192 lines
7.1 KiB
TeX
192 lines
7.1 KiB
TeX
\documentclass[12pt,a4paper,pdftex]{exam}
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\usepackage[english]{babel}
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\usepackage{pslatex}
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\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
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\usepackage{xcolor}
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\usepackage{graphicx}
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\usepackage[breaklinks=true,bookmarks=true,bookmarksopen=true,pdfpagemode=UseNone,pdfstartview=FitH,colorlinks=true,citecolor=blue]{hyperref}
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%%%%% layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[left=20mm,right=20mm,top=25mm,bottom=25mm]{geometry}
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\pagestyle{headandfoot}
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\ifprintanswers
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\newcommand{\stitle}{: Solutions}
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\else
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\newcommand{\stitle}{}
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\fi
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\header{{\bfseries\large Exercise\stitle}}{{\bfseries\large PCA}}{{\bfseries\large January 7th, 2019}}
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\firstpagefooter{Prof. Dr. Jan Benda}{Phone: 29 74573}{Email:
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jan.benda@uni-tuebingen.de}
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\runningfooter{}{\thepage}{}
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\setlength{\baselineskip}{15pt}
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\setlength{\parindent}{0.0cm}
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\setlength{\parskip}{0.3cm}
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\renewcommand{\baselinestretch}{1.15}
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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\lstset{
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language=Matlab,
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basicstyle=\ttfamily\footnotesize,
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numbers=left,
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numberstyle=\tiny,
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title=\lstname,
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showstringspaces=false,
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commentstyle=\itshape\color{darkgray},
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breaklines=true,
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breakautoindent=true,
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columns=flexible,
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frame=single,
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xleftmargin=1em,
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xrightmargin=1em,
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aboveskip=10pt
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}
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%%%%% math stuff: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{bm}
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\usepackage{dsfont}
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\newcommand{\naZ}{\mathds{N}}
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\newcommand{\gaZ}{\mathds{Z}}
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\newcommand{\raZ}{\mathds{Q}}
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\newcommand{\reZ}{\mathds{R}}
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\newcommand{\reZp}{\mathds{R^+}}
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\newcommand{\reZpN}{\mathds{R^+_0}}
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\newcommand{\koZ}{\mathds{C}}
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%%%%% page breaks %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\continue}{\ifprintanswers%
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\else
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\vfill\hspace*{\fill}$\rightarrow$\newpage%
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\fi}
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\newcommand{\continuepage}{\ifprintanswers%
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\newpage
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\else
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\vfill\hspace*{\fill}$\rightarrow$\newpage%
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\fi}
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\newcommand{\newsolutionpage}{\ifprintanswers%
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\newpage%
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\else
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\fi}
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%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\qt}[1]{\textbf{#1}\\}
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\newcommand{\pref}[1]{(\ref{#1})}
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\newcommand{\extra}{--- Zusatzaufgabe ---\ \mbox{}}
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\newcommand{\code}[1]{\texttt{#1}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\input{instructions}
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\begin{questions}
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\question \qt{Covariance and correlation coefficient\vspace{-3ex}}
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\begin{parts}
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\part Generate two vectors $x$ and $z$ with $n=1000$ Gausian distributed random numbers.
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\part Compute $y$ as a linear combination of $x$ and $z$ according to
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\[ y = r \cdot x + \sqrt{1-r^2}\cdot z \]
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where $r$ is a parameter $-1 \le r \le 1$.
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What does $r$ do?
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\part Plot a scatter plot of $y$ versus $x$ for about 10 different values of $r$.
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What do you observe?
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\part Also compute the covariance matrix and the correlation
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coefficient matrix between $x$ and $y$ (functions \texttt{cov} and
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\texttt{corrcoef}). How do these matrices look like for different
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values of $r$? How do the values of the matrices change if you generate
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$x$ and $z$ with larger variances?
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\part Do the same analysis (scatter plot, covariance, and correlation coefficient)
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for \[ y = x^2 + 0.5 \cdot z \]
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Are $x$ and $y$ really independent?
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\end{parts}
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\question \qt{Principal component analysis in 2D\vspace{-3ex}}
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\begin{parts}
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\part Generate pairs $(x,y)$ of Gaussian distributed random numbers such
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that all $x$ values have zero mean, half of the $y$ values have mean $+d$
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and the other half mean $-d$, with $d \ge0$.
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\part Plot scatter plots of the pairs $(x,y)$ for $d=0$, 1, 2, 3, 4 and 5.
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Also plot a histogram of the $x$ values.
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\part Apply PCA on the data and plot a histogram of the data projected onto
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the PCA axis with the largest eigenvalue.
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What do you observe?
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\end{parts}
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\question \qt{Principal component analysis in 3D\vspace{-3ex}}
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\begin{parts}
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\part Generate triplets $(x,y,z)$ of Gaussian distributed random numbers such
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that all $x$ values have zero mean, half of the $y$ and $z$ values have mean $+d$
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and the other half mean $-d$, with $d \ge0$.
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\part Plot 3D scatter plots of the pairs $(x,y)$ for $d=0$, 1, 2, 3, 4 and 5.
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Also plot a histogram of the $x$ values.
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\part Apply PCA on the data and plot a histogram of the data projected onto
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the PCA axis with the largest eigenvalue.
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What do you observe?
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\end{parts}
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\continue
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\question \qt{Spike sorting}
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Extracellular recordings often pick up action potentials originating
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from more than a single neuron. In case the waveforms of the action
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potentials differ between the neurons one could assign each action
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potential to the neuron it originated from. This process is called
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``spike sorting''. Here we explore this methods on a simulated
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recording that contains action potentials from two different
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neurons.
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\begin{parts}
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\part Load the data from the file \texttt{extdata.mat}. This file
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contains the voltage trace of the recording (\texttt{voltage}),
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the corresponding time vector in seconds (\texttt{time}), and a
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vector containing the times of the peaks of detected action
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potentials (\texttt{spiketimes}). Further, and in contrast to real
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data, the waveforms of the actionpotentials of the two neurons
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(\texttt{waveform1} and \texttt{waveform2}) and the corresponding
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time vector (\texttt{waveformt}) are also contained in the file.
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\part Plot the voltage trace and mark the peaks of the detected
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action potentials. Zoom into the plot and look whether you can
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differentiate between two different waveforms of action
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potentials. How do they differ?
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\part Cut out the waveform of each action potential (5\,ms before
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and after the peak). Plot all these snippets in a single
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plot. Can you differentiate the two actionpotential waveforms?
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\part Apply PCA on the waveform snippets, that is compute the
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eigenvalues and eigenvectors of their covariance matrix, and plot
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the sorted eigenvalues (the ``eigenvalue spectrum''). How many
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eigenvalues are clearly larger than zero?
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\part Plot the two eigenvectors (``features'') with the two
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largest eigenvalues.
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\part Project the waveform snippets onto these two eigenvectors
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and display them with a scatter plot. What do you observe? Can you
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separate two ``clouds'' of data points (``clusters'')?
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\part Think about a very simply way how to separate the two
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clusters. Generate a vector whose elements label the action
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potentials, e.g. that contains '1' for all snippets belonging to
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the one cluster and '2' for the waveforms of the other
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cluster. Use this vector to mark the two clusters in the previous
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plot with two different colors.
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\part Plot the waveform snippets of each cluster together with the
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true waveform obtained from the data file. Do they match?
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\part Mark the action potentials in the recording according to
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their cluster identity.
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\part Compute interspike-interval histograms of all the (unsorted)
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action potentials, and of each of the two neurons. What do they
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tell you?
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\end{parts}
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\end{questions}
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\end{document} |