  We propose a prediction procedure for the functional linear quantile
regression model by using partial quantile covariance techniques and develop a
simple partial quantile regression (SIMPQR) algorithm to efficiently extract
partial quantile regression (PQR) basis for estimating functional coefficients.
We further extend our partial quantile covariance techniques to functional
composite quantile regression (CQR) defining partial composite quantile
covariance. There are three major contributions. (1) We define partial quantile
covariance between two scalar variables through linear quantile regression. We
compute PQR basis by sequentially maximizing the partial quantile covariance
between the response and projections of functional covariates. (2) In order to
efficiently extract PQR basis, we develop a SIMPQR algorithm analogous to
simple partial least squares (SIMPLS). (3) Under the homoscedasticity
assumption, we extend our techniques to partial composite quantile covariance
and use it to find the partial composite quantile regression (PCQR) basis. The
SIMPQR algorithm is then modified to obtain the SIMPCQR algorithm. Two
simulation studies show the superiority of our proposed methods. Two real data
from ADHD-200 sample and ADNI are analyzed using our proposed methods.
