Function-on-Function Partial Quantile Regression

A function-on-function linear quantile regression model, where both the response and predictors consist of random curves, is proposed by extending the classical quantile regression setting into the functional data to characterize the entire conditional distribution of functional response. 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. Although the functional random variables belong to an infinite-dimensional space, they are observed in a finite set of discrete-time points in practice. Thus, 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. The finite-sample performance of the proposed method is compared with the functional partial least squares method. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method.