$40$ Marks

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Study the scenario and complete the questions that follow:

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Clustering customers based on shopping behaviour

You are tasked with analyzing customer shopping behaviour to segment them into three distinct groups based on their shopping patterns.

Data Points:

The data below represents $8$ customers, where each pair of numbers corresponds to:

Number of Items Purchased

Number of Store Visits

Customer $1($C$1)$: $(2$ items, $10$ visits$)$

Customer $2($C$2)$: $(2$ items, $5$ visits$)$

Customer $3($C$3)$: $(8$ items, $4$ visits$)$

Customer $4($C$4)$: $(5$ items, $8$ visits$)$

Customer $5($C$5)$: $(7$ items, $5$ visits$)$

Customer $6($C$6)$: $(6$ items, $4$ visits$)$

Customer $7($C$7)$: $(1$ item, $2$ visits$)$

Customer $8($C$8)$: $(4$ items, $9$ visits$)$

Initial Cluster Centers:

To start, use the following initial cluster centers based on specific customers:

Cluster $1$ Center: Customer $1($C$1)(2$ items, $10$ visits$)$

Cluster $2$ Center: Customer $4($C$4)(5$ items, $8$ visits$)$

Cluster $3$ Center: Customer $7($C$7)(1$ item, $2$ visits$)$

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$$In Python, run the K$-$Means algorithm from scratch for$3$ rounds $($epochs$)$ on the data points in the scenario above. In each round, you will:

Assign each point to a cluster using the Euclidean distance formula.

Calculate the new centroids of each cluster based on the points assigned.

Create a chart showing all $8$ points, indicating which points belong to each cluster. Mark the centroids with a special symbol and use different colors for each cluster. Add labels and a legend for clarity. Thus in total you will have three charts, one for each round.

Important Notes:

No built$-$in clustering functions $($like those from scikit$-$learn$)$ are allowed. You must use basic Python constructs $($mathematical operations, loops, etc$)$ to manually calculate distances, assign clusters, and update centroids.

You may use libraries like numpy for array operations and matplotlib for visualizing the clusters and centroids.