The function computes the Jacobian matrix of the residual map with respect to
the attractivenesses. The function returns NA when applied to spatial
interaction models for which the map is not defined.
Value
a \(p\times p\) matrix, where \(p\) is the number of destination
locations, or NA when the residual map is not defined
Details
The residual map \(\mathbf{G}\) is given by
$$G_j=D_j-\kappa_j Z_j,$$
where \(D_j\) is the destination flow (destination_flow()),
\(Z_j\) the attractiveness (attractiveness()) and \(\kappa_j\) the
conversion factor (sim_conversion()), all attached to destination \(j\).
The equilibria of the Harris and Wilson model (see blvim()) are the zeros
of \(\mathbf{G}\).
The derivatives of \(\mathbf{G}\) w.r.t. \(\mathbf{Z}\) can be used to analyse the stability of the solutions, among other applications.
Examples
distances <- french_cities_distances[1:10, 1:10] / 1000 ## convert to km
production <- log(french_cities$population[1:10])
attractiveness <- log(french_cities$area[1:10])
model <- static_blvim(distances, production, 1.5, 1 / 250, attractiveness)
sim_residual_jacobian(model) ## must be NA
#> [1] NA
dyn_model <- blvim(distances, production, 1.5, 1 / 250, attractiveness)
sim_residual_jacobian(dyn_model)
#> 75056 13055 69123 31555 06088
#> 75056 -6.453756e-01 -3.354732e-01 -2.088890e-08 -9.584645e-08 -2.917354e-09
#> 13055 -3.546244e-01 -6.645268e-01 -2.924378e-08 -1.748035e-07 -1.515684e-08
#> 69123 -1.814987e-23 -2.403702e-23 -9.999999e-01 -3.578083e-30 -2.359241e-31
#> 31555 -1.328065e-21 -2.291305e-21 -5.706055e-29 -9.999997e-01 -2.001951e-29
#> 06088 -6.644647e-25 -3.265733e-24 -6.184393e-32 -3.290734e-31 -1.000000e+00
#> 44109 -6.550514e-23 -1.784382e-23 -9.765843e-31 -8.666127e-30 -1.444311e-31
#> 34172 -3.886997e-23 -1.036094e-22 -2.775735e-30 -2.355295e-29 -1.087673e-30
#> 67482 -6.342707e-23 -2.242871e-23 -1.974129e-30 -2.068131e-30 -1.866761e-31
#> 33063 -1.101044e-22 -9.478708e-23 -3.507725e-30 -4.752046e-29 -7.786579e-31
#> 59350 -3.967008e-24 -6.683150e-25 -4.751822e-32 -1.656159e-31 -5.662556e-33
#> 44109 34172 67482 33063 59350
#> 75056 -5.486513e-08 -2.056465e-08 -5.079729e-08 -5.986225e-08 -1.657012e-08
#> 13055 -1.579864e-08 -5.794515e-08 -1.898808e-08 -5.447638e-08 -2.950901e-09
#> 69123 -7.107039e-31 -1.275977e-30 -1.373724e-30 -1.657035e-30 -1.724570e-31
#> 31555 -1.005749e-28 -1.726616e-28 -2.295025e-29 -3.579912e-28 -9.585351e-30
#> 06088 -2.755272e-32 -1.310654e-31 -3.405160e-32 -9.642237e-32 -5.387136e-33
#> 44109 -9.999999e-01 -1.357036e-30 -7.862847e-31 -6.737055e-30 -5.010579e-31
#> 34172 -2.148351e-30 -9.999999e-01 -1.608159e-30 -7.554412e-30 -3.021291e-31
#> 67482 -8.223087e-31 -1.062356e-30 -9.999999e-01 -9.312525e-31 -8.100000e-31
#> 33063 -1.037870e-29 -7.351220e-30 -1.371782e-30 -9.999999e-01 -7.768129e-31
#> 59350 -1.004723e-31 -3.826810e-32 -1.553060e-31 -1.011119e-31 -1.000000e+00