$3.$ In this problem, you will generate simulated data, and then perform K$-$means clus$-$ tering on the data.

$($a$)$ Generate a simulated data set with $20$ observations in each of K$=5$ well$-$separated clusters, with p $=2$ variables describing each observation. Do it in similar fash$-$ ion to K $=3$ case in $$K$-$means: Selecting K$$ lecture slides. Plot the resulting data points.

$($b$)$ Run K$-$means algorithm on your simulated data for K $=4,5$ & $6.$ Use $50$ random starts in each case. Provide the code and report the total within$-$cluster sum of squares $($WSS$)$ for each solution. Which transition caused the bigger drop in total WSS$,$ from K $=4$ to K $=5,$ or from K $=5$ to K $=6?$

$($c$)$ Proceed to plot the clustering solutions for K $=4,5$ & $6($just use the plots automatically generated by eclust$()$ function$).$ Judging by the plots, explain why the respective sizes of WSS drops in part $($b$)$ were expected. Use lecture slides $($the K $=3$ simulation study$)$ for reference.

$($d$)$ Run K$-$means clustering on your simulated data for K $=1,2,...,10,$ record total WSS for each K value. Plot the progression of WSS values. According to $$elbow$$ method logic, which K value appears as the optimal one?

$($e$)$ For a more formal approach $($alternative to elbow method from part $($d$)),$ run the gap statistic calculations for K $=1,...,10,$ with $50$ replicates each. Which K value is optimal? Does it match with the # of well$-$separated clusters in our simulated data? Provide the plot of gap statistic values.

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