K-Means Clustering & The Elbow Method

In modern commerce, treating all customers identically leads to suboptimal marketing returns and high churn. K-Means Clustering is the preeminent unsupervised machine learning algorithm used to partition unlabeled datasets into $K$ distinct, non-overlapping behavioral clusters based on geometric feature proximity.

1. How K-Means Works: The Lloyd-Forgy Algorithm

1. 1
   Initialization: Select $K$ initial centroid coordinates at random (or via K-Means++).
2. 2
   Assignment Step: Assign each data point to its nearest centroid using Euclidean distance:

$$d(x, c_j) = \sqrt{\sum_{i=1}^{p} (x_i - c_{ji})^2}$$

1. 1
   Update Step: Recalculate each centroid as the geometric mean of all data points assigned to it.
2. 2
   Convergence: Repeat steps 2 and 3 until centroids stabilize (inertia stops decreasing).

2. Choosing Optimal $K$: The Elbow Method & Inertia

Because K-Means is unsupervised, there are no ground-truth labels. Analysts use Inertia (Within-Cluster Sum of Squares - WCSS):

$$WCSS = \sum_{k=1}^{K} \sum_{x \in C_k} ||x - \mu_k||^2$$

As $K$ increases, WCSS naturally drops towards zero (if $K=N$, WCSS=0). The Elbow Method finds the point of diminishing marginal returns:

3. Validating with Silhouette Score

When the elbow curve is ambiguous, calculate the Silhouette Score (ranges from -1 to +1):

• Near +1: Clusters are dense and cleanly separated.
• Near 0: Clusters overlap significantly.
• Negative: Points are assigned to the wrong cluster.

4. Profiling Clusters for Business Strategy

After assigning cluster labels, profile each cluster by calculating unscaled feature means: