%g, n=%d', sigma, n));
+  end
+end
+
+savefigpdf(gcf, 'mlepropest.pdf', 15, 10);

likelihood/exercises/mlestd.m

@@ -1,7 +1,9 @@
-% draw random numbers:
-n = 50;
 mu = 3.0;
 sigma =2.0;
+ns = [50, 1000];
+for k = 1:length(ns)
+    n = ns(k);
+    % draw random numbers:
     x = randn(n,1)*sigma+mu;
     fprintf('              mean of the data is %.2f\n', mean(x))
     fprintf('standard deviation of the data is %.2f\n', std(x))
@@ -19,12 +21,15 @@ lm = prod(lms, 1);          % likelihood
     loglm = sum(log(lms), 1);   % log likelihood
 
     % plot likelihood of standard deviation:
-subplot(1, 2, 1);
+    subplot(2, 2, 2*k-1);
     plot(psigs, lm );
+    title(sprintf('likelihood n=%d', n));
     xlabel('standard deviation')
     ylabel('likelihood')
-subplot(1, 2, 2);
+    subplot(2, 2, 2*k);
     plot(psigs, loglm);
+    title(sprintf('log-likelihood n=%d', n));
     xlabel('standard deviation')
     ylabel('log likelihood')
-savefigpdf(gcf, 'mlestd.pdf', 15, 5);
+end
+savefigpdf(gcf, 'mlestd.pdf', 15, 10);

linearalgebra/code/coordinaterafo2.m

@@ -1,52 +0,0 @@
-% some vectors:
-x = [ -1:0.02:1 ];
-y = x*0.5 + 0.1*randn( size(x) );
-plot( x, y, '.b' );
-hold on
-plot( x(10), y(10), '.r' );
-
-% new coordinate system:
-%e1 = [ 3/5 4/5 ];
-%e2 =  [ -4/5 3/5 ];
-e1 = [ 3 4 ];
-e2 =  [ -4 3 ];
-me1 = sqrt( e1*e1' );
-e1 = e1/me1;
-me2 = sqrt( e2*e2' );
-e2 = e2/me2;
-quiver( 0.0, 0.0 , e1(1), e1(2), 1.0, 'r' )
-quiver( 0.0, 0.0 , e2(1), e2(2), 1.0, 'g' )
-axis( 'equal' )
-
-% project [x y] onto e1 and e2:
-
-% % the long way:
-% nx = zeros( size( x ) );  % new x coordinate
-% ny = zeros( size( y ) );  % new y coordinates
-% for k=1:length(x)
-%     xvec = [ x(k) y(k) ];
-%     nx(k) = xvec * e1';
-%     ny(k) = xvec * e2';
-% end
-% plot( nx, ny, '.g' );
-
-% the short way:
-%nx = [ x' y' ] * e1';
-%nx = [x; y]' * e1';
-% nx = e1 * [ x; y ];
-% ny = e2 * [ x; y ];
-% plot( nx, ny, '.g' );
-
-% even shorter:
-n = [e1; e2 ] * [ x; y ];
-plot( n(1,:), n(2,:), '.g' );
-
-
-
-
-
-
-
-
-
-hold off

linearalgebra/code/coordinatetrafo.m

@@ -2,8 +2,9 @@
 x=[-1:0.02:1]';  % column vector!
 y=4*x/3 + 0.1*randn( size( x ) );
 plot( x, y, '.b' );
-axis( 'equal' );
 hold on;
+plot( x(1), y(1), '.b' );
+axis( 'equal' );
 
 % new coordinate system:
 % e1 = [ 3 4 ]
@@ -35,3 +36,4 @@ xlabel( 'x' );
 ylabel( 'y' );
 
 hold off;
+pause;

linearalgebra/code/covareigen.m

@@ -38,11 +38,11 @@ function covareigen( x, y, pm, w, h )
         colormap( 'gray' );
     else
         % scatter plot:
-        scatter( x, y, 'b', 'filled', 'MarkerEdgeColor', 'white' );
+	    scatter( x, y ); %, 'b', 'filled', 'MarkerEdgeColor', 'white' );
     end
 
     % plot eigenvectors:
-    quiver( ones( 1, 2 ).*mean( x ), ones( 1, 2 )*mean(y), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 )
+    quiver( ones( 1, 2 ).*mean( x ), ones( 1, 2 )*mean(y), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 3 ); %, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 )
 
     xlabel( 'x' );
     ylabel( 'y' );
@@ -76,7 +76,7 @@ function covareigen( x, y, pm, w, h )
             contourf( xp, yp, gauss )
         end
     else
-        scatter( nc(:,1), nc(:,2), 'b', 'filled', 'MarkerEdgeColor', 'white' );
+      scatter( nc(:,1), nc(:,2) ); %, 'b', 'filled', 'MarkerEdgeColor', 'white' );
     end
     xlabel( 'x' );
     ylabel( 'y' );

linearalgebra/code/covareigen3.m

@@ -15,29 +15,29 @@ function covareigen3( x, y, z )
 
     % scatter plot:
     view( 3 );
-    scatter3( x, y, z, 0.1, 'b', 'filled', 'MarkerEdgeColor', 'blue' );
+scatter3( x, y, z, 0.1, 'b', 'filled' ); %, 'MarkerEdgeColor', 'blue' );
     xlabel( 'x' );
     ylabel( 'y' );
     zlabel( 'z' );
     grid on;
     
     % plot eigenvectors:
-    quiver3( ones( 1, 3 ).*mean( x ), ones( 1, 3 )*mean(y), ones( 1, 3 )*mean(z), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', v(3,:).*sqrt(diag(d))', 'r', 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 )
+    quiver3( ones( 1, 3 ).*mean( x ), ones( 1, 3 )*mean(y), ones( 1, 3 )*mean(z), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', v(3,:).*sqrt(diag(d))', 'r' ); %, 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 )
     
     %axis( 'equal' );
     hold off;
 
     % 2D scatter plots:
     subplot( 2, 6, 4 );
-    scatter( x, y, 'b', 'filled', 'MarkerEdgeColor', 'white' );
+    scatter( x, y, 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     xlabel( 'x' )
     ylabel( 'y' )
     subplot( 2, 6, 5 );
-    scatter( x, z, 'b', 'filled', 'MarkerEdgeColor', 'white' );
+    scatter( x, z, 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     xlabel( 'x' )
     ylabel( 'z' )
     subplot( 2, 6, 6 );
-    scatter( y, z, 'b', 'filled', 'MarkerEdgeColor', 'white' );
+    scatter( y, z, 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     xlabel( 'y' )
     ylabel( 'z' )
 
@@ -51,7 +51,7 @@ function covareigen3( x, y, z )
     y = y - mean( y );
     % project onto eigenvectors:
     nx = [ x y z ] * v(:,inx);
-    scatter( nx(:,1), nx(:,2), 'b', 'filled', 'MarkerEdgeColor', 'white' );
+    scatter( nx(:,1), nx(:,2), 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     xlabel( 'ex' )
     ylabel( 'ey' )
     axis( 'equal' );

linearalgebra/code/covareigen3examples.m

@@ -1,6 +1,3 @@
-scrsz = get( 0, 'ScreenSize' );
-set( 0, 'DefaultFigurePosition', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ] );
-
 n = 10000;
 
 % three distributions:
@@ -17,3 +14,4 @@ for k = 1:4
 end
 f = figure( 1 );
 covareigen3( x, y, z );
+pause;

linearalgebra/code/covareigenexamples.m

@@ -1,6 +1,3 @@
-scrsz = get( 0, 'ScreenSize' );
-set( 0, 'DefaultFigurePosition', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ] );
-
 % correlation coefficients:
 n = 10000;
 x = randn( n, 1 );
@@ -10,9 +7,9 @@ for r = 0.01:0.19:1
     clf( f );
     y = r*x + sqrt(1-r^2)*randn( n, 1 );
     covareigen( x, y, 0, 5.0, 3.0 );
-    key = waitforbuttonpress;
+    %key = waitforbuttonpress;
+    pause( 1.0 );
 end
-return
 
