K$-$means. You are given three data samples:

i$123$

xi$1-114$

xi$2004$

$($a$)(8$ points$)$ We want to have K$=2$ cluster. After K$-$mean finishes on the above training

data, what are the centers of the two clusters? What is the equation for the boundary of

the two clusters?

$($b$)(7$ points$)$ Suppose you want to use clustering for outlier detection. You find cluster

means $$i$,$i$=1,...,$ K on the training data. Then, given a new data xand a threshold

t$,$ you declare xan outlier if kx$$ik $$ tfor all i$.$ Complete the following function

to implement the outlier detection on a matrix of data X$.$ The output is out$[$i$]=1$ if the

sample X$[$i$,$:$]$ is an outlier, and out$[$i$]=0$ otherwise. You must specify the other inputs of

your function. Avoid for loops for full credit.

def outlier detect$($X$,...)$:

$...$

return out