Note:

$-$ Use $3$ centroids, K$=3.$

$-$ Your calculations should be only for the first iteration only.

Income Spending Score

$1539$

$1581$

$166$

$1677$

$1740$

$1776$

$186$

$1894$

$193$

$1972$

As the first step I will select $3$ points for initiating centroids.

C$1(16,6)$

C$2(18,6)$

C$3(19,3)$

Then we will calculate distance between C$1,$ C$2,$ C$3$ and all data. Use Squared Euclidean to calculate distance.

Squared Euclidean distance$=($x$1$x$2)2+($y$1$y$2)2$

C$1(16,6)$ C$2(18,6)$ C$3(19,3)$

I$1($Instance$1)=1090$ I$1($Instance$1)=1098$ I$1($Instance$)=916$

I$2=1626$ I$2=5634$ I$2=5200$

I$3=$ O I$3=4$ I$3=18$

I$4=5041$ I$4=5045$ I$4=4633$

I$5=1157$ I$5=1157$ I$5=965$

I$6=4901$ I$6=4901$ I$6=4493$

I$7=4$ I$7=0$ I$7=10$

I$8=7748$ I$8=7744$ I$8=7226$

I$9=18$ I$9=10$ I$9=36$

I$10=4365$ I$10=4357$ I$10=3969$

Assign each data point to the cluster corresponding to the nearest centroid.

C$1$ C$2$ C$3$

I$2=1626$ I$7=0$ I$1=916$

I$3=0$ I$8=7744$ I$4=4633$

I$9=10$ I$5=965$

I$6=4443$

I$10=3969$

Calculate mean of the data points in each cluster to determine the new centroid positions.

We will get instances for C$1,$ C$2,$ C$3$ then calculate the average.

C$1=(15+16)/2,(81+6)/2=15\mathrm{.}2,43\mathrm{.}5$

C$2=(18+18+18)/3,(6+94+3)/3=18\mathrm{.}3,34\mathrm{.}3$

C$3=(15+16+17+17+19)/5,(39+77+40+76+72)/5=16\mathrm{.}8,68\mathrm{.}6$

After each iteration, reassess the cluster assignments based on the updated centroids.

C$1=(15\mathrm{.}5),(43\mathrm{.}5)$ C$2=(18\mathrm{.}3),(34\mathrm{.}3)$ C$3=(16\mathrm{.}8),(68\mathrm{.}6)$

C$)(15$ Points$)$ Provide an example of a K$-$means limitation using the data points in Q$1.$B dataset only.