Iterative estimation of the Heteroskedastic Functional Regression Model with application to atmospheric data

In recent years there has been an increasing interest in analyzing and modeling functional data that, in environmental studies, often arise when dense sets of measurements are recorded over a period of time or over some domain, such as depth or height. Meanwhile the availability of atmospheric measurements is becoming larger and larger since high technology remote instruments, such as radiosondes or LIDAR, provide vertical atmospheric profiles of Essential Climate Variables (ECVs), like pressure, temperature, water vapour, wind and aerosol. In this work, we introduce an heteroskedastic functional regression model which extends the standard concurrent functional linear model to understand the variability of important atmospheric parameters and its relation to environmental factors, height and distance. In order to estimate functional coefficients of both the conditional mean and variance, we re-write a functional linear model as a standard generalized additive model. Then to handle heteroskedasticity, we implement an iterative algorithm which is started by a preliminary standard generalized additive model estimation for the functional mean. Then it is given by iterating, up to convergence, the following two steps: a functional regression model estimation step applied to the squared residuals, and an heteroskedastic mixed model estimation step for the functional mean, the latter obtained by pretending that the variance function estimated at the first step is the actual one. The proposed approach is illustrated by the co-location uncertainty analysis of two co-located stations involved in GCOS reference upper-air network (GRUAN).