Functional Additive Mixed Models.

Area of Science:

• Statistics
• Functional Data Analysis
• Mixed-Effects Modeling

Background:

• Analyzing correlated functional responses presents challenges due to complex dependencies and covariate effects.
• Existing methods often lack flexibility in handling multiple nested or crossed functional random effects and smooth covariate variations.

Purpose of the Study:

• To develop an extensive framework for additive regression models tailored to correlated functional responses.
• To incorporate flexible correlation structures for spatial, temporal, or longitudinal functional data.
• To model linear and nonlinear effects of functional and scalar covariates with smooth variations.

Main Methods:

• The framework utilizes additive regression models for correlated functional responses.
• It accommodates multiple partially nested or crossed functional random effects with flexible correlation structures.
• Estimation and inference are based on standard additive mixed models, employing robust and flexible algorithms.

Main Results:

• The proposed method reliably recovers relevant effects, even with small sample sizes.
• The framework demonstrates scalability to larger datasets.
• Simulations and applications show good performance for spatial and longitudinal functional data.

Conclusions:

• The developed framework provides a flexible and interpretable approach for modeling complex functional data.
• The open-source R function `pffr()` facilitates practical application of these advanced statistical methods.
• The approach is suitable for diverse functional data types, including spatial and longitudinal observations.