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Boltzmann–Lotka–Volterra Interaction Model

blvim implements A. Wilson’s Boltzmann–Lotka–Volterra (BLV) interaction model. The model is described in Wilson, A. (2008), “Boltzmann, Lotka and Volterra and spatial structural evolution: an integrated methodology for some dynamical systems”, J. R. Soc. Interface.5865–871

Installation

You can install the development version of blvim from GitHub with:

# install.packages("pak")
pak::pak("fabrice-rossi/blvim")

Spatial interaction models

Spatial interaction models try to estimate flows between locations, for instance workers commuting from residential zones to employment zones. The focus of the blvim package is on maximum entropy models developed by Alan Wilson. See vignette("theory") for a theoretical background.

In practice, if we have \(n\) origin locations and \(p\) destination locations, the aim is to compute a matrix of flows \((Y_{ij})_{1\leq i\leq n, 1\leq j\leq p}\), where \(Y_{ij}\) is the flow from origin \(i\) to destination \(j\). This is done using characteristics of the origin and destination locations, together with a matrix of exchange difficulties, a cost matrix, \((c_{ij})_{1\leq i\leq n, 1\leq j\leq p}\). For instance \(c_{ij}\) can be the distance between origin \(i\) and destination \(j\).

Usage

The package is loaded in a standard way.

Input data

To compute a spatial interaction model with blvim, one needs at least a cost matrix. In this example, we use the distance between some random locations.

set.seed(42)
## random origin locations
origins <- matrix(runif(2 * 30), ncol = 2)
## random destination locations
destinations <- matrix(runif(2 * 10), ncol = 2)
## cost matrix
full_costs <- as.matrix(dist(rbind(origins, destinations)))
cost_matrix <- full_costs[1:nrow(origins), (nrow(origins) + 1):(nrow(origins) + nrow(destinations))]

In addition, we focus on production constrained models which means that we need to specify the production of each origin location (a vector of positive values \((X_i)_{1\leq i\leq n}\)). In this example we assume a common unitary production.

X <- rep(1, nrow(origins))

Finally, the simple static model needs an attractiveness value of each destination location, a vector of positive values \((Z_j)_{1\leq j\leq p}\). We assume again a common unitary attractiveness.

Z <- rep(1, nrow(destinations))

Static models

In Wilson’s production constrained maximum entropy model, the flows are given by

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

where \(\alpha\) is a return to scale parameter and \(\beta\) is the inverse of a cost scale parameter. Notice that the flow matrix is production constrained, which means that the total outgoing flow from any origin location is equal to the production of this location, i.e.

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

The model is obtained using the static_blvim() function as follows:

a_model <- static_blvim(cost_matrix, X, alpha = 1.1, beta = 2, Z)
a_model
#> Spatial interaction model with 30 origin locations and 10 destination locations
#> • Model: Wilson's production constrained
#> • Parameters: return to scale (alpha) = 1.1 and inverse cost scale (beta) = 2

Several functions are provided to extract parts of the result. In particular flows() returns the flow matrix \(Y\).

a_model_flows <- flows(a_model)

which can be displayed using, e.g., the image() function.

par(mar = rep(0.1, 4))
image(t(a_model_flows),
  col = gray.colors(20, start = 1, end = 0),
  axes = FALSE,
  frame = TRUE
)

In this representation, each row gives the flows from one origin location to all the destination location. The package provides a ggplot2::autoplot() function that can be used as follows

b_model <- static_blvim(cost_matrix, X, alpha = 1.1, beta = 15, Z)
b_model
#> Spatial interaction model with 30 origin locations and 10 destination locations
#> • Model: Wilson's production constrained
#> • Parameters: return to scale (alpha) = 1.1 and inverse cost scale (beta) = 15

Different values of the parameters \(\alpha\) and \(\beta\) lead to more or less concentrated flows as exemplified by the two above figures.

Dynamic models

A. Wilson’s Boltzmann–Lotka–Volterra (BLV) interaction model is based on the production constrained maximum entropy model. The main idea consists in updating the attractivenesses of the destination locations based on their incoming flows. In the limit we want to have

\[Z_j =\sum_{i=1}^{n}Y_{ij}, \]

where the flows are given by the equations above. The model is estimated using the blvim() function as follows.

a_blv_model <- blvim(cost_matrix, X, alpha = 1.1, beta = 2, Z)
a_blv_model
#> Spatial interaction model with 30 origin locations and 10 destination locations
#> • Model: Wilson's production constrained
#> • Parameters: return to scale (alpha) = 1.1 and inverse cost scale (beta) = 2
#> ℹ The BLV model converged after 5800 iterations.

Notice that we start with some initial values of the attractivenesses but the final values are different. They can be obtained using the attractiveness() function as follows (we show the values using a bar plot).

par(mar = c(0.1, 4, 1, 0))
a_final_Z <- attractiveness(a_blv_model)
barplot(a_final_Z)

In this example, one destination location acts as a global attractor of all the flows. This can be seen also on the final flow matrix.

The autoplot() function can also be used to show the destination flows or the attractivenesses as follows.

autoplot(a_blv_model, "attractiveness")

Results are of course strongly influenced by the parameters, as shown by this second example.

b_blv_model <- blvim(cost_matrix, X, alpha = 1.1, beta = 15, Z)
b_blv_model
#> Spatial interaction model with 30 origin locations and 10 destination locations
#> • Model: Wilson's production constrained
#> • Parameters: return to scale (alpha) = 1.1 and inverse cost scale (beta) = 15
#> ℹ The BLV model converged after 13300 iterations.
autoplot(b_blv_model, "attractiveness")