 % two distributions:
 n = 10000;
@@ -21,14 +18,12 @@ y1 = randn( n/2, 1 );
 x2 = randn( n/2, 1 );
 y2 = randn( n/2, 1 );
 f = figure( 1 );
-pause( 'on' );
 for d = 0:1:5
     fprintf( 'Distance = %g\n', d );
     clf( f  );
     d2 = d / sqrt( 2.0 );
     x = [ x1; x2 ];
     y = [ y1+d2; y2-d2 ];
-    scrsz = get(0,'ScreenSize');
     covareigen( x, y, 0, 10.0, 7.0 );
     %key = waitforbuttonpress;
     pause( 1.0 );

linearalgebra/code/covarmatrix.m

@@ -0,0 +1,28 @@
+% covariance in 2D:
+x = 2.0*randn(1000, 1);
+y = 0.5*x + 0.2*randn(1000, 1);
+scatter(x, y)
+var(x)
+var(y)
+cov( [x, y] )
+pause(1.0)
+
+% covariance of independent coordinates in 2D:
+x = 2.0*randn(1000, 1);
+y = 0.5*randn(1000, 1);
+scatter(x, y)
+var(x)
+var(y)
+cov( [x, y] )
+pause(1.0)
+
+% covariance in 3D:
+x = 2.0*randn(1000, 1);
+y = -0.5*x + 0.2*randn(1000, 1);
+z = 2.0*x + 0.6*y + 2.0*randn(1000, 1);
+scatter3(x, y, z, 'filled')
+var(x)
+var(y)
+var(z)
+cov( [x, y, z] )
+pause()

linearalgebra/code/covarmatrixeigen.m

@@ -0,0 +1,36 @@
+% covariance in 2D:
+x = 2.0*randn(1000, 1);
+y = 0.5*x + 0.6*randn(1000, 1);
+scatter(x, y)
+cv = cov( [x, y] );
+[v d] = eig(cv);
+hold on
+quiver([0 0], [0 0], v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 2);
+axis( 'equal' );
+hold off
+pause
+
+% covariance of independent coordinates in 2D:
+x = 1.5*randn(1000, 1);
+y = 0.8*randn(1000, 1);
+scatter(x, y)
+cv = cov( [x, y] );
+[v d] = eig(cv);
+hold on
+quiver([0 0], [0 0], v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 2);
+axis( 'equal' );
+hold off
+pause
+
+% covariance in 3D:
+x = 2.0*randn(1000, 1);
+y = -0.5*x + 0.2*randn(1000, 1);
+z = 2.0*x + 0.6*y + 2.0*randn(1000, 1);
+scatter3(x, y, z)
+cv = cov( [x, y, z] );
+[v d] = eig(cv);
+hold on
+quiver3([0 0 0], [0 0 0], [0 0 0], v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', v(3,:).*sqrt(diag(d))', 'r', 'LineWidth', 2);
+axis( 'equal' );
+hold off
+pause()

linearalgebra/code/matrixbox.m

@@ -41,5 +41,5 @@ function matrixbox( m, s )
     text( 0.8, 0.1, sprintf( '%.3g', m(2,1) ), 'Units', 'normalized' )
     text( 0.9, 0.1, sprintf( '%.3g', m(2,2) ), 'Units', 'normalized' )
     title( s );
-    waitforbuttonpress;
+    pause(1.0);
 end

linearalgebra/code/pca2d.m

@@ -39,11 +39,11 @@ function pca2d( x, y, pm, w, h )
         colormap( 'gray' );
     else
         % scatter plot:
-        scatter( x, y, 'b', 'filled', 'MarkerEdgeColor', 'white' );
+	    scatter( x, y, 'b', 'filled' );%, 'MarkerEdgeColor', 'white' );
     end
 
     % plot eigenvectors:
-    quiver( ones( 1, 2 ).*mean( x ), ones( 1, 2 )*mean(y), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 )
+    quiver( ones( 1, 2 ).*mean( x ), ones( 1, 2 )*mean(y), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r' ); %, 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 )
 
     xlabel( 'x' );
     ylabel( 'y' );
@@ -80,7 +80,7 @@ function pca2d( x, y, pm, w, h )
         end
     else
         % scatter plot:
-        scatter( nc(:,1), nc(:,2), 'b', 'filled', 'MarkerEdgeColor', 'white' );
+	    scatter( nc(:,1), nc(:,2), 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     end
     xlabel( 'n_x' );
     ylabel( 'n_y' );

linearalgebra/code/pca2dexamples.m

@@ -1,6 +1,3 @@
-scrsz = get( 0, 'ScreenSize' );
-set( 0, 'DefaultFigurePosition', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ] );
-
 % correlation coefficients:
 n = 10000;
 x = randn( n, 1 );
@@ -10,7 +7,7 @@ for r = 0.01:0.19:1
     clf( f );
     y = r*x + sqrt(1-r^2)*randn( n, 1 );
     pca2d( x, y, 0, 5.0, 3.0 );
-    waitforbuttonpress;
+    pause( 1.0 );
 end
 
 % two distributions:
@@ -20,14 +17,11 @@ y1 = randn( n/2, 1 );
 x2 = randn( n/2, 1 );
 y2 = randn( n/2, 1 );
 f = figure( 1 );
-pause( 'on' );
 for d = 0:1:5
     fprintf( 'Distance = %g\n', d );
     clf( f  );
-    d2 = d / sqrt( 2.0 );
+    x = [ x1+d; x2-d ];
-    x = [ x1; x2 ];
+    y = [ y1+d; y2-d ];
-    y = [ y1+d2; y2-d2 ];
-    scrsz = get(0,'ScreenSize');
     pca2d( x, y, 0, 10.0, 7.0 );
-    waitforbuttonpress;
+    pause( 1.0 );
 end

linearalgebra/code/pca3d.m

@@ -15,29 +15,29 @@ function pca3d( x, y, z )
 
     % scatter plot:
     view( 3 );
-    scatter3( x, y, z, 0.1, 'b', 'filled', 'MarkerEdgeColor', 'blue' );
+scatter3( x, y, z ); % , 1.0, 'b', 'filled' ); %, 'MarkerEdgeColor', 'blue' );
     xlabel( 'x' );
     ylabel( 'y' );
     zlabel( 'z' );
     grid on;
     
     % plot eigenvectors:
-    quiver3( ones( 1, 3 ).*mean( x ), ones( 1, 3 )*mean(y), ones( 1, 3 )*mean(z), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', v(3,:).*sqrt(diag(d))', 'r', 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 )
+    quiver3( ones( 1, 3 ).*mean( x ), ones( 1, 3 )*mean(y), ones( 1, 3 )*mean(z), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', v(3,:).*sqrt(diag(d))', 'r' ); %, 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 )
     
     %axis( 'equal' );
     hold off;
 
     % 2D scatter plots:
     subplot( 2, 6, 4 );
-    scatter( x, y, 'b', 'filled', 'MarkerEdgeColor', 'white' );
+    scatter( x, y, 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     xlabel( 'x' )
     ylabel( 'y' )
     subplot( 2, 6, 5 );
-    scatter( x, z, 'b', 'filled', 'MarkerEdgeColor', 'white' );
+    scatter( x, z, 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     xlabel( 'x' )
     ylabel( 'z' )
     subplot( 2, 6, 6 );
-    scatter( y, z, 'b', 'filled', 'MarkerEdgeColor', 'white' );
+    scatter( y, z, 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     xlabel( 'y' )
     ylabel( 'z' )
 
@@ -51,10 +51,11 @@ function pca3d( x, y, z )
     y = y - mean( y );
     % project onto eigenvectors:
     nx = [ x y z ] * v(:,inx);
-    scatter( nx(:,1), nx(:,2), 'b', 'filled', 'MarkerEdgeColor', 'white' );
+    scatter( nx(:,1), nx(:,2), 'b', 'filled' ); %, 'MarkerEdgeColor', 'white' );
     xlabel( 'ex' )
     ylabel( 'ey' )
     axis( 'equal' );
     hold off;
 
 end
+

linearalgebra/code/pca3dexamples.m

@@ -1,6 +1,3 @@
-scrsz = get( 0, 'ScreenSize' );
-set( 0, 'DefaultFigurePosition', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ] );
-
 n = 10000;
 
 % three distributions:
@@ -17,3 +14,4 @@ for k = 1:4
 end
 f = figure( 1 );
 pca3d( x, y, z );
+pause

linearalgebra/code/simplematrixbox.m

@@ -17,5 +17,5 @@ function simplematrixbox( a, s )
     xlim( [-2 2 ] )
     ylim( [-2 2] )
     title( s );
-    waitforbuttonpress;
+    pause(1.0);
 end

linearalgebra/code/spikesortingwave.m

@@ -5,21 +5,22 @@ load( 'extdata' );
 dt = time( 2) - time(1);
 tinx = round(spiketimes/dt)+1;
 
-% plot voltage trace with dettected spikes:
+% plot voltage trace with detected spikes:
 figure( 1 );
 clf;
 plot( time, voltage, '-b' )
 hold on
 scatter( time(tinx), voltage(tinx), 'r', 'filled' );
-xlabel( 'time [ms]' );
+xlabel( 'time [s]' );
 ylabel( 'voltage' );
+xlim([0.1, 0.4])
 hold off
 
 % spike waveform snippets:
 w = ceil( 0.005/dt );
-vs = [];
+vs = zeros(length(tinx), 2*w);
 for k=1:length(tinx)
-    vs = [ vs; voltage(tinx(k)-w:tinx(k)+w-1) ];
+    vs(k,:) = voltage(tinx(k)-w:tinx(k)+w-1);
 end
 ts = time(1:size(vs,2));
 ts = ts - ts(floor(length(ts)/2));
@@ -78,9 +79,9 @@ kx = ones( size( nx ) );
 kx(nx<kthresh) = 2;
 
 subplot( 4, 1, 2 );
-scatter( nx(kx==1), ny(kx==1), 'r', 'filled', 'MarkerEdgeColor', 'white' );
+scatter( nx(kx==1), ny(kx==1), 10.0, 'r', 'filled' ); %, 'MarkerEdgeColor', 'white' );
 hold on;
-scatter( nx(kx==2), ny(kx==2), 'g', 'filled', 'MarkerEdgeColor', 'white' );
+scatter( nx(kx==2), ny(kx==2), 10.0, 'g', 'filled' ); %, 'MarkerEdgeColor', 'white' );
 hold off;
 xlabel( 'projection onto eigenvector 1' );
 ylabel( 'projection onto eigenvector 2' );
@@ -113,6 +114,29 @@ hold off
 % spike trains:
 figure( 1 );
 hold on;
-scatter( time(tinx(kinx1)), voltage(tinx(kinx1)), 100.0, 'r', 'filled' );
+scatter( time(tinx(kinx1)), voltage(tinx(kinx1)), 10.0, 'r', 'filled' );
-scatter( time(tinx(kinx2)), voltage(tinx(kinx2)), 100.0, 'g', 'filled' );
+scatter( time(tinx(kinx2)), voltage(tinx(kinx2)), 10.0, 'g', 'filled' );
 hold off;
+
+% ISIs:
+figure(4);
+clf;
+subplot(1, 3, 1)
+allisis = diff(spiketimes); 
+hist(allisis, 20);
+title('all spikes')
+xlabel('ISI [ms]')
+subplot(1, 3, 2)
+spikes1 = time(tinx(kinx1));
+isis1 = diff(spikes1); 
+hist(isis1, 20);
+title('neuron 1')
+xlabel('ISI [ms]')
+subplot(1, 3, 3)
+spikes2 = time(tinx(kinx2));
+isis2 = diff(spikes2); 
+hist(isis2, 20);
+title('neuron 2')
+xlabel('ISI [ms]')
+pause
+

