Compute flows between origin and destination locations

This function computes flows between origin locations and destination locations according to the production constrained entropy maximising model proposed by A. Wilson.

Usage

static_blvim(
  costs,
  X,
  alpha,
  beta,
  Z,
  bipartite = TRUE,
  origin_data = NULL,
  destination_data = NULL
)

Arguments

costs: a cost matrix
X: a vector of production constraints
alpha: the return to scale parameter
beta: the inverse cost scale parameter
Z: a vector of destination attractivenesses
bipartite: when TRUE (default value), the origin and destination locations are considered to be distinct. When FALSE, a single set of locations plays the both roles. This has only consequences in functions specific to this latter case such as terminals().
origin_data: NULL or a list of additional data about the origin locations (see details)
destination_data: NULL or a list of additional data about the destination locations (see details)

Value

an object of class sim (and sim_wpc) for spatial interaction model that contains the matrix of flows from the origin locations to the destination locations (see $(Y_{ij})_{1\leq i\leq n, 1\leq j\leq p}$ above) and the attractivenesses of the destination locations.

Details

The model computes flows using the following parameters:

costs ($c$) is a $n\times p$ matrix whose $(i,j )$ entry is the cost of having a "unitary" flow from origin location $i$ to destination location $j$
X ($X$) is a vector of size $n$ containing non negative production constraints for the $n$ origin locations
alpha ($\alpha$) is a return to scale parameter that enhance (or reduce if smaller that 1) the attractivenesses of destination locations when they are larger than 1
beta ($\beta$) is the inverse of a cost scale parameter, i.e., costs are multiplied by beta in the model
Z ($Z$) is a vector of size $p$ containing the positive attractivenesses of the $p$ destination locations

According to Wilson's model, the flow from origin location $i$ to destination location $j$, $Y_{ij}$, is given by

$$Y_{ij}=\frac{X_iZ_j^{\alpha}\exp(-\beta c_{ij})}{\sum_{k=1}^pZ_k^{\alpha}\exp(-\beta c_{ik})}.$$

The model is production constrained because

$$\forall i,\quad X_i=\sum_{j=1}^{p}Y_{ij},$$

that is the origin location $i$ sends a total flow of exactly $X_i$.

Location data

While models in this package do not use location data beyond X and Z, additional data can be stored and used when analysing spatial interaction models.

Origin and destination location names

Spatial interaction models can store names for origin and destination locations, using origin_names<-() and destination_names<-(). Names are taken by default from names of the cost matrix costs. More precisely, rownames(costs) is used for origin location names and colnames(costs) for destination location names.

Origin and destination location positions

Spatial interaction models can store the positions of the origin and destination locations, using origin_positions<-() and destination_positions<-().

Specifying location data

In addition to the functions mentioned above, location data can be specified directly using the origin_data and destination_data parameters. Data are given by a list whose components are not interpreted excepted the following ones:

names is used to specify location names and its content has to follow the restrictions documented in origin_names<-() and destination_names<-()
positions is used to specify location positions and its content has to follow the restrictions documented in origin_positions<-() and destination_positions<-()

References

Wilson, A. (1971), "A family of spatial interaction models, and associated developments", Environment and Planning A: Economy and Space, 3(1), 1-32 doi:10.1068/a030001

Examples

positions <- matrix(rnorm(10 * 2), ncol = 2)
distances <- as.matrix(dist(positions))
production <- rep(1, 10)
attractiveness <- c(2, rep(1, 9))
model <- static_blvim(distances, production, 1.5, 1, attractiveness,
  origin_data = list(names = letters[1:10], positions = positions),
  destination_data = list(names = letters[1:10], positions = positions)
)
model
#> Spatial interaction model with 10 origin locations and 10 destination locations
#> • Model: Wilson's production constrained
#> • Parameters: return to scale (alpha) = 1.5 and inverse cost scale (beta) = 1
location_names(model)
#> $origin
#>  [1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j"
#> 
#> $destination
#>  [1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j"
#> 
location_positions(model)
#> $origin
#>              [,1]       [,2]
#>  [1,] -1.42153475  0.9480316
#>  [2,]  1.17005617 -0.1742460
#>  [3,] -1.40471454 -1.1062360
#>  [4,]  1.10170810 -0.9459850
#>  [5,]  0.69798626  0.2890896
#>  [6,] -0.86434980  0.8769131
#>  [7,] -1.09147035 -1.1489039
#>  [8,] -0.03705146 -1.1376128
#>  [9,]  0.81005379 -1.4372467
#> [10,] -0.49935541 -0.4941435
#> 
#> $destination
#>              [,1]       [,2]
#>  [1,] -1.42153475  0.9480316
#>  [2,]  1.17005617 -0.1742460
#>  [3,] -1.40471454 -1.1062360
#>  [4,]  1.10170810 -0.9459850
#>  [5,]  0.69798626  0.2890896
#>  [6,] -0.86434980  0.8769131
#>  [7,] -1.09147035 -1.1489039
#>  [8,] -0.03705146 -1.1376128
#>  [9,]  0.81005379 -1.4372467
#> [10,] -0.49935541 -0.4941435
#>