From be35befc5b552590af540f4472bdc754ab541fcd Mon Sep 17 00:00:00 2001 From: Jan Benda Date: Tue, 8 Jan 2019 14:47:04 +0100 Subject: [PATCH] [linalg] added solutions to exercise --- linearalgebra/exercises/covariance.m | 19 +++ linearalgebra/exercises/covariance.pdf | Bin 0 -> 509392 bytes linearalgebra/exercises/exercises01.tex | 80 +++++++-- linearalgebra/exercises/extdata.mat | Bin 0 -> 2366486 bytes .../exercises/nonlinearcorrelation.m | 9 + .../exercises/nonlinearcorrelation.pdf | Bin 0 -> 57661 bytes linearalgebra/exercises/pca2d-2.pdf | Bin 0 -> 120377 bytes linearalgebra/exercises/pca2d.m | 39 +++++ linearalgebra/exercises/pca3d-2.pdf | Bin 0 -> 119545 bytes linearalgebra/exercises/pca3d.m | 44 +++++ linearalgebra/exercises/savefigpdf.m | 28 +++ linearalgebra/exercises/spikesorting.m | 161 ++++++++++++++++++ linearalgebra/exercises/spikesorting1.pdf | Bin 0 -> 382802 bytes linearalgebra/exercises/spikesorting2.pdf | Bin 0 -> 4549 bytes linearalgebra/exercises/spikesorting3.pdf | Bin 0 -> 11704 bytes linearalgebra/exercises/spikesorting4.pdf | Bin 0 -> 56453 bytes linearalgebra/exercises/spikesorting5.pdf | Bin 0 -> 369954 bytes linearalgebra/exercises/spikesorting6.pdf | Bin 0 -> 18464 bytes linearalgebra/exercises/spikesorting7.pdf | Bin 0 -> 5808 bytes 19 files changed, 366 insertions(+), 14 deletions(-) create mode 100644 linearalgebra/exercises/covariance.m create mode 100644 linearalgebra/exercises/covariance.pdf create mode 100644 linearalgebra/exercises/extdata.mat create mode 100644 linearalgebra/exercises/nonlinearcorrelation.m create mode 100644 linearalgebra/exercises/nonlinearcorrelation.pdf create mode 100644 linearalgebra/exercises/pca2d-2.pdf create mode 100644 linearalgebra/exercises/pca2d.m create mode 100644 linearalgebra/exercises/pca3d-2.pdf create mode 100644 linearalgebra/exercises/pca3d.m create mode 100644 linearalgebra/exercises/savefigpdf.m create mode 100644 linearalgebra/exercises/spikesorting.m create mode 100644 linearalgebra/exercises/spikesorting1.pdf create mode 100644 linearalgebra/exercises/spikesorting2.pdf create mode 100644 linearalgebra/exercises/spikesorting3.pdf create mode 100644 linearalgebra/exercises/spikesorting4.pdf create mode 100644 linearalgebra/exercises/spikesorting5.pdf create mode 100644 linearalgebra/exercises/spikesorting6.pdf create mode 100644 linearalgebra/exercises/spikesorting7.pdf diff --git a/linearalgebra/exercises/covariance.m b/linearalgebra/exercises/covariance.m new file mode 100644 index 0000000..2f60060 --- /dev/null +++ b/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 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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 pairs $(x,y)$ of Gaussian distributed random numbers such + \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. @@ -115,11 +123,16 @@ jan.benda@uni-tuebingen.de} \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 triplets $(x,y,z)$ of Gaussian distributed random numbers such + \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. @@ -127,15 +140,19 @@ jan.benda@uni-tuebingen.de} \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} - \continue + \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 methods on a simulated + ``spike sorting''. Here we explore this method on a simulated recording that contains action potentials from two different neurons. \begin{parts} @@ -149,42 +166,77 @@ jan.benda@uni-tuebingen.de} 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? + 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? - - \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? + \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. + 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 - separate two ``clouds'' of data points (``clusters'')? + 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} diff --git a/linearalgebra/exercises/extdata.mat b/linearalgebra/exercises/extdata.mat new file mode 100644 index 0000000000000000000000000000000000000000..619c0c1011a5c90e918cacc7637bbc0b2fd30837 GIT binary patch literal 2366486 zcmb@tWpEuq&nBA0%*-4!Guy|^%*@Qp_OTtuF;g5fvtwpvW@d)tV`k=?_wH`(_f_qW zd#hF@m872O*7K*;Ga8M8h`PLpC@Cis3#o#LI-{kHy#)iQvb~ABrIV`zAE}hQqP7SZ z2Lq{?tA&ZXg*mB{BOj@nha;(?lP4)D8z~zr-#_NyB4uM?{=85$Rtj;)2>eHc_6DNmF~Ee?6)jjCaBQP<=0yJ-GWk)eiOfyI!;|MPFBC!eOg+7OIFE4A<-;KZvXeCEPCR%G^_59ev-p&6cX{w zK^HD!yu~Cw)+wv|*8cN2(~6fm2BAaqXq4^7C9hUfhEsrxyN^k9cf&mjVr)dM`9WA* z7D|2xfO0+Amtt%%PV3CpDf(vbI(bV6N@a=@LDI*xH%nqXFy?AL@YiMt5t`?nd6wIP z1kZ7M5i=m2WS2POwEiKmeCzvaZ88L1Pb-`9dkMSyjc#w(b+Q<;vuU& z=MBrWn=}4*E&D)*AK1rg8nM=Gs41l=t5P5U-0V6XsY$Gx>GioYPx`otoWz0)dT4xX 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+ 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 diff --git a/linearalgebra/exercises/pca3d-2.pdf b/linearalgebra/exercises/pca3d-2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..57b62253d143a7d1ba1f55ff85c506c8dc4fc781 GIT binary patch literal 119545 zcmYgXV|ZL!xQ=bBu^Owf8r!yQG{(ee>@-GWHfTI)j5fB-#^#;moOAE}k>{Cb?;m?6 zd#(3-wWyTDBp6wlI1s1?j^hdtK9RDJI-1xZ@bfdvnLAjzS&?#ruBb3eSlhXoyO1(V z*crQ-ib(8cxEkk$i{tYbD`p_F;wnp0{7eHOHzdl8@TBVghh@k 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