96 lines
3.2 KiB
TeX
96 lines
3.2 KiB
TeX
\documentclass[addpoints,10pt]{exam}
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\usepackage{url}
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\usepackage{color}
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\usepackage{hyperref}
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\usepackage{graphicx}
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\usepackage{amsmath}
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\pagestyle{headandfoot}
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\runningheadrule
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\firstpageheadrule
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\firstpageheader{Scientific Computing}{Principal Component Analysis}{Oct 29, 2014}
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%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\firstpagefooter{}{}{}
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\runningfooter{}{}{}
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\pointsinmargin
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\bracketedpoints
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%\printanswers
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\shadedsolutions
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\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro
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%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{listings}
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\lstset{
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basicstyle=\ttfamily,
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numbers=left,
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showstringspaces=false,
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language=Matlab,
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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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captionpos=t,
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xleftmargin=2em,
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xrightmargin=1em,
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aboveskip=10pt,
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%title=\lstname,
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title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext}
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}
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\begin{document}
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\sffamily
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{questions}
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\question \textbf{Gaussian distribution}
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\begin{parts}
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\part Use \texttt{randn} to generate 1000000 normally (zero mean, unit variance) distributed random numbers.
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\part Plot a properly normalized histogram of these random numbers.
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\part Compare the histogram with the probability density of the Gaussian distribution
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\[ p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
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where $\mu$ is the mean and $\sigma^2$ is the variance of the Gaussian distribution.
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\part Generate Gaussian distributed random numbers with mean $\mu=2$ and
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standard deviation $\sigma=\frac{1}{2}$.
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\end{parts}
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\question \textbf{Covariance and correlation coefficient}
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\begin{parts}
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\part Generate two vectors $x$ and $z$ with 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 \textbf{Principal component analysis}
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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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\end{questions}
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\end{document}
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