This function identifies terminals among the locations underlying the given spatial interaction model. Terminals are locally dominant 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"), ...)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 the seminal work of J. D. Nystuen and M. F. Dacey (Nystuen & Dacey, 1961), as well as on the follow-up variation by Rihll & Wilson (1987 and 1991). We are 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
$$D_j=\sum_{i=1}^{n}Y_{ij},$$
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 detail in Rihll and Wilson (1991). With this relaxed version, location \(j\) is a terminal if
$$\forall l,\quad D_j \geq Y_{jl}.$$
In words, \(j\) is a terminal if its total incoming flow is at least as
large as the flow it sends to any other location. It is easy to see that each
Nystuen & Dacey terminal is a Rihll & Wilson terminal, but the converse does
not hold in general. The function uses 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
distances <- french_cities_distances[1:10, 1:10] / 1000 ## convert to km
production <- rep(1, 10)
attractiveness <- rep(1, 10)
model <- blvim(distances, production, 1.3, 1 / 250, attractiveness,
bipartite = FALSE
)
destination_names(model) <- french_cities$name[1:10]
terminals(model)
#> Paris Montpellier
#> 1 7
dist_times <- french_cities_times[1:10, 1:10]
tmodel <- blvim(dist_times, production, 1.3, 1 / 5000, attractiveness,
bipartite = FALSE
)
destination_names(tmodel) <- french_cities$name[1:10]
terminals(tmodel)
#> Paris Nice Nantes Montpellier Strasbourg Bordeaux
#> 1 5 6 7 8 9
#> Lille
#> 10
terminals(tmodel, definition = "RW")
#> Paris Nice Nantes Montpellier Strasbourg Bordeaux
#> 1 5 6 7 8 9
#> Lille
#> 10