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Nonlinear functional models for functional responses in reproducing kernel Hilbert spaces

Heng Lian

TL;DR

The paper addresses nonlinear functional regression with functional responses by developing a nonlinear functional regression framework within an operator-valued RKHS and deriving a representer-based estimator. It implements a Gaussian operator kernel with grid-based discretization and uses generalized cross-validation to select smoothing, demonstrating advantages over linear and simple kernel methods in nonlinear settings. Through simulations and a weather data application, it shows that the RKHS approach can capture complex x→y mappings and provides a practical tool for functional response modeling, albeit with higher computational cost. Overall, the work broadens functional data analysis by enabling nonparametric handling of functional responses and paves the way for future scalability improvements.

Abstract

An extension of reproducing kernel Hilbert space (RKHS) theory provides a new framework for modeling functional regression models with functional responses. The approach only presumes a general nonlinear regression structure as opposed to previously studied linear regression models. Generalized cross-validation (GCV) is proposed for automatic smoothing parameter estimation. The new RKHS estimate is applied to both simulated and real data as illustrations.

Nonlinear functional models for functional responses in reproducing kernel Hilbert spaces

TL;DR

The paper addresses nonlinear functional regression with functional responses by developing a nonlinear functional regression framework within an operator-valued RKHS and deriving a representer-based estimator. It implements a Gaussian operator kernel with grid-based discretization and uses generalized cross-validation to select smoothing, demonstrating advantages over linear and simple kernel methods in nonlinear settings. Through simulations and a weather data application, it shows that the RKHS approach can capture complex x→y mappings and provides a practical tool for functional response modeling, albeit with higher computational cost. Overall, the work broadens functional data analysis by enabling nonparametric handling of functional responses and paves the way for future scalability improvements.

Abstract

An extension of reproducing kernel Hilbert space (RKHS) theory provides a new framework for modeling functional regression models with functional responses. The approach only presumes a general nonlinear regression structure as opposed to previously studied linear regression models. Generalized cross-validation (GCV) is proposed for automatic smoothing parameter estimation. The new RKHS estimate is applied to both simulated and real data as illustrations.

Paper Structure

This paper contains 7 sections, 3 theorems, 24 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reproducing kernel Hilbert spaces
  3. Models for functional data
  4. Examples
  5. Simulation study
  6. Application to the weather data
  7. Conclusions

Key Result

Theorem 1

Given the observations $\{(x_i,y_i)\}_{i=1}^n$, the solution to (hyperopt) has the following representation with functional coefficients $\alpha_i\in H$.

Figures (3)

  • Figure 1: Comparison of the GCV with the error computed from validation data. The solid curves are the GCV estimates, the dashed curves are the error computed from validation set. The curves are shifted and normalized to show the shape of the curves rather than its absolute magnitudes.
  • Figure 2: Daily weather data for 35 Canadian stations, the curves plotted here result from using smoothing splines to fit the raw data.
  • Figure 3: Raw data (points) and predictions (solid) of log precipitation for four weather stations.

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3