$($a$)$ How do we select cluster centers initially in k$-$means algorithm?

$($b$)$ What is best way to determine the number of clusters $($k$)$ that is needed?

$($c$)$ If we increase the number of clusters, does the overall$/$sum error $($error is how far a point from its assigned

center$)$ increase or decrease? Why?

$($d$)$ How does an outlier affect the cluster centers? How do you propose to minimize the impact from outliers?

$($e$)$ If we are clustering $m$ points into $k$ clusters using $100$ iterations, write down the big $O$ complexity of the

k$-$means algorithm using $m$ and $k.$ Do not simplify the terms. Justify your answer.The k$-$means algorithm is an unsupervised learning model used in clustering applications. Given below is the

algorithm.

Algorithm: K$-$means

Given: Data set $x^{(i)},i=1,$dots,$m$

Initialize:

${}^{(j)}$:$=$RandomChoice$(x^{(1\text{:}m)}),j=1,$dots,$k$

Repeat until convergence:

Assign point clusters: $y^{(i)}$:$=argmi{n}_j|{}^{(j)}-x^{(i)}{|}_2^2,i=1,$dots,$m$

Compute new centers: ${}^{(j)}$:$={\displaystyle \underset{i=1}{\overset{m}{}}}(x^{(i)}1\frac{y^{(i)}}{1\{y^{(i)})}=j=j\}=j$

Answer the following questions based on k$-$means algorithm.

$($a$)$ How do we select cluster centers initially in k$-$means algorithm?