IDENTIFYING THE NUMBER OF COMPONENTS IN GAUSSIAN MIXTURE MODELS USING NUMERICAL ALGEBRAIC GEOMETRY

Area of Science:

• Data Science
• Statistical Modeling
• Numerical Algebraic Geometry

Background:

• Gaussian mixture models (GMMs) are widely used for clustering in science and engineering.
• Estimating GMM parameters typically requires the number of components beforehand using the Expectation-Maximization algorithm.
• Existing methods like AIC and BIC can be unreliable when the Gaussian assumption is violated.

Purpose of the Study:

• To propose and evaluate two new algorithms for identifying the optimal number of components in GMMs.
• To compare the performance of these novel algorithms against established methods (AIC, BIC).
• To demonstrate the robustness of the proposed methods, particularly under non-Gaussian conditions.

Main Methods:

• Developed two algorithms based on numerical algebraic geometry: an area-based algorithm and a local maxima algorithm.
• The area-based algorithm uses polynomial regression splines and homotopy continuation.
• The local maxima algorithm fits smoothing splines and solves for derivatives to find component centers and counts.

Main Results:

• The proposed algorithms effectively identify the optimal number of Gaussian components.
• In a case study and simulations, the new methods showed superior robustness compared to AIC and BIC when the Gaussian assumption was not met.
• The local maxima algorithm also successfully estimated the centers of Gaussian components.

Conclusions:

• The novel numerical algebraic geometry algorithms offer a more robust approach to determining the number of components in GMMs.
• These methods provide a valuable alternative to AIC and BIC, especially in real-world scenarios with potential deviations from ideal Gaussian distributions.
• The findings have significant implications for accurate clustering in diverse scientific and engineering applications.