[linalg] added exercise sheet

linearalgebra/code/spikesortingwave.m

@@ -117,4 +117,26 @@ hold on;
 scatter( time(tinx(kinx1)), voltage(tinx(kinx1)), 10.0, 'r', '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/exercises01.tex

@@ -0,0 +1,192 @@
+\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?
+    \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?
+  \end{parts}
+
+  \question \qt{Principal component analysis in 2D\vspace{-3ex}}
+  \begin{parts}
+    \part Generate 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?
+  \end{parts}
+
+  \question \qt{Principal component analysis in 3D\vspace{-3ex}}
+  \begin{parts}
+    \part Generate 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?
+  \end{parts}
+
+  \continue
+  \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 methods 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. 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?
+
+    \part Apply PCA on the waveform snippets, that is compute the
+    eigenvalues and eigenvectors of their covariance matrix, and plot
+    the sorted eigenvalues (the ``eigenvalue spectrum''). How many
+    eigenvalues are clearly larger than zero?
+
+    \part Plot the two eigenvectors (``features'') with the two
+    largest eigenvalues.
+
+    \part Project the waveform snippets onto these two eigenvectors
+    and display them with a scatter plot. What do you observe? Can you
+    separate two ``clouds'' of data points (``clusters'')?
+
+    \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.
+
+    \part Plot the waveform snippets of each cluster together with the
+    true waveform obtained from the data file. Do they match?
+
+    \part Mark the action potentials in the recording according to
+    their cluster identity.
+
+    \part Compute interspike-interval histograms of all the (unsorted)
+    action potentials, and of each of the two neurons. What do they
+    tell you?
+  \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}