Mixture of Gaussian Distributions Model
In the realm of machine learning, Gaussian Mixture Models (GMM) have proven to be a versatile tool, finding use-cases in clustering, anomaly detection, image segmentation, density estimation, and more. This article delves into the workings of GMM, offering insights into its applications and limitations.
At its core, a GMM is a probabilistic model that assumes data points are generated from a mixture of several Gaussian (normal) distributions with unknown parameters. The model starts with initial guesses for the means, covariances, and mixing coefficients of each Gaussian distribution. It then alternates between the Expectation Step (E-step) and Maximization Step (M-step) of the Expectation-Maximization (EM) algorithm until the log-likelihood of the data converges.
In the E-step, the algorithm calculates the probability that each data point belongs to each cluster based on the current parameter estimates. Conversely, in the M-step, the algorithm updates the parameters (mean, covariance, and mixing coefficients) to better fit the data. The mean represents the central point or average location of the cluster in the feature space, while the covariance matrix describes the shape, size, and orientation of the cluster.
The overall likelihood of observing a data point under all Gaussians in GMM is achieved by summing over all possible clusters for each point: P(x|GMM) = Σk P(k|x) * P(x|k). The probability of a data point belonging to a specific cluster k in GMM is calculated using the formula: P(k|x) = πk * f(x|μk, Σk), where πk is the mixing probability of the k-th Gaussian, f(x|μk, Σk) is the Gaussian distribution with mean μk and covariance Σk.
Visualizing these Gaussian components helps to understand how GMM fits flexible, overlapping clusters in real-world data. Scatter plots show the raw data points clustered around their respective means, while overlaid kernel density estimate (KDE) contours represent the smooth shape of each Gaussian.
One of the advantages of GMM is its ability to model ellipsoidal and overlapping clusters, making it more powerful than simpler methods like K-Means. GMM performs soft clustering by assigning each point a probability of belonging to multiple clusters, providing a more nuanced understanding of the data distribution.
However, GMM is not without its limitations. It is sensitive to initialization, computationally intense, assumes Gaussian distributions, requires specifying the number of components/clusters before fitting, and cannot handle non-Gaussian cluster shapes. Despite these limitations, GMM remains a valuable tool in the machine learning arsenal due to its ability to handle complex data distributions and its interpretable parameters.
The development of the GMM algorithm can be traced back to significant foundational work by Carl Friedrich Gauss on Gaussian distributions and later advancements by various researchers in the 20th century. The specific attribution to a single person or institute is not clearly documented.
This article is an excerpt from the journal "Tufan Gupta, Machine Learning". For a comprehensive understanding of GMM and its applications, we encourage readers to explore the full journal.
