Compute terminals for a spatial interaction model

This function identifies terminals in the locations underlying the given spatial interaction model. Terminals are locally dominating locations that essentially send less to other locations than they receive (see details for formal definitions). As we compare incoming flows to outgoing flows, terminal computation is restricted to interaction models in which origin and destination locations are identical, i.e. models that are not bipartite.

Usage

terminals(sim, definition = c("ND", "RW"), ...)

Arguments

sim: a spatial interaction model object
definition: terminal definition, either "ND" (for Nystuen & Dacey, default) or "RW" (for Rihll & Wilson), see details.
...: additional parameters

Value

a vector containing the indexes of the terminals identified from the flow matrix of the interaction model.

Details

The notion of terminal used in this function is based on seminal work by J. D. Nystuen and M. F. Dacey (Nystuen & Dacey, 1961), as well as on the follow up variation from Rihll & Wislon (1987 and 1991). We assume given a square flow matrix $(Y_{ij})_{1\leq i\leq n, 1\leq j\leq n}$. The incoming flow at location $j$ is given by

$$D_j=\sum_{j=i}^{p}Y_{ij},$$

and is used as a measure of importance of this location. Then in Nystuen & Dacey (1961), location $j$ is a "terminal point" (or a "central city") if

$$D_j \geq D_{m(j)},$$

where $m(j)$ is such that

$$\forall l,\quad Y_{jl}\leq Y_{jm(j)}.$$

In words, $j$ is a terminal if the location $m(j)$ to which it sends its largest flow is less important than $j$ itself, in terms of incoming flows. This is the definition used by the function when definition is "ND".

Rihll & Wilson (1987) use a modified version of this definition described in details in Rihll and Wilson (1991). With this relaxed version, location $j$ is a terminal if

$$\forall i,\quad D_j \geq Y_{ij}.$$

In words, $j$ is a terminal if it receives more flows than it is sending to each other location. It is easy to see that each Nystuen & Dacey terminal is a Rihll & Wilson terminal, but the reverse is false in general. The function use the Rihll & Wilson definition when definition is "RW"

References

Nystuen, J.D. and Dacey, M.F. (1961), "A graph theory interpretation of nodal regions", Papers and Proceedings of the Regional Science Association 7: 29–42. doi:10.1007/bf01969070

Rihll, T.E., and Wilson, A.G. (1987). "Spatial interaction and structural models in historical analysis: some possibilities and an example", Histoire & Mesure 2: 5–32. doi:10.3406/hism.1987.1300

Rihll, T., and Wilson, A. (1991), "Modelling settlement structures in ancient Greece: new approaches to the polis", In City and Country in the Ancient World, Vol. 3, Edited by J. Rich and A. Wallace-Hadrill, 58–95. London: Routledge.

Examples

positions <- matrix(rnorm(10 * 2), ncol = 2)
distances <- as.matrix(dist(positions))
production <- rep(1, 10)
attractiveness <- rep(1, 10)
model <- blvim(distances, production, 1.3, 2, attractiveness, bipartite = FALSE)
terminals(model)
#> [1] 2