Problem $3($K$-$Means Clustering with MATLAB$)$: Implement and analyze the k$-$means clustering algorithm to partition a given set of data points into three clusters.

Data Points

Cluster $1$: $(1,2),(2,1),(1,1)$

Cluster $2$: $(5,6),(6,5),(5,5),(6,6),(5,7)$

Cluster $3$: $(9,10),(10,9),(9,9),(10,10)$

Initial Centers

Center $1$: $(1,2)$

Center $2$: $(5,6)$

Center $3$: $(9,10)$

Tasks

a$)$ Analytical Calculation:

Apply k$-$means in batch mode to update centers until convergence.

Document the center updates at each iteration.

b$)$ MATLAB Coding:

Code the k$-$means process starting with the given points and centers.

Iterate to update centers in batch mode until convergence.

Display centers at each iteration and plot final clusters.

Deliverables

Analytical calculation of center updates.

MATLAB code implementing k$-$means, with iteration displays and final plot.