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\noindent
{\it The Final Exam (Pattern Recognition) ... 17 Jan 2001}\\
\noindent
Name [ \hspace{2in} ] \hspace{.1in} Student-ID [ \hspace{1in} ]\\

\begin{itemize}
\item[1.] As for a two-class, two-dimensional Bayesian classifier where 
$p(\omega_1)=p(\omega_2),\hspace{.1in} \mu_1=(0, 0), 
\hspace{.1in}\mu_2=(1, 1)$ and
\begin{eqnarray}
\nonumber
\Sigma_1 = \Sigma_2 = \Sigma =
\left(
\begin{array}{cc}
1.9 & 0.3             \\
0.3 & 1.1
\end{array}
\right)
\end{eqnarray}
answer the following questions.
   \begin{itemize}
   \item[(1)] Obtain $d_M$ of $\mbox{\bf x}=(1.1, 2.1)$ from $\mu_2$.
\vspace{1.0in}
   \item[(2)] Obtain the boundary equation between the two classes.
\vspace{2.0in}
   \item[(3)] Transform the equation obtained in (2) so that the 
              $x_1 \cdot x_2$ term vanishes.
   \end{itemize}
\vspace{2.0in}
\item[2.]Obtain the Haar transform in the case of $n=1$.
\clearpage
\item[3.] We have a feed-forward neural network 
with two neurons in input layer, 
one neuron in hidden layer, and one neuron in output layer.
The task is to realize XOR function.
Assume that all the weights are initialized to 0.1 and
$\epsilon=0.01$. Also assume that transfer function $f(x) = x^3$
and threshold of all the neurons are fixed to zero,
which are just for the sake of simplicity. 
Of course neurons here are all analog ones, not binary ones.
We now start to train the neural network by the Back-propagation algorithm.
Calculate the following quantities to modify the weights 
when $x_1 = -1$, $x_2 = -1$ are given as a training sample.
The notations are those we used in the course.
\vspace{2.2in}
\begin{displaymath}
\begin{array}{lllll}
h^2_1=\hspace{.6in}& y^2_1=\hspace{.6in}& y^3_1=\hspace{.6in}& 
z_1=\hspace{.6in}& e_1= \\
e^2_1=\hspace{.6in}&{\delta}^2_1=\hspace{.6in}& 
\triangle w^1_{12}=\hspace{.6in}& w^1_{12}=
\hspace{.6in}& w^2_{11}=
\end{array}
\end{displaymath}
\item[4.]Obtain the K-L Transform for the correlation matrix:
\begin{eqnarray}
\nonumber
R = \left(
\begin{array}{rrr}
 0.3 &  0.1 &   0.1 \\
 0.1 &  0.3 &  -0.1 \\
 0.1 & -0.1 &  0.3 \\
\end{array}
\right)
\end{eqnarray}
\vspace{2.5in}

\end{itemize}
\end{document}