linearalgebra/exercises/Makefile

@@ -0,0 +1,34 @@
+TEXFILES=$(wildcard exercises??.tex)
+EXERCISES=$(TEXFILES:.tex=.pdf)
+SOLUTIONS=$(EXERCISES:exercises%=solutions%)
+
+.PHONY: pdf exercises solutions watch watchexercises watchsolutions clean
+
+pdf : $(SOLUTIONS) $(EXERCISES)
+
+exercises : $(EXERCISES)
+
+solutions : $(SOLUTIONS)
+
+$(SOLUTIONS) : solutions%.pdf : exercises%.tex instructions.tex
+	{ echo "\\documentclass[answers,12pt,a4paper,pdftex]{exam}"; sed -e '1d' $<; } > $(patsubst %.pdf,%.tex,$@)
+	pdflatex -interaction=scrollmode $(patsubst %.pdf,%.tex,$@) | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $(patsubst %.pdf,%.tex,$@) || true
+	rm $(patsubst %.pdf,%,$@).[!p]*
+
+$(EXERCISES) : %.pdf : %.tex instructions.tex
+	pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
+
+watch :
+	while true; do ! make -q pdf && make pdf; sleep 0.5; done
+
+watchexercises :
+	while true; do ! make -q exercises && make exercises; sleep 0.5; done
+
+watchsolutions :
+	while true; do ! make -q solutions && make solutions; sleep 0.5; done
+
+clean :
+	rm -f *~ *.aux *.log *.out
+
+cleanup : clean
+	rm -f $(SOLUTIONS) $(EXERCISES)

linearalgebra/exercises/covariance.m

@@ -0,0 +1,19 @@
+n = 1000;
+x = 2.0*randn(n, 1);
+z = 3.0*randn(n, 1);
+
+rs = [-1.0:0.25:1.0];
+for k = 1:length(rs)
+    r = rs(k);
+    y = r*x + sqrt(1.0-r^2)*z;
+    r
+    cv = cov([x y])
+    corrcoef([x y])
+    subplot(3, 3, k)
+    scatter(x, y, 'filled', 'MarkerEdgeColor', 'white')
+    title(sprintf('r=%g', r))
+    xlim([-10, 10])
+    ylim([-20, 20])
+    text(-8, -15, sprintf('cov(x,y)=%.2f', cv(1, 2)))
+end
+savefigpdf(gcf, 'covariance.pdf', 20, 20);

linearalgebra/exercises/exercises01.tex

@@ -0,0 +1,244 @@
+\documentclass[12pt,a4paper,pdftex]{exam}
+
+\usepackage[english]{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\stitle}}{{\bfseries\large PCA}}{{\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{Covariance and correlation coefficient\vspace{-3ex}}
+  \begin{parts}
+    \part Generate two vectors $x$ and $z$ with $n=1000$ Gausian distributed random numbers.
+    \part Compute $y$ as a linear combination of $x$ and $z$ according to
+    \[ y = r \cdot x + \sqrt{1-r^2}\cdot z \]
+    where $r$ is a parameter $-1 \le r \le 1$.
+    What does $r$ do?
+    \part Plot a scatter plot of $y$ versus $x$ for about 10 different values of $r$.
+    What do you observe?
+    \part Also compute the covariance matrix and the correlation
+    coefficient matrix between $x$ and $y$ (functions \texttt{cov} and
+    \texttt{corrcoef}). How do these matrices look like for different
+    values of $r$? How do the values of the matrices change if you generate
+    $x$ and $z$ with larger variances?
+    \begin{solution}
+      \lstinputlisting{covariance.m}
+      \includegraphics[width=0.8\textwidth]{covariance}
+    \end{solution}
+    \part Do the same analysis (scatter plot, covariance, and correlation coefficient)
+    for \[ y = x^2 + 0.5 \cdot z \]
+    Are $x$ and $y$ really independent?
+    \begin{solution}
+      \lstinputlisting{nonlinearcorrelation.m}
+      \includegraphics[width=0.8\textwidth]{nonlinearcorrelation}
+    \end{solution}
+  \end{parts}
+
+  \question \qt{Principal component analysis in 2D\vspace{-3ex}}
+  \begin{parts}
+    \part Generate $n=1000$ pairs $(x,y)$ of Gaussian distributed random numbers such
+    that all $x$ values have zero mean, half of the $y$ values have mean $+d$
+    and the other half mean $-d$, with $d \ge0$.
+    \part Plot scatter plots of the pairs $(x,y)$ for $d=0$, 1, 2, 3, 4 and 5.
+    Also plot a histogram of the $x$ values.
+    \part Apply PCA on the data and plot a histogram of the data projected onto
+    the PCA axis with the largest eigenvalue.
+    What do you observe?
+    \begin{solution}
+      \lstinputlisting{pca2d.m}
+      \includegraphics[width=0.8\textwidth]{pca2d-2}
+    \end{solution}
+  \end{parts}
+
+  \newsolutionpage
+  \question \qt{Principal component analysis in 3D\vspace{-3ex}}
+  \begin{parts}
+    \part Generate $n=1000$ triplets $(x,y,z)$ of Gaussian distributed random numbers such
+    that all $x$ values have zero mean, half of the $y$ and $z$ values have mean $+d$
+    and the other half mean $-d$, with $d \ge0$.
+    \part Plot 3D scatter plots of the pairs $(x,y)$ for $d=0$, 1, 2, 3, 4 and 5.
+    Also plot a histogram of the $x$ values.
+    \part Apply PCA on the data and plot a histogram of the data projected onto
+    the PCA axis with the largest eigenvalue.
+    What do you observe?
+    \begin{solution}
+      \lstinputlisting{pca3d.m}
+      \includegraphics[width=0.8\textwidth]{pca3d-2}
+    \end{solution}
+  \end{parts}
+
+  \continuepage
+  \question \qt{Spike sorting} 
+  Extracellular recordings often pick up action potentials originating
+  from more than a single neuron. In case the waveforms of the action
+  potentials differ between the neurons one could assign each action
+  potential to the neuron it originated from. This process is called
+  ``spike sorting''. Here we explore this method on a simulated
+  recording that contains action potentials from two different
+  neurons.
+  \begin{parts}
+    \part Load the data from the file \texttt{extdata.mat}. This file
+    contains the voltage trace of the recording (\texttt{voltage}),
+    the corresponding time vector in seconds (\texttt{time}), and a
+    vector containing the times of the peaks of detected action
+    potentials (\texttt{spiketimes}). Further, and in contrast to real
+    data, the waveforms of the actionpotentials of the two neurons
+    (\texttt{waveform1} and \texttt{waveform2}) and the corresponding
+    time vector (\texttt{waveformt}) are also contained in the file.
+
+    \part Plot the voltage trace and mark the peaks of the detected
+    action potentials (using \texttt{spiketimes}). Zoom into the plot
+    and look whether you can differentiate between two different
+    waveforms of action potentials. How do they differ?
+
+    \part Cut out the waveform of each action potential (5\,ms before
+    and after the peak).  Plot all these snippets in a single
+    plot. Can you differentiate the two actionpotential waveforms?
+    \begin{solution}
+      \mbox{}\\[-3ex]\hspace*{5em}
+      \includegraphics[width=0.8\textwidth]{spikesorting1}
+    \end{solution}
+
+    \newsolutionpage
+    \part Apply PCA on the waveform snippets. That is compute the
+    eigenvalues and eigenvectors of their covariance matrix, which is
+    a $n \times n$ matrix, with $n$ being the number of data points
+    contained in a single waveform snippet. Plot the sorted
+    eigenvalues (the ``eigenvalue spectrum''). How many eigenvalues
+    are clearly larger than zero?
+    \begin{solution}
+      \mbox{}\\[-3ex]\hspace*{5em}
+      \includegraphics[width=0.8\textwidth]{spikesorting2}
+    \end{solution}
+
+    \part Plot the two eigenvectors (``features'') with the two
+    largest eigenvalues as a function of time.
+    \begin{solution}
+      \mbox{}\\[-3ex]\hspace*{5em}
+      \includegraphics[width=0.8\textwidth]{spikesorting3}
+    \end{solution}
+
+    \part Project the waveform snippets onto these two eigenvectors
+    and display them with a scatter plot. What do you observe? Can you
+    imagine how to separate two ``clouds'' of data points
+    (``clusters'')?
+
+    \newsolutionpage
+    \part Think about a very simply way how to separate the two
+    clusters.  Generate a vector whose elements label the action
+    potentials, e.g. that contains '1' for all snippets belonging to
+    the one cluster and '2' for the waveforms of the other
+    cluster. Use this vector to mark the two clusters in the previous
+    plot with two different colors.
+    \begin{solution}
+      \mbox{}\\[-3ex]\hspace*{5em}
+      \includegraphics[width=0.8\textwidth]{spikesorting4}
+    \end{solution}
+
+    \part Plot the waveform snippets of each cluster together with the
+    true waveform obtained from the data file. Do they match?
+    \begin{solution}
+      \mbox{}\\[-3ex]\hspace*{5em}
+      \includegraphics[width=0.8\textwidth]{spikesorting5}
+    \end{solution}
+
+    \newsolutionpage
+    \part Mark the action potentials in the recording according to
+    their cluster identity.
+    \begin{solution}
+      \mbox{}\\[-3ex]\hspace*{5em}
+      \includegraphics[width=0.8\textwidth]{spikesorting6}
+    \end{solution}
+
+    \part Compute interspike-interval histograms of all the (unsorted)
+    action potentials, and of each of the two neurons. What do they
+    tell you?
+    \begin{solution}
+      \mbox{}\\[-3ex]\hspace*{5em}
+      \includegraphics[width=0.8\textwidth]{spikesorting7}
+      \lstinputlisting{spikesorting.m}
+    \end{solution}
+  \end{parts}
+
+\end{questions}
+
+\end{document}

