Interaction Models for Functional Regression.

Area of Science:

• Statistics
• Functional Data Analysis
• Statistical Modeling

Background:

• Traditional regression models often fail to capture complex interactions in functional data.
• Existing methods may lack the flexibility to model two-way interactions between multiple functional predictors.
• Accurate modeling of interactions is crucial for robust statistical inference and prediction.

Purpose of the Study:

• To propose a novel functional regression model capable of handling scalar responses with multiple functional predictors.
• To incorporate and effectively model two-way interaction effects alongside main effects.
• To develop an estimation and hypothesis testing procedure for functional interaction effects.

Main Methods:

• Utilized penalized regression splines for modeling main effects.
• Employed a tensor product basis for modeling the functional interaction effect.
• Extended the model to generalized linear models and addressed sparse/noisy data.

Main Results:

• Numerical studies demonstrated that ignoring interactions leads to poor estimation and prediction.
• Fitting an interaction model when no interaction exists resulted in minimal performance loss.
• The proposed method showed superior performance compared to additive models when interactions are present.

Conclusions:

• The proposed functional regression model effectively captures two-way interactions between functional predictors.
• The methodology provides a flexible and implementable approach for analyzing complex functional data.
• The developed method offers significant advantages in estimation and prediction accuracy.