Here’s a step-by-step approach using Python:

**Generating Sample Data**: We’ll create a synthetic dataset.**Applying K-means**: We’ll apply the K-means algorithm to segment customers.**Visualization**: We’ll visualize the clusters.

First, you’ll need to install the required package, if you haven’t already:

pip install matplotlib scikit-learn

Now, let’s write the Python code:

import numpy as np import matplotlib.pyplot as plt from sklearn.cluster import KMeans from sklearn.datasets import make_blobs # Step 1: Generate synthetic data # We create a dataset with 200 samples, 2 features (annual income and spending score) and roughly 4 clusters X, _ = make_blobs(n_samples=200, centers=4, cluster_std=0.60, random_state=0) # Step 2: Apply K-means clustering kmeans = KMeans(n_clusters=4) kmeans.fit(X) y_kmeans = kmeans.predict(X) # Step 3: Visualization of clusters plt.scatter(X[:, 0], X[:, 1], c=y_kmeans, s=50, cmap='viridis') centers = kmeans.cluster_centers_ plt.scatter(centers[:, 0], centers[:, 1], c='red', s=200, alpha=0.75) # Red dots are cluster centers plt.title('Customer Segmentation') plt.xlabel('Annual Income') plt.ylabel('Spending Score') plt.show()

### Detailed Explanation of the Python Code

**Import Libraries** ` `

`import numpy as np `

`import matplotlib.pyplot as plt `

`from sklearn.cluster import KMeans `

`from sklearn.datasets import make_blobs`

**Generate Synthetic Data**

`X, _ = make_blobs(n_samples=200, centers=4, cluster_std=0.60, random_state=0)`

`make_blobs`

: generates 200 samples (data points), spread across 4 centers (clusters), with a standard deviation of 0.60 for each cluster.

`X`

contains the coordinates for each point (here representing features like “Annual Income” and “Spending Score”).

`_`

(underscore) is a placeholder for the cluster labels assigned by `make_blobs`

which we don’t use here since we want to perform our own clustering.

**Apply K-means Clustering**

`kmeans = KMeans(n_clusters=4) `

`kmeans.fit(X) `

`y_kmeans = kmeans.predict(X)`

`KMeans(n_clusters=4)`

creates a K-means clustering model specifying 4 clusters, based on our initial assumption or analysis (e.g., the Elbow Method might have suggested this number).

`kmeans.fit(X)`

fits the K-means model on the dataset `X`

. This method computes the centroids of the clusters, trying to minimize the within-cluster variance.

`kmeans.predict(X)`

assigns each sample in `X`

to one of the 4 clusters, returning an array of cluster indices which we store in `y_kmeans`

.

**Visualization of Clusters**

`plt.scatter(X[:, 0], X[:, 1], c=y_kmeans, s=50, cmap='viridis') centers = kmeans.cluster_centers_ `

`plt.scatter(centers[:, 0], centers[:, 1], c='red', s=200, alpha=0.75) `

`plt.title('Customer Segmentation') `

`plt.xlabel('Annual Income') `

`plt.ylabel('Spending Score') `

`plt.show()`

`plt.scatter(X[:, 0], X[:, 1], c=y_kmeans, s=50, cmap='viridis')`

plots all the points in `X`

. The parameters:

`X[:, 0]`

and `X[:, 1]`

are the coordinates for each point (annual income and spending score, respectively).

`c=y_kmeans`

colors each point based on its cluster assignment, providing visual differentiation of clusters.

`s=50`

sets the size of the points.

`cmap='viridis'`

uses the ‘viridis’ color map for coloring the different clusters.

`centers = kmeans.cluster_centers_`

extracts the centroids of the clusters calculated by the K-means algorithm.

`plt.scatter(centers[:, 0], centers[:, 1], c='red', s=200, alpha=0.75)`

plots the centroids as red dots, where:

`s=200`

makes the centroid dots larger than the data points.

`alpha=0.75`

makes the red color slightly translucent.

The remaining `plt`

functions set the title and labels for the axes, and `plt.show()`

displays the plot.

`numpy`

is imported to handle data in array format efficiently.

`matplotlib.pyplot`

is used for plotting graphs to visualize data and clustering results.

`KMeans`

from `sklearn.cluster`

is the K-means clustering implementation in the scikit-learn library.

`make_blobs`

from `sklearn.datasets`

is used to generate synthetic datasets with a Gaussian distribution, ideal for demonstrating clustering.