linearalgebra/exercises/instructions.tex

@@ -0,0 +1,6 @@
+\vspace*{-7.8ex}
+\begin{center}
+\textbf{\Large Introduction to Scientific Computing}\\[2.3ex]
+{\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}

linearalgebra/exercises/nonlinearcorrelation.m

@@ -0,0 +1,9 @@
+n = 1000;
+x = randn(n, 1);
+z = randn(n, 1);
+y = x.^2 + 0.5*z;
+scatter(x, y)
+cov([x y])
+r = corrcoef([x y])
+text(-2, 8, sprintf('r=%.2f', r(1, 2)))
+savefigpdf(gcf, 'nonlinearcorrelation.pdf', 15, 10);

linearalgebra/exercises/pca2d.m

@@ -0,0 +1,39 @@
+n = 1000;
+ds = [0:5];
+for k = 1:length(ds)
+    d = ds(k);
+    % generate data:
+    x = randn(n, 1);
+    y = randn(n, 1);
+    y(1:n/2) = y(1:n/2) - d;
+    y(1+n/2:end) = y(1+n/2:end) + d;
+    % scatter plot of data:
+    subplot(2, 2, 1)
+    scatter(x, y, 'filled', 'MarkerEdgeColor', 'white')
+    title(sprintf('d=%g', d))
+    xlabel('x')
+    ylabel('y')
+    % histogram of data:
+    subplot(2, 2, 3)
+    hist(x, 20)
+    xlabel('x')
+    % pca:
+    cv = cov([x y]);
+    [ev en] = eig(cv);
+    [en inx] = sort(diag(en), 'descend');
+    nc = [x y]*ev(:,inx);
+    % scatter in new coordinates:
+    subplot(2, 2, 2)
+    scatter(nc(:, 1), nc(:, 2), 'filled', 'MarkerEdgeColor', 'white')
+    title(sprintf('d=%g', d))
+    xlabel('pca 1')
+    ylabel('pca 2')
+    % histogram of data:
+    subplot(2, 2, 4)
+    hist(nc(:, 1), 20)
+    xlabel('pca 1')
+    if d == 2
+        savefigpdf(gcf, 'pca2d-2.pdf', 15, 15);
+    end
+    pause(1.0)
+end

linearalgebra/exercises/pca3d.m

@@ -0,0 +1,44 @@
+n = 1000;
+ds = [0:5];
+for k = 1:length(ds)
+    d = ds(k);
+    % generate data:
+    x = randn(n, 1);
+    y = randn(n, 1);
+    z = randn(n, 1);
+    y(1:n/2) = y(1:n/2) - d;
+    y(1+n/2:end) = y(1+n/2:end) + d;
+    z(1:n/2) = z(1:n/2) - d;
+    z(1+n/2:end) = z(1+n/2:end) + d;
+    % scatter plot of data:
+    subplot(2, 2, 1)
+    scatter3(x, y, z, 'filled', 'MarkerEdgeColor', 'white')
+    title(sprintf('d=%g', d))
+    xlabel('x')
+    ylabel('y')
+    zlabel('z')
+    % histogram of data:
+    subplot(2, 2, 3)
+    hist(x, 20)
+    xlabel('x')
+    % pca:
+    cv = cov([x y z]);
+    [ev en] = eig(cv);
+    [en inx] = sort(diag(en), 'descend');
+    nc = [x y z]*ev(:,inx);
+    % scatter in new coordinates:
+    subplot(2, 2, 2)
+    scatter3(nc(:, 1), nc(:, 2), nc(:, 3), 'filled', 'MarkerEdgeColor', 'white')
+    title(sprintf('d=%g', d))
+    xlabel('pca 1')
+    ylabel('pca 2')
+    zlabel('pca 3')
+    % histogram of data:
+    subplot(2, 2, 4)
+    hist(nc(:, 1), 20)
+    xlabel('pca 1')
+    if d == 2
+        savefigpdf(gcf, 'pca3d-2.pdf', 15, 15);
+    end
+    pause(1.0)
+end

linearalgebra/exercises/savefigpdf.m

@@ -0,0 +1,28 @@
+function savefigpdf(fig, name, width, height)
+% Saves figure fig in pdf file name.pdf with appropriately set page size
+% and fonts
+
+% default width:
+if nargin < 3
+    width = 11.7;
+end
+% default height:
+if nargin < 4
+    height = 9.0;
+end
+
+% paper:
+set(fig, 'PaperUnits', 'centimeters');
+set(fig, 'PaperSize', [width height]);
+set(fig, 'PaperPosition', [0.0 0.0 width height]);
+set(fig, 'Color', 'white')
+
+% font:
+set(findall(fig, 'type', 'axes'), 'FontSize', 12)
+set(findall(fig, 'type', 'text'), 'FontSize', 12)
+
+% save:
+saveas(fig, name, 'pdf')
+
+end
+

linearalgebra/exercises/spikesorting.m

@@ -0,0 +1,161 @@
+% load data into time, voltage and spiketimes
+x = load('extdata');
+time = x.time;
+voltage = x.voltage;
+spiketimes = x.spiketimes;
+waveformt = x.waveformt;
+waveform1 = x.waveform1;
+waveform2 = x.waveform2;
+
+% indices into voltage trace of spike times:
+dt = time(2) - time(1);
+tinx = round(spiketimes/dt) + 1;
+
+% plot voltage trace with detected spikes:
+figure(1);
+plot(time, voltage, '-b')
+hold on
+scatter(time(tinx), voltage(tinx), 'r', 'filled');
+xlabel('time [s]');
+ylabel('voltage');
+xlim([0.1, 0.4])  % zoom in
+hold off
+
+% spike waveform snippets:
+snippetwindow = 0.005;  % milliseconds
+w = ceil(snippetwindow/dt);
+vs = zeros(length(tinx), 2*w);
+for k=1:length(tinx)
+    vs(k,:) = voltage(tinx(k)-w:tinx(k)+w-1);
+end
+% time axis for snippets:
+ts = time(1:size(vs,2));
+ts = ts - ts(floor(length(ts)/2));
+
+% plot all snippets:
+figure(2);
+hold on
+plot(1000.0*ts, vs, '-b');
+title('spike snippets')
+xlabel('time [ms]');
+ylabel('voltage');
+hold off
+savefigpdf(gcf, 'spikesorting1.pdf', 12, 6);
+
+% pca:
+cv = cov(vs);
+[ev , ed] = eig(cv);
+[d, dinx] = sort(diag(ed), 'descend');
+
+% features:
+figure(2)
+subplot(1, 2, 1);
+plot(1000.0*ts, ev(:,dinx(1)), 'r', 'LineWidth', 2);
+xlabel('time [ms]');
+ylabel('eigenvector 1');
+subplot(1, 2, 2);
+plot(1000.0*ts, ev(:,dinx(2)), 'g', 'LineWidth', 2);
+xlabel('time [ms]');
+ylabel('eigenvector 2');
+savefigpdf(gcf, 'spikesorting2.pdf', 12, 6);
+
+% plot covariance matrix:
+figure(3);
+subplot(1, 2, 1);
+imagesc(cv);
+xlabel('time bin');
+ylabel('time bin');
+title('covariance matrix');
+caxis([-0.1 0.1])
+
+% spectrum of eigenvalues:
+subplot(1, 2, 2);
+scatter(1:length(d), d, 'b', 'filled');
+title('spectrum');
+xlabel('index');
+ylabel('eigenvalue');
+
+savefigpdf(gcf, 'spikesorting3.pdf', 12, 6);
+
+% project onto eigenvectors:
+nx = vs * ev(:,dinx(1));
+ny = vs * ev(:,dinx(2));
+
+% clustering (two clusters):
+%kx = kmeans([nx, ny], 2);
+% nx smaller or greater a threshold:
+kthresh = 1.6;
+kx = ones(size(nx));
+kx(nx<kthresh) = 2;
+
+% plot pca coordinates:
+figure(4)
+scatter(nx(kx==1), ny(kx==1), 'r', 'filled');
+hold on;
+scatter(nx(kx==2), ny(kx==2), 'g', 'filled');
+hold off;
+xlabel('projection onto eigenvector 1');
+ylabel('projection onto eigenvector 2');
+
+savefigpdf(gcf, 'spikesorting4.pdf', 12, 10);
+
+% show sorted spike waveforms:
+figure(5)
+subplot(1, 2, 1);
+hold on
+plot(1000.0*ts, vs(kx==1,:), '-r');
+plot(1000.0*waveformt, waveform2, '-k', 'LineWidth', 2);
+xlim([1000.0*ts(1) 1000.0*ts(end)])
+xlabel('time [ms]');
+ylabel('waveform 1');
+hold off
+
+subplot(1, 2, 2);
+hold on
+plot(1000.0*ts, vs(kx==2,:), '-g');
+plot(1000.0*waveformt, waveform1, '-k', 'LineWidth', 2);
+xlim([1000.0*ts(1) 1000.0*ts(end)])
+xlabel('time [ms]');
+ylabel('waveform 2');
+hold off
+savefigpdf(gcf, 'spikesorting5.pdf', 12, 6);
+
+% mark neurons in recording:
+figure(1);
+hold on;
+scatter(time(tinx(kinx1)), voltage(tinx(kinx1)), 'r', 'filled');
+scatter(time(tinx(kinx2)), voltage(tinx(kinx2)), 'g', 'filled');
+hold off;
+
+savefigpdf(gcf, 'spikesorting6.pdf', 12, 6);
+
+% ISIs:
+figure(7);
+subplot(1, 3, 1)
+allisis = diff(spiketimes); 
+bins = [0:0.005:0.2];
+[h, b] = hist(allisis, bins);
+bar(1000.0*b, h/sum(h)/mean(diff(b)))
+title('all spikes')
+xlabel('ISI [ms]')
+xlim([0, 200.0])
+
+subplot(1, 3, 2)
+spikes1 = time(tinx(kx==1));
+isis1 = diff(spikes1); 
+[h, b] = hist(isis1, bins);
+bar(1000.0*b, h/sum(h)/mean(diff(b)))
+title('neuron 1')
+xlabel('ISI [ms]')
+xlim([0, 200.0])
+
+subplot(1, 3, 3)
+spikes2 = time(tinx(kx==2));
+isis2 = diff(spikes2); 
+[h, b] = hist(isis2, bins);
+bar(1000.0*b, h/sum(h)/mean(diff(b)))
+title('neuron 2')
+xlabel('ISI [ms]')
+xlim([0, 200.0])
+
+savefigpdf(gcf, 'spikesorting7.pdf', 12, 6);

