A Theoretical Basis for Entropy-Scaling Effects in Human Mobility Patterns

Abstract

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.

Comparing Regional Patterns of Individual Movement Using Corrected Mobility Entropy

“…Lin et al (2012: 386) noted this volatility when their experiments suggested "changing temporal scales has similar effects on the predictability of different individuals, while changing the spatial scale has different effects, depending on the mobility characteristics of each individual." Osgood et al (2016: 3) state: comparing different individuals in the same dataset could be problematic if there is heterogeneity in the geographic bin size or sampling rate; for example, in a study comparing the mobility of rural and urban populations through cell phone records, where the rural Voronoi cells were systemically and significantly larger than their urban counterparts. Osgood et al (2016) are, to the best of the authors' knowledge, the first and only to provide a theoretical derivation of a scaling law for mobility entropy.…”

“…Osgood et al (2016: 3) state: comparing different individuals in the same dataset could be problematic if there is heterogeneity in the geographic bin size or sampling rate; for example, in a study comparing the mobility of rural and urban populations through cell phone records, where the rural Voronoi cells were systemically and significantly larger than their urban counterparts. Osgood et al (2016) are, to the best of the authors' knowledge, the first and only to provide a theoretical derivation of a scaling law for mobility entropy. Using Lempel-ziv compression on non-repeating straight-line paths to estimate mobility entropy, they show how ME's scaling behavior can be described by four terms: the length of the path, the average velocity of the agent, the width of the spatial bin, and the period of the sampling rate.…”

On entropy research analysis: cross-disciplinary knowledge transfer

“… 2006 ; Srivastav and Simonovic 2015 ; Osgood et al. 2016 ; Hirsh et al. 2012 ; Almeida 2001 ; Cabrera et al.…”

Multiscale entropy rate analysis of complex mobile agents

“…Because of its reliance on cellular call records, as in subsequent works [ 2 – 4 ], their dataset was subject to bias both by its focus on a particular demographic, and adherence to a particular spatial and temporal resolution. More recent empirical research has established that the estimated predictability of human mobility is contingent on the scale [ 5 ] and structure [ 6 , 7 ] of the data, and underlying mobility model assumptions [ 8 ]. While Song et al made a foundational contribution to quantifying mobility predictability in complex systems, their results are only applicable to the population and spatio-temporal resolution [ 8 ] represented in the data.…”

“…If the combination of these properties is deterministic, then a model could be derived which separated the system error (spatio-temporal resolution) from the observation (path properties). This paper builds upon the results of [ 8 ] to derive a general solution to the scaling of mobility entropy rate estimates in a discretized space, based on observations about the structure of paths through a discrete grid. The model shows considerable agreement with empirical data for agents as diverse as university students, taxicabs, moose and ocean buoys.…”