Gravity and spatial interaction models

R arm of the Python modules morie.fn.igrav*. Igrav: log-linear gravity OLS on the positive flows. Igravbl: iterative proportional fitting to origin and destination totals (as stats::loglin with a start table). Igravcl: distance-decay exponent by Poisson ML with the mass product as offset (reproduces total flow and mean log distance). Igraver: RMSE and standardised RMSE. Igravfe: PPML with origin and destination fixed effects. Igravlm: LM error test on gravity residuals. Igravnb: negative-binomial (NB2) gravity by ML, as MASS::glm.nb. Igravrt: Huff retail probabilities. Igravsq: symmetric gravity on a square OD matrix with an asymmetry index. Igravvf: variance function \(\phi\mu^p\). Igravwl: Wilson's doubly constrained entropy model.

Usage

Igrav(flows, mass_o, mass_d, dist)

Igravbl(flow_matrix, row_totals, col_totals, tol = 1e-12, max_iter = 10000)

Igravcl(flows, mass_o, mass_d, dist)

Igraver(flows, flows_hat)

Igravfe(flows, origin_id, dest_id, dist)

Igravlm(resid, W)

Igravnb(flows, mass_o, mass_d, dist)

Igravrt(mass, dist, beta = 2, demand = NULL)

Igravsq(flow_matrix, mass, dist_matrix)

Igravvf(flows_hat, phi = 1, power = 1)

Igravwl(flows, mass_o, mass_d, dist, beta = 1, tol = 1e-12, max_iter = 10000)

Arguments

flows

Observed flows (vector of pairs; a matrix for Igravwl, or NULL there).

mass_o, mass_d

Origin and destination masses (totals for Igravwl).

dist

Distances (a cost matrix for Igravwl, a consumer by centre matrix for Igravrt).

flow_matrix

Square or rectangular flow matrix.

row_totals, col_totals

Target totals.

tol

Relative tolerance.

max_iter

Iteration cap.

flows_hat

Predicted flows.

origin_id, dest_id

Origin and destination labels.

resid

Gravity residuals.

W

Weights between origin-destination pairs.

mass

Attractiveness (Igravrt) or masses of the places (Igravsq).

beta

Distance- or cost-decay parameter.

demand

Demand at each consumer location.

dist_matrix

Distances between the places.

phi

Dispersion.

power

Variance power.

Value

A list whose statistic is the headline value, with the components of the Python result.

References

Tinbergen, J. (1962). Shaping the World Economy. Twentieth Century Fund, New York.

Santos Silva, J. M. C. and Tenreyro, S. (2006). The log of gravity. Review of Economics and Statistics 88, 641-658.

Wilson, A. G. (1967). A statistical theory of spatial distribution models. Transportation Research 1, 253-269.

Deming, W. E. and Stephan, F. F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Annals of Mathematical Statistics 11, 427-444.

Huff, D. L. (1963). A probabilistic analysis of shopping center trade areas. Land Economics 39, 81-90.

Hyman, G. M. (1969). The calibration of trip distribution models. Environment and Planning 1, 105-112.

Anderson, J. E. and van Wincoop, E. (2003). Gravity with gravitas. American Economic Review 93, 170-192.

Knudsen, D. C. and Fotheringham, A. S. (1986). Matrix comparison, goodness-of-fit, and spatial interaction modeling. International Regional Science Review 10, 127-147.

Examples

Fl <- c(12, 3, 30, 7, 55, 4, 9, 21)
mo <- c(5, 5, 9, 9, 20, 20, 7, 7)
md <- c(9, 20, 5, 20, 5, 9, 20, 9)
d <- c(1, 3, 1, 2, 3, 2, 2.5, 1.5)
Igrav(Fl, mo, md, d)$coefficients
#> [1]  7.4646102 -0.5715425 -1.7577700  0.5894246
Igravcl(Fl, mo, md, d)$statistic
#> [1] 0.9079358
Igravbl(matrix(1:4, 2, byrow = TRUE), c(5, 5), c(6, 4))$balanced
#>          [,1]     [,2]
#> [1,] 2.757651 2.242349
#> [2,] 3.242349 1.757651