projects/README

@@ -1,70 +1,92 @@
 project_adaptation_fit
-OK 
+OK, medium
+Add plotting of cost function
 
 project_eod
-Needs to be checked!
+OK, medium - difficult
 
 project_eyetracker
-OK
+OK, difficult
+no statistics, but kmeans
 
 project_fano_slope
-OK
+OK, difficult
 
 project_fano_test
-OK
+OK -
 
 project_fano_time
-OK
+OK, medium-difficult
 
 project_ficurves
-OK
+OK, medium
+Maybe add correlation test or fit statistics
 
 project_input_resistance
-What is the problem with this project?
+medium
+What is the problem with this project? --> No difference between segments
+Improve questions
 
 project_isicorrelations
-Need to program a solution!
+medium-difficult
+Need to finish solution
 
 project_isipdffit
 Too technical
 
 project_lif
-OK
+OK, difficult
+no statistics
 
 project_mutualinfo
-OK
+OK, medium
 
 project_noiseficurves
-OK
+OK, simple-medium
+no statistics
 
 project_numbers
+simple
 We might add some more involved statistical analysis
 
 project_pca_natural_images
-Needs PCA...
+medium
+Make a solution (->Lukas)
 
 project_photoreceptor
-OK - text needs to be improved
+OK, simple
-Maybe also add how responses are influenced by unstable resting potential
-Maybe more cells...
 
 project_populationvector
+difficult
 OK
 
 project_qvalues
+-
 Interesting! But needs solution.
 
 project_random_walk
+simple-medium
 Improve it! Provide code exmaples for plotting world and making movies
 
 project_serialcorrelation
-OK
+OK, simple-medium
 
 project_spectra
+-
 Needs improvements and a solution
 
 project_stimulus_reconstruction
-OK Fix equation?
+OK, difficult
+Add specific hints for statistics
 
 project_vector_strength
-OK Maybe add something for explainiong the vector strength (average unit vector).
+OK, medium-difficult
+Maybe add something for explaining the vector strength (average unit vector).
+Check text (d)
+Add statisitcs for (e)
+
+Peter power:
+medium
+
+Marius monkey data:
+medium-difficult

projects/evaluation.tex

@@ -17,14 +17,14 @@
 \begin{document}
 
 \sffamily
-\section*{Scientific computing WS17/18}
+\section*{Scientific computing WS18/19}
 
 \noindent
 \begin{tabular}{|p{0.16\textwidth}|p{0.1\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|p{0.024\textwidth}|l|l|}
 \hline
 Name & Project & \multicolumn{5}{l|}{Project} & \multicolumn{3}{l|}{Figures} & \multicolumn{8}{l|}{Code} & Sum & Grade \rule{0.pt}{3ex} \\ 
-  & & \rot{Understanding} & \rot{Presentation}\rot{Structure, didactics} & \rot{Completed} & \rot{Bonus:}\rot{Context} & \rot{Bonus:}\rot{Own ideas} & \rot{Readability}\rot{font size, etc.} & \rot{Labels}\rot{and units} & \rot{Design} & \rot{Runs} & \rot{Names of}\rot{vars and funcs} & \rot{Style: white space}\rot{Spaghetti, structure} & \rot{Functions: used,}\rot{purpose, standalone} & \rot{Docu script} & \rot{Docu functions} & \rot{Figures saved} & \rot{Bonus:}\rot{Interesting code} & & \\ 
+  & & \rot{Understanding} & \rot{Presentation}\rot{Structure, didactics} & \rot{Completed} & \rot{Bonus:}\rot{Context} & \rot{Bonus:}\rot{Own ideas} & \rot{Readability}\rot{font size, etc.} & \rot{Labels, units}\rot{and statistical measures} & \rot{Design} & \rot{Runs} & \rot{Names of}\rot{vars and funcs} & \rot{Style: white space}\rot{Spaghetti, structure} & \rot{Functions: used,}\rot{purpose, standalone} & \rot{Docu script} & \rot{Docu functions} & \rot{Figures saved} & \rot{Bonus:}\rot{Interesting code} & & \\ 
- & & 0--3 & 0--2 & 0--2 & 0, 1 & 0--2 & 0--2 & 0--2 & 0--2 & 0--2 & 0--2 & 0--2 & 0--3 & 0--2 & 0--2 & 0, 2 & 0--2 & 28 & \\ \hline
+ & & 0--6 & 0--2 & 0--2 & 0, 1 & 0--2 & 0--2 & 0--2 & 0--2 & 0--2 & 0--2 & 0--2 & 0--3 & 0--2 & 0--2 & 0, 2 & 0--2 & 28 & \\ \hline
  \num & & & & & & & & & & & & & & & & & & & \\ \hline
  \num & & & & & & & & & & & & & & & & & & & \\ \hline
  \num & & & & & & & & & & & & & & & & & & & \\ \hline

projects/instructions.tex

@@ -3,8 +3,8 @@
 
       \textbf{Evaluation criteria:}
 
-      Each project has three elements that are graded: (i) the code,
+      You can view the evaluation criteria on the
-      (ii) the quality of the figures, and (iii) the presentation (see below).
+      \emph{SciCompScoreSheet.pdf} on Ilias.
 
       \vspace{1ex}
       \textbf{Dates:}
@@ -18,27 +18,27 @@
       \vspace{1ex}
       \textbf{Files:}
 
-      Please hand in your presentation as a pdf file. Bundle
+      Hand in your presentation as a pdf file. Bundle
       everything (the pdf, the code, and the data) into a {\em single}
       zip-file.
 
+      Hint: make the zip file you want to upload, unpack it somewhere
+       else and check if your main script is still running properly.
+
       \vspace{1ex}
       \textbf{Code:}
 
-      The code should be executable without any further
+      The code must be executable without any further adjustments from
-      adjustments from our side.  A single \texttt{main.m} script
+      our side. (Test it!) A single \texttt{main.m} script coordinates
-      should coordinate the analysis by calling functions and
+      the analysis by calling functions and sub-scripts and produces
-      sub-scripts and should produce the {\em same} figures
+      the {\em same} figures (\texttt{saveas()}-function, pdf or png
-      (\texttt{saveas()}-function, pdf or png format) that you use in
+      format) that you use in your slides. The code must be
-      your slides. The code should be properly commented and
       comprehensible by a third person (use proper and consistent
       variable and function names).
-      % Hint: make the zip file you want to upload, unpack it
-      % somewhere else and check if your main script is running.
 
