# Quadratic regression for functional response models

## TL;DR

This paper introduces a quadratic functional regression model that captures interactions in functional data, estimated via penalized likelihood, and demonstrates its effectiveness through simulations and meteorological data analysis.

## Contribution

It extends the functional linear model to include quadratic terms with interaction effects, using basis expansions and penalized likelihood estimation.

## Key findings

- The quadratic model outperforms linear models in simulations.
- The method effectively analyzes meteorological data.
- Penalized likelihood provides robust parameter estimation.

## Abstract

We consider the problem of constructing a regression model with a functional predictor and a functional response. We extend the functional linear model to the quadratic model, where the quadratic term also takes the interaction between the argument of the functional data into consideration. We assume that the predictor and the coefficient functions are expressed by basis expansions, and then parameters included in the model are estimated by the penalized likelihood method assuming that the error function follows a Gaussian process. Monte Carlo simulations are conducted to illustrate the efficacy of the proposed method. Finally, we apply the proposed method to the analysis of meteorological data and explore the results.