Truncated Linear Models for Functional Data

TL;DR

This paper develops methods to estimate the unknown support of the coefficient function in a functional linear model, with applications to particulate emissions data, including theoretical analysis and practical algorithms.

Contribution

It introduces novel techniques for jointly estimating the support, the coefficient function, and the intercept in truncated functional linear models.

Findings

Effective estimation of support boundaries demonstrated through simulations.

Application to real particulate emissions data shows practical utility.

Theoretical properties and identifiability conditions established.

Abstract

A conventional linear model for functional data involves expressing a response variable $Y$ in terms of the explanatory function $X(t)$, via the model: $Y=a+\int_I b(t)X(t)dt+\hbox{error}$, where $a$ is a scalar, $b$ is an unknown function and $I=[0, \alpha]$ is a compact interval. However, in some problems the support of $b$ or $X$, $I_1$ say, is a proper and unknown subset of $I$, and is a quantity of particular practical interest. In this paper, motivated by a real-data example involving particulate emissions, we develop methods for estimating $I_1$. We give particular emphasis to the case $I_1=[0,\theta]$, where $\theta \in(0,\alpha]$, and suggest two methods for estimating $a$, $b$ and $\theta$ jointly; we introduce techniques for selecting tuning parameters; and we explore properties of our methodology using both simulation and the real-data example mentioned above. Additionally, we derive theoretical properties of the methodology, and discuss implications of the theory. Our theoretical arguments give particular emphasis to the problem of identifiability.