Nonlinear function-on-scalar regression via functional universal approximation.

Area of Science:

• Statistics
• Functional Data Analysis
• Machine Learning

Background:

• General nonlinear function-on-scalar (FOS) regression models are challenging due to complex predictor-response relationships.
• Existing methods often assume specific model forms or struggle with high-dimensional scalar predictors.

Purpose of the Study:

• To develop a novel method for fitting general nonlinear FOS regression models.
• To overcome the limitations of existing methods in handling numerous scalar predictors and unknown nonlinear forms.

Main Methods:

• Developed a functional universal approximation theorem inspired by neural network theory.
• Utilized smoothness regularity and estimated a sequence of bivariate functions to avoid high-dimensional estimation.
• The method avoids specific model form assumptions.

Main Results:

• The proposed method successfully fits general nonlinear FOS regression models without restrictive assumptions.
• It effectively handles models with a relatively large number of scalar predictors.
• Empirical studies on simulated and real datasets demonstrate the method's good performance.

Conclusions:

• The novel method provides a flexible and powerful tool for analyzing complex FOS relationships.
• It advances functional data analysis by enabling the modeling of intricate nonlinearities with multiple predictors.
• The approach offers improved prediction accuracy and broader applicability in statistical modeling.