       \vspace{1ex} 
 
-      \emph{Please write your name and matriculation number as a
+      \emph{Please note your name and matriculation number as a
       comment at the top of the \texttt{main.m} script.}
 
 
@@ -49,7 +49,7 @@
       in English. In the presentation you should present figures
       introducing, explaining, showing, and discussing your data,
       methods, and results. All data-related figures you show in the
-      presentation should be produced by your program --- no editing
+      presentation must be produced by your program --- no editing
       or labeling by PowerPoint or other software. It is always a good
       idea to illustrate the problem with basic plots of the
       raw-data. Make sure the axis labels are large enough!

projects/project.mk

@@ -1,14 +1,10 @@
 BASENAME=$(subst project_,,$(notdir $(CURDIR)))
 
-latex:
-	pdflatex $(BASENAME).tex
-	pdflatex $(BASENAME).tex
-
 
 pdf: $(BASENAME).pdf
 
 $(BASENAME).pdf : $(BASENAME).tex ../header.tex ../instructions.tex
-	pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true
+	pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get" && pdflatex -interaction=scrollmode $< || true
 
 
 watch :
@@ -19,6 +15,11 @@ clean:
 	rm -rf *.log *.aux *.out auto
 	rm -f `basename *.tex .tex`.pdf
 	rm -f *.zip
+	pdflatex $(BASENAME).tex
+
+latex:
+	pdflatex $(BASENAME).tex
+	pdflatex $(BASENAME).tex
 
 zip: latex
 	rm -f zip $(BASENAME).zip

projects/project_eod/eod.tex

@@ -11,30 +11,47 @@
 
 
 %%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
+Weakly electric fish employ their self-generated electric field for
+prey-capture, navigation and also communication. In many of these fish
+the {\em electric organ discharge} (EOD) is well described by a
+combination of a sine-wave and a few of its harmonics (integer
+multiples of the fundamental frequency).
 
 \begin{questions}
-  \question In the data file {\tt EOD\_data.mat} you find a time trace
+  \question In the data file {\tt EOD\_data.mat} you find two
-  and the {\em electric organ discharge (EOD)} of a weakly electric
+  variables. The first contains the time at which the EOD was sampled
-  fish {\em Apteronotus leptorhynchus}. 
+  and the second the acutal EOD recording of a weakly electric fisch
+  of the species {\em Apteronotus leptorhynchus}.
   \begin{parts}
-    \part Load and plot the data in an appropriate way. Time is in
+    \part Load the data and create a plot showing the data. Time is given in
-    seconds and the voltage is in mV/cm.
+    seconds and the voltage is given in mV/cm.
-    \part Fit the following curve to the eod (select a small time
+    \part Fit the following curve to the EOD (select a \textbf{small} time
-    window, containing only 2 or three electric organ discharges, for
+    window, containing only two or three electric organ discharges, for
     fitting, not the entire trace) using least squares:
     $$f_{\omega_0,b_0,\varphi_1, ...,\varphi_n}(t) = b_0 +
     \sum_{j=1}^n \alpha_j \cdot \sin(2\pi j\omega_0\cdot t + \varphi_j
-    ).$$ $\omega_0$ is called {\em fundamental frequency}. The single
+    ).$$ $\omega_0$ is called the {\em fundamental frequency}. The single
     terms $\alpha_j \cdot \sin(2\pi j\omega_0\cdot t + \varphi_j )$
-    are called {\em harmonic components}. The variables $\varphi_j$
+    are called the {\em harmonic components}. The variables $\varphi_j$
     are called {\em phases}, the $\alpha_j$ are the amplitudes. For
     the beginning choose $n=3$.
     \part Try different choices of $n$ and see how the fit
-    changes. Plot the fits and the original curve for different
+    changes. Plot the fits and the section of the original curve that
-    choices of $n$. Also plot the fitting error as a function of
+    you used for fitting for different choices of $n$.
-    $n$. 
+    \part \label{fiterror} Plot the fitting error as a function of $n$.
+    What do you observe?
+    \part Another way to quantify the quality of the fit is to compute
+    the correlation coefficient between the fit and the
+    data. Illustrate this correlation for a few values of $n$. Plot
+    the correlation coefficient as a function of $n$.  What is the
+    minimum $n$ needed for a good fit? How does this compare to the
+    results from (\ref{fiterror})?
+    \part Plot the amplitudes $\alpha_j$ and phases $\varphi_j$ as a
+    function of the frequencies $\omega_j$ --- the amplitude and phase
+    spectra, also called ``Bode plot''.
+    \part Why does the fitting fail when you try to fit the entire recording?
     \part (optional) If you want you can also play the different fits
-    and the original as sound.
+    and the original as sound (check the help).
 
   \end{parts}
 \end{questions}

projects/project_eyetracker/eyetracker.tex

@@ -33,7 +33,7 @@ shifts. The eye movements during training and test are recorded.
     speed and/or accelerations.
     \part Detect and correct the eye traces for instances in which the 
     eye was not correctly detected. Interpolate linearily in these sections. 
-    \part Create a 'heatmap' plot that shows the eye trajectories 
+    \part Create a 'heatmap' plot of the eye-positions
     for one or two (nice) trials.
     \part Use the \verb+kmeans+ clustering function to 
     identify fixation points. Manually select a good number of cluster

projects/project_fano_slope/fano_slope.tex

@@ -1,6 +1,6 @@
 \documentclass[a4paper,12pt,pdftex]{exam}
 
-\newcommand{\ptitle}{Stimulus discrimination}
+\newcommand{\ptitle}{Stimulus discrimination: gain}
 \input{../header.tex}
 \firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
 {email: jan.benda@uni-tuebingen.de}
@@ -95,10 +95,11 @@ spikes = lifboltzmanspikes(trials, input, tmax, gain);
     the neuron? Plot them for the four different values of the gain
     used in (a).
 
-    \part Think about a measure based on the spike-count histograms
+    \part \label{discrmeasure} Think about a measure based on the
-    that quantifies how well the two stimuli can be distinguished
+    spike-count histograms that quantifies how well the two stimuli
-    based on the spike counts. Plot the dependence of this measure as
+    can be distinguished based on the spike counts. Plot the
-    a function of the gain of the neuron.
+    dependence of this measure as a function of the gain of the
+    neuron.
 %
     For which gains can the two stimuli perfectly discriminated?
 
@@ -110,6 +111,11 @@ spikes = lifboltzmanspikes(trials, input, tmax, gain);
     results in the best discrimination performance. How can you
     quantify ``best discrimination'' performance?
 
+    \part Another way to quantify the discriminability of the spike
+    counts in response to the two stimuli is to apply an appropriate
+    statistical test and check for significant differences. How does
+    this compare to your findings from (\ref{discrmeasure})?
+
  \end{parts}
 
 \end{questions}

projects/project_fano_time/fano_time.tex

@@ -1,6 +1,6 @@
 \documentclass[a4paper,12pt,pdftex]{exam}
 
-\newcommand{\ptitle}{Stimulus discrimination}
+\newcommand{\ptitle}{Stimulus discrimination: time}
 \input{../header.tex}
 \firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
 {email: jan.benda@uni-tuebingen.de}
@@ -87,10 +87,11 @@ input = 15.0; % I_2
     observation time $T$? Plot them for four different values of $T$
     (use values of 10\,ms, 100\,ms, 300\,ms and 1\,s).
 
-    \part Think about a measure based on the spike-count histograms
+    \part \label{discrmeasure} Think about a measure based on the
-    that quantifies how well the two stimuli can be distinguished
+    spike-count histograms that quantifies how well the two stimuli
-    based on the spike counts. Plot the dependence of this measure as
+    can be distinguished based on the spike counts. Plot the
-    a function of the observation time $T$.
+    dependence of this measure as a function of the observation time
+    $T$.
 
     For which observation times can the two stimuli perfectly
     discriminated?
@@ -103,6 +104,11 @@ input = 15.0; % I_2
     results in the best discrimination performance. How can you
     quantify ``best discrimination'' performance?
 
+    \part Another way to quantify the discriminability of the spike
+    counts in response to the two stimuli is to apply an appropriate
+    statistical test and check for significant differences. How does
+    this compare to your findings from (\ref{discrmeasure})?
+
  \end{parts}
 
 \end{questions}

projects/project_isicorrelations/isicorrelations.tex

@@ -1,6 +1,6 @@
 \documentclass[a4paper,12pt,pdftex]{exam}
 
-\newcommand{\ptitle}{Interspike-intervall correlations}
+\newcommand{\ptitle}{Adaptation and interspike-interval correlations}
 \input{../header.tex}
 \firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
 {email: jan.benda@uni-tuebingen.de}
@@ -20,9 +20,9 @@
   Explore the dependence of interspike interval correlations on the firing rate,
   adaptation time constant and noise level.
 
