Part $3$: Clustering $(50$pts total$)$

For Part $3,$ you will apply $k-$means clustering to two different simulated datasets and compare

its performance. Both datasets are generated with $N=500$ observations, but each were

generated with a different number of "clusters." These datasets, ps$2-1.$csv and ps$2-2.$csv

will be labeled as "one" and "two," respectively.

$3\mathrm{.}1$ Vanilla $k-$means $(25$ pts total$)$

$3\mathrm{.}1\mathrm{.}1$ Visualizing the data $(5$pts$)$

Make a scatterplot of each dataset. Visually, guess at how many groups were used to simulate

the data and include the name of the dataset and your guess in the title of each.

# TODO: Two plots $3\mathrm{.}2$ Implementing $k-$means with Manhattan distance $(25$ pts total$)$

$3\mathrm{.}2\mathrm{.}1$ Computing Manhattan distance $(5$pts$)$

Implement the Manhattan distance metric in a function that you can use in your $k-$means

implementation. Write three test cases to be sure your function works correctly.

# TODO: Manhattan distance

$3\mathrm{.}2\mathrm{.}2$ Running k$-$Means again $(5$ pts$)$

Using your implementation from $3\mathrm{.}1\mathrm{.}2,$ implement the $k-$means algorithm with Manhattan

distance distance. Recall that you will be assigning points to clusters based on minimum

Manhattan distance. Be sure to produce the within$-$cluster sum of squared Manhattan

distance $($WCSS$).$ For each dataset, run the algorithm $M=10$ times using a for$()$ loop for

your optimal values of $k$ from $3\mathrm{.}1\mathrm{.}3.$ Plot the best solution for that value of $k,$ with points

colored by cluster assignment. Include the dataset, $k,$ and the final WCSS in the title of each

plot.