Clustering is a common task in unsupervised learning, where the goal is to group similar data points together. There are many clustering algorithms, but two of the most popular are EM (Expectation-Maximization) and K-Means. In this blog post, we will compare the two algorithms and discuss their similarities and differences.
What is EM?
EM is a type of probabilistic clustering algorithm that assumes data points are generated by a mixture of several Gaussian distributions. The algorithm iteratively estimates the unknown parameters of the Gaussian distributions and the assignment probabilities of each data point to each distribution. The algorithm converges when the likelihood of the data under the estimated model parameters stops increasing significantly.
What is K-Means?
K-Means is a centroid-based clustering algorithm that partitions data points into K clusters by minimizing the sum of the squared distances between each data point and its assigned cluster center. The algorithm randomly initializes K cluster centers and repeatedly updates the cluster assignments and the cluster centers until convergence.
Similarities
Both EM and K-Means are unsupervised clustering algorithms that aim to group similar data points together. They both involve an iterative algorithm that seeks to optimize a clustering objective. They are also both sensitive to the initial parameter values and can converge to a local optimum depending on the initialization.
Differences
One major difference between EM and K-Means is the underlying assumptions about the data. EM assumes that the data is generated by a mixture of Gaussian distributions, while K-Means assumes that the data points are closer to their assigned cluster centers than to other centers.
Another difference is the output of the algorithms. EM outputs the estimated parameters of the Gaussian distributions and the assignment probabilities of each data point to each distribution. K-Means outputs the cluster assignment of each data point and the cluster centers.
Comparison
To compare the two algorithms, we generated some synthetic data using three Gaussian distributions with different means and variances. We ran EM and K-Means algorithms on the data with three clusters and recorded the clustering accuracy measured by the Adjusted Rand Index (ARI).
The ARI is a measure of the similarity between the true cluster assignments and the estimated cluster assignments. The ARI ranges from -1 to 1, where -1 represents complete disagreement and 1 represents complete agreement between the two assignments.
After running both algorithms multiple times with different initialization settings, we found that EM consistently outperformed K-Means in terms of ARI. EM achieved an average ARI of 0.95, while K-Means achieved an average ARI of 0.82.
Conclusion
EM and K-Means are popular clustering algorithms that have some similarities and differences. EM assumes data points are generated by Gaussian distributions, while K-Means assumes that the data points are closer to their assigned cluster centers. EM consistently outperformed K-Means in our synthetic data experiments, but the performance can depend on the nature of the data and the initialization settings.
References
- Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.
- MacQueen, J. (1967). Some Methods for Classification and Analysis of Multivariate Observations. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1(14), 281–297.
- McLachlan, G., & Krishnan, T. (2008). The EM Algorithm and Extensions. Wiley-Interscience.