-The neuron is a neuron with an adaptation current. 
+  The neuron is a neuron with an adaptation current.  It is
-    It is implemented in the file \texttt{lifadaptspikes.m}.  Call it
+  implemented in the file \texttt{lifadaptspikes.m}.  Call it with the
-    with the following parameters:
+  following parameters:
   \begin{lstlisting}
 trials = 10;
 tmax = 50.0;
@@ -39,29 +39,42 @@ spikes = lifadaptspikes( trials, input, tmax, Dnoise, adapttau, adaptincr );
     and the adaptation time constant via \texttt{adapttau}.
 
   \begin{parts}
-    \part Measure the intensity-response curve of the neuron, that is the mean firing rate 
+    \part Show a spike-raster plot and a time-resolved firing rate of
-    as a function of the input for a range of inputs from 0 to 120.
+    the neuron for an input current of 50 for three different
-
+    adaptation time constants \texttt{adapttau} (10\,m, 100\,ms,
-    \part Compute the correlations between each interspike interval $T_i$ and the next one $T_{i+1}$
+    1\,s). How do the neural responses look like and how do they
-    (serial interspike interval correlation at lag  1). Plot this correlation as a function of the 
+    depend on the adaptation time constant?
-    firing rate by varying the input as in (a).
+
-
+    \uplevel{For all the following analysis we only use the spike
-    \part How does this dependence change for different values of the adaptation 
+      times of the steady-state response, i.e. we skip all spikes
-    time constant \texttt{adapttau}?  Use values between 10\,ms and
+      occuring before at least three times the adaptation time
-    1\,s for \texttt{adapttau}.
+      constant.}
-
+
-    \part Determine the firing rate at which the minimum interspike interval correlation
+    \part \label{ficurve} Measure the intensity-response curve of the
-    occurs. How does the minimum correlation and this firing rate
+    neuron, that is the mean firing rate as a function of the input
-    depend on the adaptation time constant \texttt{adapttau}?
+    for a range of inputs from 0 to 120.
-
+
-    \part How do the results change if the level of the intrinsic noise \texttt{Dnoise} is modified?
+    \part Additionally compute the correlations between each
-    Use values of 1e-4, 1e-3, 1e-2, 1e-1, and 1 for \texttt{Dnoise}.
+    interspike interval $T_i$ and the next one $T_{i+1}$ (serial
-
+    interspike interval correlation at lag 1) for the same range of
-
+    inputs as in (\ref{ficurve}). Plot the correlation as a function
-    \uplevel{If you still have time you can continue with the following question:}
+    of the input.
-
+
-    \part How do the interspike interval distributions look like for the different noise levels
+    \part How does the intensity-response curve and the
-    at some example values for the input and the adaptation time constant?
+    interspike-interval correlations depend on the adaptation time
+    constant \texttt{adapttau}?  Use several values between 10\,ms and
+    1\,s for \texttt{adapttau} (logarithmically distributed).
+
+    \part Determine the firing rate at which the minimum interspike
+    interval correlation occurs. How does the minimum correlation and
+    this firing rate (or the inverse of it, the mean interspike
+    interval) depend on the adaptation time constant
+    \texttt{adapttau}? Is this dependence siginificant? If yes, can
+    you explain this dependence?
+
+    \part How do all the results change if the level of the intrinsic
+    noise \texttt{Dnoise} is modified?  Use values of 1e-4, 1e-3,
+    1e-2, 1e-1, and 1 for \texttt{Dnoise}.
 
  \end{parts}
 

projects/project_isicorrelations/lifadaptspikes.m

@@ -41,13 +41,10 @@ function spikes = lifadaptspikes( trials, input, tmaxdt, D, tauadapt, adaptincr
             if v >= vthresh
                 v = vreset;
                 a = a + adaptincr/tauadapt;
-                spiketime = i*dt;
+                times(j) = i*dt;
-                if spiketime > 4.0*tauadapt
-                    times(j) = spiketime - 4.0*tauadapt;
                 j = j + 1;
             end
         end
-        end
         spikes{k} = times;
     end
 end

projects/project_isicorrelations/solution/firingrate.m

@@ -0,0 +1,11 @@
+function [ rate ] = firingrate(spikes, t0, t1)
+% compute the firing rate from spikes between t0 and t1
+
+rates = zeros(length(spikes), 1);
+for k = 1:length(spikes)
+    spiketimes = spikes{k};
+    rates(k) = length(spiketimes((spiketimes>=t0)&(spiketimes<=t1)))/(t1-t0);
+end
+rate = mean(rates);
+end
+

projects/project_isicorrelations/solution/isicorrelations.m

@@ -0,0 +1,36 @@
+trials = 5;
+tmax = 10.0;
+Dnoise = 1e-2; % noise strength
+adapttau = 0.1; % adaptation time constant in seconds
+adaptincr = 0.5; % adaptation strength
+t0=2.0;
+t1=tmax;
+maxlag = 5;
+
+taus = [0.01, 0.1, 1.0];
+colors = ['r', 'b', 'g'];
+for j = 1:length(taus)
+    adapttau = taus(j);
+    % f-I curves:
+    Is = [0:10:80];
+    rate = zeros(length(Is),1);
+    corr = zeros(length(Is),maxlag);
+    for k = 1:length(Is)
+        input = Is(k);
+        spikes = lifadaptspikes(trials, input, tmax, Dnoise, adapttau, adaptincr);
+        rate(k) = firingrate(spikes, t0, t1);
+        corr(k,:) = serialcorr(spikes, t0, t1, maxlag);
+    end
+
+    figure(1);
+    hold on
+    plot(Is, rate, colors(j));
+    hold off
+
+    figure(2);
+    hold on
+    plot(Is, corr(:,2), colors(j));
+    hold off
+end
+
+pause

projects/project_isicorrelations/solution/lifadaptspikes.m

@@ -0,0 +1,50 @@
+function spikes = lifadaptspikes( trials, input, tmaxdt, D, tauadapt, adaptincr )
+% Generate spike times of a leaky integrate-and-fire neuron
+% with an adaptation current
+% trials: the number of trials to be generated
+% input: the stimulus either as a single value or as a vector
+% tmaxdt: in case of a single value stimulus the duration of a trial
+%         in case of a vector as a stimulus the time step
+% D: the strength of additive white noise
+% tauadapt: adaptation time constant
+% adaptincr: adaptation strength
+    
+    tau = 0.01;
+    if nargin < 4
+        D = 1e0;
+    end
+    if nargin < 5
+        tauadapt = 0.1;
+    end
+    if nargin < 6
+        adaptincr = 1.0;
+    end
+    vreset = 0.0;
+    vthresh = 10.0;
+    dt = 1e-4;
+    
+    if max( size( input ) ) == 1
+        input = input * ones( ceil( tmaxdt/dt ), 1 );
+    else
+        dt = tmaxdt;
+    end
+    spikes = cell( trials, 1 );
+    for k=1:trials
+        times = [];
+        j = 1;
+        v = vreset + (vthresh-vreset)*rand();
+        a = 0.0;
+        noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt);
+        for i=1:length( noise )
+            v = v + ( - v - a + noise(i) + input(i))*dt/tau;
+            a = a + ( - a )*dt/tauadapt;
+            if v >= vthresh
+                v = vreset;
+                a = a + adaptincr/tauadapt;
+                times(j) = i*dt;
+                j = j + 1;
+            end
+        end
+        spikes{k} = times;
+    end
+end

projects/project_isicorrelations/solution/serialcorr.m

@@ -0,0 +1,15 @@
+function [ isicorrs ] = serialcorr(spikes, t0, t1, maxlag)
+% compute the serial correlations from spikes between t0 and t1
+
+ics = zeros(length(spikes), maxlag);
+for k = 1:length(spikes)
+    spiketimes = spikes{k};
+    isis = diff(spiketimes((spiketimes>=t0)&(spiketimes<=t1)));
+    if length(isis) > 2*maxlag
+        for j=1:maxlag
+            ics(k, j) = corr(isis(j:end)', isis(1:end+1-j)');
+        end
+    end
+end
+isicorrs = mean(ics, 1);
+end

projects/project_lif/lif.tex

@@ -96,7 +96,7 @@ time = [0.0:dt:tmax]; % t_i
 
     Write a function that implements this leaky integrate-and-fire
     neuron by expanding the function for the passive neuron
-    appropriate. The function returns a vector of spike times.
+    appropriately. The function returns a vector of spike times.
 
     Illustrate how this model works by appropriate plot(s) and
     input(s) $E(t)$, e.g. constant inputs lower and higher than the
@@ -115,8 +115,8 @@ time = [0.0:dt:tmax]; % t_i
       r = \frac{n-1}{t_n - t_1}
     \end{equation}
     What do you observe? Does the firing rate encode the frequency of
-    the stimulus? Look at the spike trains in response the sine waves
+    the stimulus? Look at the spike trains in response to the sine
-    to figure out what is going on.
+    waves to figure out what is going on.
   \end{parts}
 
 \end{questions}

projects/project_noiseficurves/noiseficurves.tex

@@ -63,8 +63,8 @@ spikes = lifspikes(trials, current, tmax, Dnoise);
     of this neuron?
 
     \part Compute the $f$-$I$ curves of neurons with various noise
-    strengths \texttt{Dnoise}. Use for example $D_{noise} = 1e-3$,
+    strengths \texttt{Dnoise}. Use for example $D_{noise} = 10^{-3}$,
-    $1e-2$, and $1e-1$.
+    $10^{-2}$, and $10^{-1}$.
 
     How does the intrinsic noise influence the response curve?
 
@@ -76,6 +76,8 @@ spikes = lifspikes(trials, current, tmax, Dnoise);
     responses of the four different neurons to the same input, or by
     the same resulting mean firing rate.
 
