Estimating Mixture of Gaussian Processes by Kernel Smoothing.

Area of Science:

• Statistics
• Machine Learning
• Data Analysis

Background:

• Traditional statistical methods struggle with datasets containing multiple classes of functional curves (inhomogeneous data).
• Existing functional data analysis techniques may not adequately capture complex data structures.
• The need for robust methods that handle both functional and inhomogeneous properties is critical.

Purpose of the Study:

• To propose a new estimation procedure for Mixture of Gaussian Processes (MGP).
• To develop a method that incorporates both functional and inhomogeneous data characteristics.
• To extend high-dimensional normal mixture models to functional data with smoothed structures.

Main Methods:

• The proposed method extends high-dimensional normal mixtures by imposing smoothed structures on mean and covariance functions.
• It utilizes a combination of the Expectation-Maximization (EM) algorithm, kernel regression, and functional principal component analysis (FPCA).
• Model identifiability is theoretically established.

Main Results:

• The Mixture of Gaussian Processes model is shown to be identifiable.
• Efficient estimation is achieved through the integrated algorithmic approach.
• Empirical justification is provided via Monte Carlo simulations.

Conclusions:

• The proposed Mixture of Gaussian Processes method offers a powerful tool for analyzing inhomogeneous functional data.
• The methodology effectively handles complex data structures, outperforming traditional approaches.
• The method is validated and demonstrated through simulations and a real-world supermarket dataset analysis.