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\firstpageheader{Scientific Computing}{Principal Component Analysis}{Oct 29, 2014}
%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\begin{document}

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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%

\begin{questions}
  \question \textbf{Gaussian distribution}
  \begin{parts}
    \part Use \texttt{randn} to generate 1000000 normally (zero mean, unit variance) distributed random numbers.
    \part Plot a properly normalized histogram of these random numbers.
    \part Compare the histogram with the probability density of the Gaussian distribution
    \[ p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
    where $\mu$ is the mean and $\sigma^2$ is the variance of the Gaussian distribution.
    \part Generate Gaussian distributed random numbers with mean $\mu=2$ and
    standard deviation $\sigma=\frac{1}{2}$.
  \end{parts}

  \question \textbf{Covariance and correlation coefficient}
  \begin{parts}
    \part Generate two vectors $x$ and $z$ with 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 \textbf{Principal component analysis}
  \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}
  
\end{questions}


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