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

Abstract

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.

• Additive model
• Conditional covariance
• Conditional distribution in function space
• Functional data analysis
• Functional regression
• Prediction region
• Velocity curve

ACKNOWLEDGMENTS

The authors gratefully acknowledge UC Berkeley for releasing the Mobile Century data for research purpose, and want to thank the editor, an associate editor, and two referees for the very constructive comments. This research was supported by NSF grants DMS-1104426 and DMS-1228369.