On the Number of Independent Pieces of Information in a Functional Linear Model with a Scalar Response

Eduardo L Montoya

doi:10.3390/stats3040032

Abstract

In a functional linear model (FLM) with scalar response, the parameter curve quantifies the relationship between a functional explanatory variable and a scalar response. While these models can be ill-posed, a penalized regression spline approach may be used to obtain an estimate of the parameter curve. The penalized regression spline estimate will be dependent on the value of a smoothing parameter. However, the ability to obtain a reasonable parameter curve estimate is reliant on how much information is present in the covariate functions for estimating the parameter curve. We propose to quantify the information present in the covariate functions to estimate the parameter curve. In addition, we examine the influence of this information on the stability of the parameter curve estimator and on the performance of smoothing parameter selection methods in a FLM with a scalar response.

Highlights

Functional data analysis (FDA) continues to be an active and growing area of research as measurements from continuous processes are increasingly becoming prevalent in many fields.This type of data is functional data because they can be viewed as samples from curves
Ζ (x(t)), for a functional linear model (FLM) with a scalar response to determine how much information is present in the covariate curves for estimating the parameter curve β(t) when the parameter curve is identifiable
To estimate the parameter curve in model (1), penalized regression spline estimation is used, and we summarize several commonly used methods for selecting the smoothing parameter

Summary

Introduction

Functional data analysis (FDA) continues to be an active and growing area of research as measurements from continuous processes are increasingly becoming prevalent in many fields. We review the performance of different smoothing parameter selection methods based on the amount of information present in the covariate functions for estimating β(t).

Number of Independent Pieces of Information in a FLM

A Simulation Study

A Real Data Illustration

Discussion