An introduction to spatial interaction models: from first principles
In simodels: Flexible Framework for Developing Spatial Interaction Models

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

What are SIMs?

Spatial Interaction Models (SIMs) are mathematical models for estimating movement between spatial entities developed by Alan Wilson in the late 1960s and early 1970, with considerable uptake and refinement for transport modelling since then @boyce_forecasting_2015. There are four main types of traditional SIMs [@wilson_family_1971]:

Unconstrained
Production-constrained
Attraction-constrained
Doubly-constrained

An early and highly influential type of SIM was the 'gravity model', defined by @wilson_family_1971 as follows (in a paper that explored many iterations on this formulation):

$$ T_{i j}=K \frac{W_{i}^{(1)} W_{j}^{(2)}}{c_{i j}^{n}} $$

"where $T_{i j}$ is a measure of the interaction between zones $i$ and $W_{i}^{(1)}$ is a measure of the 'mass term' associated with zone $z_i$, $W_{j}^{(2)}$ is a measure of the 'mass term' associated with zone $z_j$, and $c_{ij}$ is a measure of the distance, or generalised cost of travel, between zone $i$ and zone $j$". $K$ is a 'constant of proportionality' and $n$ is a parameter to be estimated.

Redefining the $W$ terms as $m$ and $n$ for origins and destinations respectively [@simini_universal_2012], this classic definition of the 'gravity model' can be written as follows:

$$ T_{i j}=K \frac{m_{i} n_{j}}{c_{i j}^{n}} $$

For the purposes of this project, we will focus on production-constrained SIMs. These can be defined as follows [@wilson_family_1971]:

$$ T_{ij} = A_iO_in_jf(c_{ij}) $$

where $A$ is a balancing factor defined as:

$$ A_{i}=\frac{1}{\sum_{j} m_{j} \mathrm{f}\left(c_{i j}\right)} $$

$O_i$ is analogous to the travel demand in zone $i$, which can be roughly approximated by its population.

More recent innovations in SIMs including the 'radiation model' @simini_universal_2012. See @lenormand_systematic_2016 for a comparison of alternative approaches.

Implementation in R

Before using the functions in this or other packages, it may be worth implementing SIMs from first principles, to gain an understanding of how they work. The code presented below was written before the functions in the simodels package were developed, building on @dennett_estimating_2012. The aim is to demonstrate a common way of running SIMs, in a for loop, rather than using vectorised operations (used in the simodels package) which can be faster.

library(tmap)
library(dplyr)
library(ggplot2)

zones = simodels::si_zones
centroids = simodels::si_centroids
od = simodels::si_od_census
tm_shape(zones) + tm_polygons("all", palette = "viridis")

od_df = od::points_to_od(centroids)
od_sfc = od::odc_to_sfc(od_df[3:6])
sf::st_crs(od_sfc) = 4326
od_df$length = sf::st_length(od_sfc)
od_df = od_df %>% transmute(
  O, D, length = as.numeric(length) / 1000,
  flow = NA, fc = NA
  )
od_df = sf::st_sf(od_df, geometry = od_sfc, crs = 4326)

An unconstrained spatial interaction model can be written as follows, with a more-or-less arbitrary value for beta which can be optimised later:

beta = 0.3
i = 1
j = 2
for(i in seq(nrow(zones))) {
  for(j in seq(nrow(zones))) {
    O = zones$all[i]
    n = zones$all[j]
    ij = which(od_df$O == zones$geo_code[i] & od_df$D == zones$geo_code[j])
    od_df$fc[ij] = exp(-beta * od_df$length[ij])
    od_df$flow[ij] = O * n * od_df$fc[ij]
  }
}
od_top = od_df %>% 
  filter(O != D) %>% 
  top_n(n = 2000, wt = flow)

tm_shape(zones) +
  tm_borders() +
  tm_shape(od_top) +
  tm_lines("flow")

We can plot the 'distance decay' curve associated with this SIM is as follows:

summary(od_df$fc)
od_df %>% 
  ggplot() +
  geom_point(aes(length, fc))

We can make this production constrained as follows:

od_dfj = left_join(
  od_df,
  zones %>% select(O = geo_code, all) %>% sf::st_drop_geometry()
)
od_dfj = od_dfj %>% 
  group_by(O) %>% 
  mutate(flow_constrained = flow / sum(flow) * first(all)) %>%
  ungroup()
sum(od_dfj$flow_constrained) == sum(zones$all)
od_top = od_dfj %>% 
  filter(O != D) %>% 
  top_n(n = 2000, wt = flow_constrained)

tm_shape(zones) +
  tm_borders() +
  tm_shape(od_top) +
  tm_lines("flow_constrained")

Validation

od_dfjc = inner_join(od_dfj %>% select(-all), od)
od_dfjc %>% 
  ggplot() +
  geom_point(aes(all, flow_constrained))
cor(od_dfjc$all, od_dfjc$flow_constrained)^2