Modeling Conditional Distributions for Functional Responses, With Application to Traffic Monitoring via GPS-Enabled Mobile Phones

Motivated by problems involving a traffic monitoring system in which trajectory data are obtained from Global Positioning System-enabled mobile phones, we propose a novel approach to functional regression modeling, where instead of the usual mean regression the entire distribution of functional responses is modeled conditionally on predictors. An approach that sensibly balances flexibility and stability is obtained by assuming that the response functions are drawn from a Gaussian process, the mean and covariance function of which depend on predictors. The dependence of the mean function and covariance function of the response on the predictors is modeled additively. We demonstrate the proposed methods by constructing predicted curves and corresponding prediction regions for traffic velocity trajectories for a future time period, using current traffic velocity fields as predictor functions. The proposed functional regression and conditional distribution approach is of general interest for functional response settings, where in addition to predicting the conditional mean response function one is also interested in predicting the covariance surface of the random response functions, conditional on predictor curves.