Good estimates of how many people frequently travel between two places is related to several areas of study. It is not only a key ingredient in modeling the spreading of infectious diseases1,2,3,4, but also a fundamental problem in geo-spatial economics of facility distribution and transportation planning. Clues on the statistics of human movements and their changes at different scales are thus fundamental. Here we explore statistical expressions to calculate the number of commuting trips within areas of sizes ranging from a few kilometers (one city or town) to over one thousand kilometers (one country). We focus on commuting trips because they are stable in time and account for the largest fraction of the total flows in a population.#R##N##R##N#In order to describe the commuting flow patterns of people, various aggregated trip distribution models have been proposed5,6,7,8,9,10. Among all these efforts, the gravity model, which assumes that the number of movements between an origin-destination pair decays with their distance, is the most widely used one6,11,12. The gravity model can be further divided into the unconstrained gravity model5, the singly constrained gravity model (such as the B.P.R model13,14, or the Voorhees gravity model11) and the doubly constrained gravity model15. More constraints generally leads to better model performance. The doubly constrained gravity model fits parameters in an iterative way to best reconstruct the empirical OD matrix. But the iterative fitting complicates the calibration of the distance decaying function, which is essential for the gravity model. However, the most important limitation of the doubly constrained gravity model is the calibrated parameters from the empirical trip data in one area lack meaningful interpretation and thus are not transferable to other regions. As a consequence, in current epidemiology studies the more generalizable but less accurate unconstrained gravity model is often used1.#R##N##R##N#From another perspective, the recently proposed radiation model16,17 has an analytical formulation that estimates trip volumes without parameters; using as input the distribution of population only. It takes inspiration from the intervening opportunity model, which assumes that the trip volume is more related to the number of opportunities between the origin and the destination instead of just to their distance8,18. As is recently shown by Simini et al.17, the radiation model represents a general framework that includes the intervening opportunity model as a degenerated case and it is linked with the the gravity model through the spatial distribution of opportunities. The model was originally developed to predict inter-county flows, which not necessarily account for daily trips. Some recent works19,20,21 have shown that the parameter-free radiation model16 does not work well at predicting intra-city trips. Two of these works19,20 introduced models with a calibrating parameter as a cost function of the distance to reproduce intra city trips. In the gravity model based approach19 a scaling parameter depending on region size is introduced; while in intervening opportunity model based approaches introducing such an interpretable parameter that can be estimated without trip data remains an open question.#R##N##R##N#The present work seeks to answer two questions: How the value of the parameter that imposes the scale dependency can be interpreted and estimated without trip data? Under which conditions the prediction of models not calibrated with empirical trip data would work? The extended radiation model that we present in this study is distinct from previous formulations, in that it is a stochastic model that depends on the distributions of opportunities and population only. The one scaling parameter α depends on the region size and the heterogeneity of opportunity distribution, which makes it interpretable and estimable in many cases even without trip data. The extended radiation model is derived by relating the trip production and attraction process to survival analysis. We propose an analogy between survival time t and the number of opportunities a a commuter has considered. Another important ingredient for modelling trips within small scales (intra urban trips) is the separation between population density and trip attraction rates. Most models on intra-city trips use the density of population20,21 as a proxy for both trip generation and trip attraction rates. While this approximation is reasonable at large scales, at inner city scale trip attraction is better represented by the density of point of interests (POIs), defined as geolocated non residential establishments presented on a digital map.#R##N##R##N#The proposed model combines the closed analytical form of the original radiation model and the flexibility of gravity-like models. While it can be calibrated when empirical trip data is available, it also provides an analytical parameter estimation when there is no trip data for calibration. The model is validated in the U.S. by the census commuting data and in three other countries (Portugal, Dominican Republic, and Rwanda) by cell phone records.
['Yingxiang Yang', 'Carlos Herrera', 'Nathan Eagle', 'Marta C. González']
Limits of Predictability in Commuting Flows in the Absence of Data for Calibration