Table 1.
The generalized radiation models with scaling exponent, searching direction and trip OD constraint.
From left to right, we generalize the model by categorizing it as intervention-based and competition-based with regard to the motivations of individual human travels. From top to bottom, we further generalize the model by adding trip OD constraints into the model, and obtain the production-constrained and the attraction-constrained forms of the radiation model.
Fig 1.
Parameter estimation for the generalized radiation model.
In the top panels, we vary the scaling exponent λ for each model, and compute the Sørensen-Dice similarity between the model output and the real data. The λ yielding the peak similarity value is the estimated parameter of the adopted model. In the middle panels, the Pearson Correlation Coefficient (PCC) is calculated between the model output and the real data. In the bottom panels, the correct normalization factor κ is derived for models with different scaling exponents. Note that the estimated scaling exponent λ is 1.0, 0.3, 0.4, 0.6, 0.8, 0.2 for US, UK, Beijing, Shenzhen, Abidjan and Chicago, respectively.
Fig 2.
Comparing the predictions of the generalized radiation model and the real data.
In the first column, these panels display the overall travel distance distribution reported in the empirical data and the fitted models. It reveals the probability of a trip between two locations that are at distance r (in km) from each other, Pdist(r). These distributions are generally collapsed with each other, indicating the predictions of the model are acceptable. In the second column, the panels display the distributions of fluxes associated with given population at destination or origin. It denotes the probability of a trip from or towards a location with population n, Pdist(n). Again, agreements between the model output and the real data are observed. In columns 3 to 6, these panels compares the observed flux, Tdata, with the predicted flux, Tmodel, for each pair of i, j counties where real travel flux exists. Note that gray points are scatter plot for each pair of locations. A box is colored green if the diagonal line lies between the 5th and the 95th percentiles in that bin and is red otherwise. The black circles correspond to the mean number of predicted travelers in that bin. In general, the generalized radiation model predicts travel fluxes in well agreement with the real data, except for flows with large volumes.
Table 2.
Agreements between the travel fluxes predicted by the generalized radiation model and the travel fluxes observed in the real data.
We conduct the two-sample Kolmogorov-Smirnov test to verify whether the distributions of the model and the data are from an identical distribution at the 5% significance level in terms of overall travel distance distribution and travels towards (or from) given population. Values in red indicates non-agreements between the model prediction and the real data.
Fig 3.
Model error as a function of spatial distance and population at destination.
In the left panels, we show the normalized probability of traveling associated with given distance and population observed in the real data. The lighter the color is, the higher the probability is, which implies where travel fluxes in the mobility network are concentrated. In other panels, we show the Sørensen-Dice coefficient of the generalized radiation models. Note that the coefficient is in the range of 0 and 1 for all the case studies, and is assigned the same color for the same value. The lighter the color is, the higher the similarity between the model output and the real data is, which indicates the power of the predicting models over different distances and origin or destination population. The dashed line shows the trend of travel flux and the similarity index distributed along with distance and population.