Characterizing how people move through space has been an important component of many disciplines. With the advent of automated data collection through GPS and other location sensing systems, researchers have the opportunity to examine human mobility at spatio-temporal resolution heretofore impossible. However, the copious and complex data collected through these logging systems can be difficult for humans to fully exploit, leading many researchers to propose novel metrics for encapsulating movement patterns in succinct and useful ways. A particularly salient proposed metric is the mobility entropy rate of the string representing the sequence of locations visited by an individual. However, mobility entropy rate is not scale invariant: entropy rate calculations based on measurements of the same trajectory at varying spatial or temporal granularity do not yield the same value, limiting the utility of mobility entropy rate as a metric by confounding inter-experimental comparisons. In this paper, we derive a scaling relationship for mobility entropy rate of non-repeating straight line paths from the definition of Lempel-Ziv compression. We show that the resulting formulation predicts the scaling behavior of simulated mobility traces, and provides an upper bound on mobility entropy rate under certain assumptions. We further show that this formulation has a maximum value for a particular sampling rate, implying that optimal sampling rates for particular movement patterns exist.

Accurate prediction of the motion of objects is a central scientific goal. For deterministic or stochastic processes, models exist which characterize motion with a high degree of reliability. For complex systems, or those where objects have a degree of agency, characterizing motion is far more challenging. The information entropy rate of motion through a discrete space can place a limit on the predictability of even the most complex or history-dependent actor, but the variability in measured encountered locations is inexorably tied to the spatial and temporal resolutions of those measurements. This relation depends on the path of the actor in ways that can be used to derive a general law in closed form relating the mobility entropy rate to different spatial and temporal resolutions, and the path properties within each cell along the path. Correcting for spatial and temporal effects through regression yields the path properties and a measure of mobility entropy rate robust to changes in dimension, allowing comparison of mobility entropy rates between datasets. Employing this measure on empirical datasets yields novel findings, from the similarity of taxicabs to drifters, to the predictable motions of undergraduates, to the browsing habits of Canadian moose.

Wedding mobile phone sensor technology and human spatial behaviour has great potential. The ubiquity of Global Positioning Systems (GPS) technology has made gathering data about human mobility simpler, more precise, and with higher fidelity, providing minute-by-minute records of the locations of cohorts from dozens of participants. While this data provides a strong basis for Geographic Information Science research, it also constitutes an invasion of the participants’ privacy and can provide more information than researchers require to answer their questions. As an ethical and practical consideration, researchers should gather only as much data as they need. In this paper, we take three weeks of GPS traces from over a hundred student participants in mobile phone-based tracking studies and show that fewer than 14 days of data is necessary to establish complete activity spaces. We define ‘complete’ as the point at which marginal information gains become negligible according to a pairwise temporal analysis of the Kullback–Leibler (KL) divergence of the spatial (bivariate) histogram through time. For the fixed level of information difference, observable in the data, impacts due to individual variability, population composition, and spatial resolution are evident. However, all populations at each level of resolution examined in the paper demonstrated convergence to low divergence levels occurred within a matter of days, and to negligible information gain in less than two weeks. The methods described in the paper represent a novel metric useful to understand the interaction between measurements and information in human mobility.

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