$K-$means code in Matlab is provided to you, which performs K$-$means clustering on a

given dataset of observations. Check what parameters this function takes as input

and apply it for partitioning the Old Faithful data$($given to you in $.$txt file$)$ into $K=2$

clusters.

a$)$ Determine the cluster centre values

$(2$ marks$).$

b$)$ Using a higher number of clusters, $K=4,$ determine the cluster centre

values.

$(2$ marks$)$

One way to measure the quality of the $K-$means clustering solution is to compute the

sum$-$squared error. Error function for $k-$means clustering with $k$ clusters, sample points

$x_1,$dots,$x_n,$ and centers of clusters $m_1,$dots,$m_k$ is given as:

$E=\frac{1}{N}{\displaystyle \underset{k=1}{\overset{K}{}}}{\displaystyle \underset{n=1}{\overset{N}{}}}{z}_{kn}||x_n-m_k|{|}^2$

where $Z_{kn}$ is treated as $1$ when point $x_n$ belongs to cluster $k,$ otherwise $Z_{kn}=0$

c$)$ Modify your k$-$means implementation so as to compute the sum$-$squared

error on each iteration. Does this error decrease after each iteration?

Increase the number of clusters $K,$ and list the values of error function.

$(5$ marks$)$

Instead of updating $m_j$ by computing the mean, we use stochastic gradient descent,

to remodel the mi such that :

$m_i=m_i+z_{kn}(x_n-m_i)$

with learning rate $($step size$).$

Recall that in the update step of the standard K$-$mean algorithm, we assign each cluster

centre to be the mean $($centroid$)$ of the data points closest to that centre. It turns out that a

particular choice of the learning rate $($which may be different for each cluster$)$ makes the

two algorithms $($batch gradient descent and the standard k$-$means algorithm$)$ have identical

update steps. Let's focus on the update for the first cluster, with center $m_1.$

d$)$ Implement the equation instead of mean in the standard algorithm and find the

learning rate value, $$ such that the value of both algorithms perform the same

update for $m_1.$