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\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}
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\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}