Bayesian Framework for Simultaneous Registration and Estimation of Noisy, Sparse and Fragmented Functional Data

Area of Science:

• Statistics
• Data Science
• Functional Data Analysis

Background:

• Functional data often suffers from observational challenges like sparse sampling and errors.
• Existing methods for function registration and estimation lack formal uncertainty propagation and are often application-specific.

Purpose of the Study:

• To develop a unified Bayesian framework for simultaneous registration and estimation of functional data.
• To accommodate diverse observational regimes, including sparse and error-contaminated data.
• To improve uncertainty quantification and visualization in functional data analysis.

Main Methods:

• A unified Bayesian framework is proposed, building on elastic functional data analysis.
• Amplitude and phase variability are modeled separately.
• Two prior strategies are used for amplitude variability: data-driven empirical basis and shape-restricted priors.

Main Results:

• The framework allows flexible inference on individual functions under general observational regimes.
• It effectively handles sparse, fragmented, and error-contaminated data.
• Uncertainty quantification and visualization of amplitude and phase components are emphasized.

Conclusions:

• The proposed Bayesian framework offers a unified and flexible approach to functional data analysis.
• It enhances the ability to estimate and register functions accurately, even with imperfect data.
• The method's utility is demonstrated through simulations and real-world applications.