Fig 1.
Flowchart of the Bayesian model selection process under the fine-scale model approach.
The input stages are shown in yellow, and the processing stages are shown in orange, while the output stages are in green.
Table 1.
List of normalized covariates used in selecting our migration model and their corresponding distributions.
Fig 2.
Schematic of the three spatial interaction modeling approaches and subsequent validation.
In the first approach (fine-scale), model coefficients are estimated at the municipality level (n = 1122) (C) and used to estimate municipality level migrations, which are subsequently aggregated to the corresponding departments of origin and destination. The estimated department level flows are then used to calculate likelihood (select the best model) based on the observed migrations at the department level (n = 32) (A). In the second approach (broad-scale), model coefficients are estimated at the department level (n = 32) (D) yielding in predicted migration proportions which are used to calculate the likelihood based on observed department level migrations (A). In the third approach (intermediate-scale), model coefficients are estimated at the intermediate level (n = 276) (E), migration proportions are predicted at the intermediate level, and likelihood calculated based on observed intermediate level migrations (n = 276, including 147 single municipality units) (B). To validate our fine-scale model estimates, we used a subset of the intermediate level observation (B) that only includes migrations to and from single-municipality units. Bold lines represent observed data, while deemed lines represent estimates corresponding to each approach namely: fine-scale (red), broad-scale (green) and intermediate (blue).
Table 2.
Iterative model selection based on Deviance Information Criterion (DIC) using forward step-wise inclusion for fine-scale models, with a single variable added at each iteration.
Fig 3.
Posterior distribution of coefficients selected based on the best fine-scale model (red) compared to parameter values we would get when fitting using the broad-scale modeling approach while covariates are restricted to those selected in the best fine-scale model (red); and compared to parameter values we would get when fitting using the intermediate-scale modeling approach while covariates are restricted to those selected in the best fine-scale (blue).
All coefficients were significant at least at the 0.05 level.
Table 3.
Coefficients of the best fine-scale model.
Fig 4.
Estimated versus observed migration flows (in log scale) between each pair of departments with estimated flows (A) based on the results of the fine-scale model, aggregated to the department level, and (B) based on the results of the broad-scale model.
Fig 5.
Observed (A) and estimated (B and C) migration flows between eight selected departments having the highest observed migration flows either as destination or origin. Estimated flows are (B) based on the results of the fine-scale model, aggregated to the department level, and (C) based on the results of the broad-scale model. Centroid points are weighted by the spatial distribution of population within each department.
Fig 6.
Predicted versus observed migration flows (sorted by magnitude) in eight departments characterized by (A & B) high, (C & D) medium, (E & F) low, and (G & H) very low incoming migration flows. The shaded violin plots show the 95% confidence interval based on the joint posterior sample parameters, while the black dots represent the observed flows. The four categories were selected based on the maximum number of migrants each department would receive based on our estimates, with each of the four departments selected randomly from each quartile.
Fig 7.
Estimated migration flows (A) between each pair of municipalities with lines going in both directions, (B) between municipalities that have the 20 highest migration flows and have distances of more than 100 km from each other, and (C) between municipalities that have the 10 highest migration flows and are within 100 km from Bogota. Centroid points are weighted by the spatial distribution of population in each municipality.
Fig 8.
Estimated versus observed migration flows (in log scale) between each pair of intermediate level census units, with estimates based (A) on the intermediate-scale model using the intermediate level migration data, (B) on the broad-scale model by assigning the predicted department level migration proportions to the corresponding single municipalities and multi-municipalities, (C) on the fine-scale model by aggregating the estimated municipality level migration flows to the corresponding intermediate level census units, and (D) on the fine-scale model as in C, but considering only single-municipalities. Red lines represent the identity line, while E through H show histograms of the residuals (in log scale) for the corresponding plots in A-D.