Function-on-function partial quantile regression

TL;DR

This paper introduces a novel functional partial quantile regression method for estimating function-on-function models, utilizing basis expansions and an iterative component extraction process, with applications to air quality data.

Contribution

It proposes a new functional partial quantile regression approach that handles multiple predictors, uses basis expansions, and incorporates variable selection and bootstrap prediction intervals.

Findings

Effective in Monte Carlo simulations across various data scenarios.

Successfully applied to real air quality data demonstrating practical utility.

Outperforms existing methods in predictive accuracy and variable selection.

Abstract

In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. In our proposal, the functional forms of the discretely observed random variables are first constructed via a finite-dimensional basis function expansion method. The functional partial quantile regression constructed using the functional random variables is approximated via the partial quantile regression constructed using the basis expansion coefficients. The proposed method uses an iterative procedure to extract the partial quantile regression components. A Bayesian information criterion is used to determine the optimum number of retained components. The proposed functional partial quantile regression model allows for more than one functional predictor in the model. However, the true form of the proposed model is unspecified, as the relevant predictors for the model are unknown in practice. Thus, a forward variable selection procedure is used to determine the significant predictors for the proposed model. Moreover, a case-sampling-based bootstrap procedure is used to construct pointwise prediction intervals for the functional response. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments under different data generation processes and error distributions. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method.