Spatial interaction models
Spatial interaction models try to estimate flows between locations, for instance workers commuting from residential zones to employment zones. Models are built from:
- a collection of \(n\) origin locations described by a some characteristics \((X_i)_{1\leq i\leq n}\)
- a collection of \(p\) destination locations described by a some characteristics \((Z_j)_{1\leq j\leq p}\)
- a collection of \(n\times p\) characteristics of the difficulty of travelling (in a broad sense) from origin \(i\) to destination \(j\), \(c_{ij}\)
The goal is to estimate flows \((Y_{ij})_{1\leq i\leq n, 1\leq j\leq p}\) from origin locations to destinations locations. A typical hypothesis is for the flows to depend on characteristics as follows
\[ Y_{ij}=f(X_i, Z_j, c_{ij}), \] for a well chosen function \(f\). The most well known spatial interaction model is the so called gravity model which takes the form \[ Y_{ij}\propto \frac{X_iZ_j}{c_{ij}^2}, \] where \(\propto\) means proportional to, \(c_{ij}\) is supposed to be the distance between origin \(i\) and destination \(j\), and the characteristics \(X_i\) and \(Z_j\) are assumed to be numerical.
Maximum entropy models
In the late 60s, Alan Wilson developed a collection of spatial interaction models based of a maximum entropy principle. In those models, the flow is given by \[ Y_{ij} \propto X_iZ_j^{\alpha}\exp(-\beta c_{ij}), \] where \(\alpha\) and \(\beta\) are two parameters interpreted as follows:
- \(Z_j\) is the attractiveness of the destination location \(j\)
- \(\alpha\) is then a return to scale parameter: if \(Z_j\) grows above \(1\), its actual attractiveness can increase in a super-linear way (\(\alpha>1\)) or in a sub-linear way \(\alpha<1\)
- \(\beta\) acts as the inverse of the scale of the costs \(c_{ij}\). If those costs are distances, for instance, \(\frac{1}{\beta}\) can be seen as a cut off distance.