Functional data analysis with covariate-dependent mean and covariance structures.

Area of Science:

• Statistics
• Functional Data Analysis

Background:

• Functional data analysis is increasingly used for continuous data.
• Existing models often assume a common covariance structure, limiting covariate linkage.
• There's a need for models that connect covariance structures to external covariates.

Purpose of the Study:

• To propose a new functional regression model with covariate-dependent mean and covariance structures.
• To enhance the interpretability and prediction power of functional models.
• To develop a robust method for model selection and estimation.

Main Methods:

• Developed a functional regression model allowing covariate-dependent variances of random scores.
• Introduced a penalized quasi-likelihood procedure combining regularization and B-spline smoothing.
• Established convergence rates and asymptotic normality for the proposed estimators.

Main Results:

• The proposed model successfully links covariance structures to external covariates.
• Identified individual-specific eigenfunctions, improving interpretability.
• Demonstrated utility through simulations and real-world data analysis.

Conclusions:

• The new functional regression model offers improved interpretability and prediction.
• The penalized quasi-likelihood procedure provides effective model selection and estimation.
• The method yields biologically relevant insights, as shown in the Avon Longitudinal Study.