Consider the 5 data points shown below:
\begin{tabular}{|c|c|c|c|}
\hline
$P_1$ & 1 & 3 & 2 \\
\hline
$P_2$ & 0 & 1 & 2 \\
\hline
$P_3$ & 3 & 0 & 5 \\
\hline
$P_4$ & 4 & 1 & 3 \\
\hline
$P_5$ & 2 & 0 & 1 \\
\hline
\end{tabular}
a. Apply the k-means clustering algorithm, to group those data points into 2 clusters, using the manhattan distance measure (the sum of differences in each dimension). Suppose that the initial centroids are $C_1$: (1, 0, 0) and $C_2$: (0, 1, 1).
b. Use the within-cluster variations (WC) measure to evaluate the quality of each resulting cluster. Which clusters have better quality than the others?
c. Calculate the \"between cluster\" score (BC) and the overall quality.
Hints: $d(x, y) = |x_1 - y_1| + |x_2 - y_2| + ... + |x_p - y_p|$
$WC(C_i) = \frac{1}{|C_i|} \sum_{x \in C_i} d(x, r_i)^2$
$BC = \sum_{1 \le i \le k} d(r_i, r)^2$
