ML Interview Q Series: How do you determine the appropriate number of clusters when using a K-Medoids clustering approach?
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Comprehensive Explanation
K-Medoids is a popular clustering technique that, unlike K-Means, selects actual data points (medoids) as cluster centers. Determining the optimal number of clusters (K) is crucial because a poorly chosen K can yield subpar cluster quality, either merging distinct patterns or artificially splitting natural groups.
There are several strategies to choose K:
Domain Knowledge and Practical Constraints
Often, an important factor is the domain context in which you are applying clustering. Some scenarios might have natural group boundaries or limited resources (for instance, you might only be able to handle a certain number of clusters operationally). Using practical constraints and domain insights can guide or constrain the range of K values to test.
Elbow Method
A common heuristic is to run K-Medoids over a range of K values, compute a cost function (such as total within-cluster dissimilarity), and plot this cost against K. You usually observe a curve that decreases rapidly at first and then tapers off. The "elbow" or point of inflection on this curve often suggests a good K. However, the elbow might not always be obvious, and you may need to compare results across a few different candidate K values.
Silhouette Analysis
Another well-known approach is the Silhouette Coefficient, which, for each data point, compares the average distance to other points in the same cluster with the average distance to points in other clusters. You then examine the mean silhouette score across all data points, and a higher value typically indicates better clustering performance.
Below is the core formula for the silhouette coefficient for each data point i:
where:
a(i) is the average distance (according to your chosen distance metric) between point i and all other points within the same cluster.
b(i) is the minimum average distance between point i and points in any other cluster (i.e., the smallest average distance to any other cluster).
After computing s(i) for all i, you average them to obtain the overall silhouette score. A higher average silhouette score indicates better-defined clusters. You repeat this for different K values, and choose K that yields a high silhouette score.
Gap Statistic
The Gap Statistic compares the within-cluster dissimilarity for the actual dataset with that for a uniformly distributed reference dataset. The general principle is: if your chosen K is too small, the actual cluster dispersion might be much bigger than the reference's; if your K is too large, it might start overfitting. The K that maximizes the gap between the actual data’s clustering metrics and the reference is often a suitable choice.
Stability-Based Methods
You can also assess clustering stability by repeatedly running K-Medoids with different random seeds or bootstrapped samples of the data. The logic is that if K is correct, the clusters remain relatively stable across different runs. If the clustering drastically changes, it may indicate you need to adjust K.
Practical Implementation
In a typical workflow, you might:
Decide on a range of K based on domain knowledge (e.g., K=2 to K=10).
For each K, fit K-Medoids and compute internal evaluation metrics (such as total within-cluster dissimilarity, silhouette score, or the gap statistic).
Visualize how these metrics change over the range of K values.
Look for an elbow point or a peak in silhouette or gap scores to identify K.
Validate or refine that choice by cross-checking with domain knowledge or other considerations (like interpretability or practical operational constraints).
Below is a small illustrative example in Python using the KMedoids implementation from the scikit-learn-extra library:
import numpy as np
from sklearn_extra.cluster import KMedoids
from sklearn.metrics import silhouette_score
# Example dataset (replace with actual data)
X = np.array([
[1, 2], [2, 2], [2, 3],
[8, 7], [8, 8], [25, 80]
])
best_k = 2
best_score = -1
# Suppose we try different K values
for k in range(2, 6):
kmed = KMedoids(n_clusters=k, random_state=42)
labels = kmed.fit_predict(X)
s_score = silhouette_score(X, labels)
print(f"K={k}, Silhouette Score={s_score:.3f}")
if s_score > best_score:
best_score = s_score
best_k = k
print("Chosen K:", best_k)
In practice, you would also consider metrics like total within-cluster distance or the gap statistic, and decide which K is most appropriate.
Follow-up Questions
How does K-Medoids differ from K-Means in determining the optimal number of clusters?
K-Medoids focuses on minimizing distance to actual data points (medoids) instead of centroids, which can make it more robust to outliers. Despite this difference, many of the methods for choosing K (elbow method, silhouette score, etc.) remain similar. In practice, both algorithms still require experimentation with multiple K values, but K-Medoids may be preferable when you want each cluster center to be an actual data point, when you have robust distance metrics (e.g., Manhattan distance), or when outliers could skew K-Means centroids.
What are common pitfalls with the Elbow Method?
One pitfall is that the elbow may not be well-defined or visible. You might see a curve that smoothly decreases without a clear angle. Additionally, sometimes there may be multiple points that appear as "elbows," and subjective judgment is needed to choose the most appropriate one. Another pitfall is that the elbow method focuses only on the within-cluster dispersion and does not consider how well-separated the clusters are from each other.
Can you discuss limitations of the Silhouette Score?
The Silhouette Score might not always reflect the best real-world grouping if clusters are heavily imbalanced or if the distance metric used does not capture the true similarity pattern. High-dimensional data can also pose challenges for the silhouette score because distances tend to become less meaningful in very high dimensions. Furthermore, a single global score might overlook local structures where certain subclusters are tightly separated while others are not.
When would you use the Gap Statistic over the Silhouette Method?
The Gap Statistic can be particularly helpful when you suspect that the dataset might have a complex structure that is difficult to evaluate using only distances within and between clusters. The Gap Statistic effectively compares your data against a reference distribution, providing a baseline that can be more robust in certain scenarios. It can capture nuanced structures that silhouette analysis might miss, such as identifying whether your data truly forms clusters or is essentially random-like.
How do you reconcile domain knowledge with quantitative metrics when they disagree on K?
If domain experts suggest a certain number of clusters (for example, marketing segments) but metrics like silhouette or elbow suggest a different number, you typically investigate why the discrepancy exists. Perhaps the domain-specific guidance stems from a particular business constraint or interpretability requirement. You might choose to override the metric-based decision to align with practical, real-world needs while acknowledging the trade-off in statistical clustering validity. Alternatively, you can refine the domain-based assumptions or re-check data quality and pre-processing to see if any hidden structures were missed.
What about computational complexity concerns?
K-Medoids can be more computationally expensive than K-Means because it involves pairwise distance computations among data points. If your dataset is extremely large, experimenting with a very wide range of K might become prohibitively expensive. Techniques like sampling, mini-batch strategies, or approximate methods can help scale your search for the optimal K. You might also use faster proxy metrics or approximate cluster centers, then refine the best candidate K values on the full dataset.
How would you handle outliers when choosing K?
K-Medoids is inherently more robust to outliers compared to K-Means, because medoids must be actual data points rather than potentially skewed centroids. Nonetheless, outliers can still inflate distances in certain metrics. You might remove extreme outliers or cap distances to reduce their effect. Proper data cleaning and domain-based outlier detection might be part of the pipeline before clustering. If outliers are meaningful, you might consider an additional "outlier" cluster or adopt alternative distance metrics suitable for your domain.
These considerations collectively guide the process of determining the optimal number of clusters for K-Medoids in a wide range of real-world scenarios.