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\begin{document}

\title{Introduction to Machine Learning\\
Homework 9: Principle Component Analysis and Feature Dimension Reduction}
\author{Prof. Yao Wang}
\date{}

\maketitle

\begin{enumerate}

\item Assume that you have 4 samples each with dimension 3, described in the data matrix $X$,
\[
    X = \left[ \begin{array}{ccc}
        3 & 2 & 1 \\
        2 & 4 & 5 \\
        1 & 2 &3 \\
        0 & 2 &5 \\
          \end{array} \right] \quad
 \]
 \begin{enumerate}
 \item Find the sample mean.
 \item Find the sample covariance matrix $Q$. 
 \item Find the eigenvalues and eigenvectors. You can use the numpy.linalg.eig to compute eigenvalues and eigenvectors from $Q$.
 \item Find the PCA coefficients corresponding to samples in $X$.
 \item Reconstruct the original samples from the PCA coefficients.
 \item Approximate the samples using principle components corresponding to the two largest eigenvalues.
 \item Verify the sum of reconstruction error squares = sum of squares of skipped PCA coefficients.
 
 You could do the above calculation using a simple python code.
 \end{enumerate}
 
 \item
Show that PCA serves the purpose of decorrelating the elements in the original samples. That is, the covariance matrix of the coefficient data is diagonal, and that the variance of the coefficients are equal to the eigenvalues.

\item
For machine learning, we often transform original sample features using PCA. Is it beneficial if you keep all the coefficients? What is the benefit if you keep a subset of coefficients? What coefficients should you keep?


\item
What is the problem of using the PCA coefficients directly as the transformed features for machine learning?
How should you fix that?

 \end{enumerate}
 \end{document}