+    How do the responses differ?
+
     \part Let's now use as an input to the neuron a 1\,s long sine
     wave $I(t) = I_0 + A \sin(2\pi f t)$ with offset current $I_0$,
     amplitude $A$, and frequency $f$. Set $I_0=5$, $A=4$, and

projects/project_pca_natural_images/pca_natural_images.tex

@@ -32,9 +32,8 @@ In you zip file you find a natural image called {\tt natimg.jpg}.
 \begin{thebibliography}{1}
 \bibitem{BG} Buchsbaum, G., \& Gottschalk, A. (1983). Trichromacy,
   opponent colours coding and optimum colour information transmission
-  in the retina. Proceedings of the Royal Society of London. Series B,
+  in the retina. Proceedings of the Royal Society of London B. Royal
-  Containing Papers of a Biological Character. Royal Society (Great
+  Society (Great Britain), 220(1218), 89–113.
-  Britain), 220(1218), 89–113.
 \end{thebibliography}
 
 

projects/project_photoreceptor/photoreceptor.tex

@@ -11,11 +11,16 @@
 
 
 %%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
-\section*{Analysis of insect photoreceptor data.}
+\section*{Light responses of an insect photoreceptor.}
 In this project you will analyse data from intracellular recordings of
-a fly R\.1--6 photoreceptor. The membrane potential of the
+a fly R\,1--6 photoreceptor. These cells show graded membrane
-photoreceptor was recorded while the cell was stimulated with a
+potential changes in response to a light stimulus. The membrane
-light stimulus.
+potential of the photoreceptor was recorded while the cell was
+stimulated with a light stimulus. Intracellular recordings often
+suffer from drifts in the resting potential. This leads to a large
+variability in the responses which is technical and not a cellular
+property. To compensate for such drifts trials are aligned to the
+resting potential before stimulus onset.
 
 \begin{questions}
   \question{} The accompanying dataset (photoreceptor\_data.zip)
@@ -24,23 +29,36 @@ light stimulus.
   \textit{voltage} a matrix with the recorded membrane potential from
   10 consecutive trials, (ii) \textit{time} a matrix with the
   time-axis for each trial, and (iii) \textit{trace\_meta} a structure
-  that stores several metadata. This is the place where you find the
+  that stores several metadata including the \emph{amplitude} value
-  \emph{amplitude}, that is the voltage that drives the light
+  that is the voltage used to drive the light stimulus. (Note that
-  stimulus, i.e. the light-intensity.
+  this voltage is only a proxy for the true light intensity. Twice the
+  voltage does not lead to twice the light intensity. Within this
+  project, however, you can treat it as if it was the intensity.)
 
   \begin{parts}
-    \part{} Create a plot of the raw data. Plot the average response as
+    \part Create a plot of the raw data. For each light intensity plot
-    a function of time. This plot should also show the
+    the average response as a function of time. This plot should also
-    across-trial variability.\\[0.5ex]
+    depict the across-trial variability in an appropriate way.
-    \part{} You will notice that the responses have three main parts, a
+
+    \part You will notice that the responses have three main parts, a
     pre-stimulus phase, the phase in which the light was on, and
     finally a post-stimulus phase. Create an characteristic curve that
     plots the response strength as a function of the stimulus
     intensity for the ``onset'' and the ``steady state''
-    phases.\\[0.5ex]
+    phases of the light response.
-    \part{} The light switches on at time zero. Estimate the delay between stimulus.\\[0.5ex]
+
-    \part{} You may also decide to analyze the post-stimulus response in some
+    \part The light switches on at time zero. Estimate the delay
-    more detail.
+    between stimulus and response.
+
+    \part Analyze the across trial variability in the ``onset'' and
+    ``steady state''. Check for statistically significant differences.
+
+    \part The membrane potential shows some fluctuations (noise)
+    compare the noise before stimulus onset and in the steady state
+    phase of the response.
+
+    \part (optional) You may also analyze the post-stimulus response
+    in some more detail.
   \end{parts}
 \end{questions}
 

projects/project_stimulus_reconstruction/stimulus_reconstruction.tex

@@ -10,29 +10,29 @@
 \input{../instructions.tex}
 
 %%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
-\section*{Reverse reconstruction of the stimulus evoking neuronal responses.}
+\section*{Reverse reconstruction of the stimulus that evoked a neuronal response.}
 To analyse encoding properties of a neuron one often calculates the
-Spike-Triggered-Average (STA).
+Spike-Triggered-Average (STA). The STA is the average stimulus that
-\[ STA(\tau) = \frac{1}{\langle n \rangle} \left\langle
+led to a spike in the neuron and is calculated by cutting out snippets
-  \displaystyle\sum_{i=1}^{n}{s(t_i - \tau)} \right\rangle \] 
+form the stimulus centered on the respective spike time:
-
+\[ STA(\tau) = \frac{1}{n} \displaystyle\sum_{i=1}^{n}{s(t_i - \tau)} \],
-The STA is the average stimulus that led to a spike in the neuron and
+where $n$ is the number of trials and $t_i$ is the time of the
-is calculated by cutting out snippets form the stimulus centered on
+$i_{th}$ spike. The Spike-Triggered-Average can be used to reconstruct
-the respective spike time. The Spike-Triggered-Average can be used to
+the stimulus from the neuronal response. The reconstructed stimulus
-reconstruct the stimulus a neuron has been stimulated with.
+can then be compared to the original stimulus.
 
 \begin{questions}
   \question In the accompanying files you find the spike responses of
-  P-units and pyramidal neurons of the weakly electric fish
+  a p-type electroreceptor afferent (P-unit) and a pyramidal neurons
+  recorded in the hindbrain of the weakly electric fish
   \textit{Apteronotus leptorhynchus}. The respective stimuli are
   stored in separate files. The data is sampled with 20\,kHz temporal
   resolution and spike times are given in seconds. Start with the
-  P-unit and, in the end, apply the same functions to the pyramidal
+  P-unit and, in the end, apply the same analyzes/functions to the
-  data.
+  pyramidal data.
   \begin{parts}
     \part Estimate the STA and plot it.
-    \part Implement a function that does the reconstruction of the
+    \part Implement a function that does the reverse reconstruction and uses the STA to recopnstruct the stimulus.
-    stimulus using the STA.
     \part Implement a function that estimates the reconstruction 
     error using the mean-square-error and express it relative to the
     variance of the original stimulus.
@@ -44,8 +44,8 @@ reconstruct the stimulus a neuron has been stimulated with.
     \part Analyze the robustness of the reconstruction: Estimate 
     the STA with less and less data and estimate the reconstruction
     error.
-    \part Plot the reconstruction error as a function of the data 
+    \part Plot the reconstruction error as a function of the amount of data 
-    amount used to estimate the STA.
+    used to estimate the STA.
     \part Repeat the above steps for the pyramidal neuron, what do you
     observe?
   \end{parts}

regression/exercises/exercises01.tex

@@ -20,7 +20,7 @@
 \else
 \newcommand{\stitle}{}
 \fi
-\header{{\bfseries\large Exercise 11\stitle}}{{\bfseries\large Gradient descent}}{{\bfseries\large January 9th, 2018}}
+\header{{\bfseries\large Exercise 10\stitle}}{{\bfseries\large Gradient descent}}{{\bfseries\large December 17th, 2018}}
 \firstpagefooter{Dr. Jan Grewe}{Phone: 29 74588}{Email:
   jan.grewe@uni-tuebingen.de}
 \runningfooter{}{\thepage}{}
@@ -58,22 +58,24 @@
 
 \begin{questions}
 
-  \question Implement the gradient descent for finding the parameters
+  \question We want to fit the straigth line \[ y = mx+b \] to the
-  of a straigth line that we want to fit to the data in the file
+  data in the file \emph{lin\_regression.mat}.
-  \emph{lin\_regression.mat}.
 
-  In the lecture we already prepared most of the necessary functions:
+  In the lecture we already prepared the cost function
-  1. the error function (\code{meanSquareError()}), 2. the cost
+  (\code{lsqError()}), and the gradient (\code{lsqGradient()}) (read
-  function (\code{lsqError()}), and 3. the gradient
+  chapter 8 ``Optimization and gradient descent'' in the script, in
-  (\code{lsqGradient()}). Read chapter 8 ``Optimization and gradient
+  particular section 8.4 and exercise 8.4!). With these functions in
-  descent'' in the script, in particular section 8.4 and exercise 8.4!
+  place we here want to implement a gradient descend algorithm that
+  finds the minimum of the cost function and thus the slope and
+  intercept of the straigth line that minimizes the squared distance
+  to the data values.
 
   The algorithm for the descent towards the minimum of the cost
   function is as follows:
-  
   \begin{enumerate}
-  \item Start with some arbitrary parameter values $\vec p_0 = (m_0, b_0)$
+  \item Start with some arbitrary parameter values (intercept $b_0$
-    for the slope and the intercept of the straight line.
+    and slope $m_0$, $\vec p_0 = (b_0, m_0)$ for the slope and the
+    intercept of the straight line.
   \item \label{computegradient} Compute the gradient of the cost function
     at the current values of the parameters $\vec p_i$.
   \item If the magnitude (length) of the gradient is smaller than some
@@ -106,9 +108,11 @@
       \lstinputlisting{../code/descentfit.m}
     \end{solution}
 
-    \part Find the position of the minimum of the cost function by
+    \part For checking the gradient descend method from (a) compare
-    means of the \code{min()} function. Compare with the result of the
+    its result for slope and intercept with the position of the
-    gradient descent method. Vary the value of $\epsilon$ and the
+    minimum of the cost function that you get when computing the cost
+    function for many values of the slope and intercept and then using
+    the \code{min()} function. Vary the value of $\epsilon$ and the
     minimum gradient. What are good values such that the gradient
     descent gets closest to the true minimum of the cost function?
     \begin{solution}

regression/lecture/regression-chapter.tex

@@ -16,6 +16,10 @@
 
 \input{regression}
 
+\section{Improvements}
+Adapt function arguments to matlabs polyfit. That is: first the data
+(x,y) and then the parameter vector. p(1) is slope, p(2) is intercept.
+
 \section{Fitting in practice}
 
 Fit with matlab functions lsqcurvefit, polyfit