Choosing the optimal number of clusters (k) in k-means clustering is an important decision, as it can significantly impact the quality and interpretability of the clustering results. There are several methods and techniques that can help determine an appropriate value for k. Here are a few commonly used approaches:
Elbow Method (see: https://www.youtube.com/watch?v=ht7geyMAFfA)
- Run k-means clustering for a range of k values (e.g., 1 to 10).
- For each k, calculate the within-cluster sum of squared distances (WCSS) or the average distance between data points and their cluster centroid.
- Plot the WCSS or average distance against the number of clusters (k).
- Look for the "elbow point" in the plot, where the rate of decrease in WCSS or average distance starts to level off. This point suggests a good number of clusters.
Silhouette Analysis (see https://youtu.be/AtxQ0rvdQIA?si=C3cKdvJXon2B1jN0&t=174)
- Run k-means clustering for different values of k.
- For each data point, calculate its silhouette coefficient, which measures how well it fits into its assigned cluster compared to other clusters.
- Compute the average silhouette coefficient for each k.
- Choose the k that maximizes the average silhouette coefficient, indicating good separation between clusters and cohesion within clusters.
Gap Statistic:
- Generate reference datasets by randomly sampling from the original data.
- Run k-means clustering on both the original data and the reference datasets for different values of k.
- Calculate the gap statistic, which measures the difference between the log of the WCSS of the original data and the average log of the WCSS of the reference datasets.
- Choose the k that maximizes the gap statistic, indicating a significant improvement in clustering quality compared to random clustering.
Domain Knowledge and Interpretability:
- Consider the specific context and domain knowledge of the problem.
- Assess the interpretability and usefulness of the clustering results for different values of k.
- Choose a k that aligns with the desired level of granularity and provides meaningful insights for the given application.
Hierarchical Clustering:
- Perform hierarchical clustering on the data.
- Analyze the resulting dendrogram, which visualizes the hierarchical structure of the clusters.
- Identify the number of distinct clusters based on the dendrogram by considering the height at which the clusters are merged.
The disadvantage of elbow and average silhouette methods is that, they measure a global clustering characteristic only. A more sophisticated method is to use the gap statistic which provides a statistical procedure to formalize the elbow/silhouette heuristic in order to estimate the optimal number of clusters.
It's important to note that there is no single universally optimal method for determining the number of clusters. The choice of k often depends on the specific characteristics of the data, the goals of the analysis, and the trade-off between simplicity and capturing fine-grained patterns. It's recommended to use a combination of these methods, along with domain expertise and visual inspection, to make an informed decision about the appropriate number of clusters for a given problem.