Measuring human activity spaces from GPS data with density ranking and summary curves

Activity spaces are fundamental to the assessment of individuals' dynamic exposure to social and environmental risk factors associated with multiple spatial contexts that are visited during activities of daily living. In this paper we survey existing approaches for measuring the geometry, size and structure of activity spaces based on GPS data, and explain their limitations. We propose addressing these shortcomings through a nonparametric approach called density ranking, and also through three summary curves: the mass-volume curve, the Betti number curve, and the persistence curve. We introduce a novel mixture model for human activity spaces, and study its asymptotic properties. We prove that the kernel density estimator which, at the present time, is one of the most widespread methods for measuring activity spaces is not a stable estimator of their structure. We illustrate the practical value of our methods with a simulation study, and with a recently collected GPS dataset that comprises the locations visited by ten individuals over a six months period.