Spatial Interaction Models

Spatial interaction models describe flows between locations, such as commuters, shoppers, migrants, freight, telephone calls or hospital admissions, as a function of the size of the origin, the size of the destination, and the impedance between them.

They matter for accessibility work in two ways. They are the theoretical foundation of the potential (gravity) measures in Accessibility indicators, and they are the framework in which decay parameters are estimated from observed flow data rather than borrowed.

The gravity model

The oldest form, borrowed by analogy from Newtonian gravity in the nineteenth century, states that the flow $T_{ij}$ from zone $i$ to zone $j$ is proportional to the product of the two masses and inversely proportional to a power of the distance between them:

$$T_{ij} = k \cdot \frac{O_i \cdot D_j}{d_{ij}^{,b}}$$

Hansen (1959) turned the same expression into an accessibility measure by dropping the origin mass and summing over destinations, which is the potential measure still in use today.

The analogy is suggestive rather than explanatory. It says nothing about why people behave this way, and nothing guarantees that the predicted flows add up to the totals that are actually observed leaving each origin and arriving at each destination.

Wilson's family

Wilson (1967, 1971) put the model on a firmer footing by deriving it from entropy maximisation: of all flow patterns consistent with what is known about the system, the model picks the most probable one. The known quantities are imposed as constraints, and which constraints are imposed defines four distinct models.

Model	Constrained to reproduce	Balancing factors
Unconstrained	Only the grand total of all flows	None
Production constrained	Observed outflow from each origin	$A_i$
Attraction constrained	Observed inflow into each destination	$B_j$
Doubly constrained	Both outflows and inflows	$A_i$ and $B_j$

The doubly constrained form is

$$T_{ij} = A_i O_i B_j D_j f(d_{ij}), \qquad A_i = \left[\sum_j B_j D_j f(d_{ij})\right]^{-1}, \qquad B_j = \left[\sum_i A_i O_i f(d_{ij})\right]^{-1}$$

The two balancing factors are defined in terms of each other, so they are found by iterating until they stop changing.

Two points are worth carrying over to accessibility work.

The balancing factor $A_i^{-1}$ is exactly a potential accessibility measure: the decay weighted sum of attractions reachable from $i$. And $B_j^{-1}$ is the decay weighted demand pressing on destination $j$. This is the same quantity that appears in the denominator of the competition corrected measures in Accessibility indicators. Competition correction is not an add on to the gravity measure; it is what the doubly constrained model has been doing all along.

Which model to use is a statement about what you are willing to assume. A production constrained model assumes destination capacity is unlimited, which is reasonable for shops and not for hospital beds. A doubly constrained model assumes both totals are known in advance, which makes it a distribution model rather than a forecasting one.

Alonso's General Theory of Movement

Wilson's models take the marginal totals as given. A new road cannot change how many people leave a zone, only where they go. Alonso (1973, 1978) proposed a more general system, in which flows, outflows and inflows are modelled together and connected through the balancing factors, so that a change in impedance on a single link can propagate to every flow and every total in the system. The model is known under several names, including the Three Component Model and the Extended Gravity Model; Object Vision follows the review literature in calling it the General Theory of Movement. Essentially the same model was developed independently for international trade and hospital admissions by Bikker (1987).

Wilson's four models are special cases of the GTM, obtained by fixing particular elasticities to zero or one. The reverse does not hold, which is where the extra generality comes from.

The GTM has been discussed far more than it has been applied. The review literature attributes this mainly to the two balancing factors, which Alonso calls systemic variables. They carry the coherence of the whole system, they appear in nearly every equation, and there is no single obvious way to write the model down, since equivalent representations can be derived from one another by substitution. Several interpretations have been offered: as an interpolation between the members of Wilson's family, as inverted prices or shadow prices for access, and as the outcome of a choice process in which actors respond to those prices.

Computation is straightforward and can be done on a spreadsheet. Estimation is not. The balancing factors are unobserved endogenous variables, so they have to be constructed first, typically by estimating the allocation part of the model and deriving them from the result. Because they are endogenous, the system is simultaneous, and ordinary least squares gives biased and inconsistent estimates. Instrumental variables or maximum likelihood are required.

What to take from this in practice

• If you only need an accessibility indicator, the potential measure in Accessibility indicators is the unconstrained model with the origin mass removed. You do not need the rest of the machinery.
• If supply is capacity limited, you need a constraint on the destination side, whether you call it a balancing factor or a competition correction.
• If you are estimating a decay function rather than borrowing one, you are fitting a spatial interaction model, and the choice of constraints will affect the parameter you get.
• If you are evaluating infrastructure that could change how much people travel and not only where, the doubly constrained model cannot represent that, and the GTM is the reason the distinction exists.

In the GeoDMS

This is not only theory here. The GeoDMS implements the GTM directly, as the interaction section of the impedance options, so a single impedance call can produce routing, accessibility and assigned flows in one pass. Its central expression is

$$M_{ij} = v_i \cdot w_j \cdot t_{ij} \cdot D_i^{\alpha-1}, \qquad D_i = \sum_j w_j \cdot t_{ij}$$

with $v_i$ a trip generation weight per origin, $w_j$ a trip distribution weight per destination and $t_{ij}$ the facility of movement, which is the decay function of Distance decay functions. Alongside the flows it returns $M_{ix}$ and $M_{xj}$ per zone and $LF_l$, the assigned flow per link.

Three things are worth knowing before reaching for it.

The destination side is not constrained: the implementation fixes $\beta_j = 1.0$. What the elasticity $\alpha$ then varies is the origin side. At $\alpha = 1$ the model is free, and at $\alpha = 0$ it is origin constrained, since $M_{ix} = v_i \cdot D_i^{\alpha}$ reduces to $v_i$. In Wilson's terms that spans the unconstrained and production constrained models and everything between, but not the attraction constrained or doubly constrained ones. If your destinations have hard capacities, this is the boundary to be aware of, and the competition corrected measures in Accessibility indicators are the practical alternative.

$D_i$ is a potential accessibility measure, computed as a by-product. If an accessibility indicator is all you need, you do not need the flows.

The notation differs from the review. GeoDMS follows Alonso's symbols rather than Wilson's, so the balancing factors appear as $D_i^{-1}$ and $C_j^{-1}$ where the review writes $A_i$ and $B_j$. The full set of rewrite rules is listed in impedance links, and impedance interaction potential shows how to recover $t_{ij}$ and $M_{ij}$ per od-pair from the returned results.

Sources

This summary follows the review by De Vries, Nijkamp & Rietveld (2000), which sets out the GTM in a notation consistent with the Wilson tradition and analyses its statistical properties and estimation problems. The primary sources are Alonso (1973, 1978), Wilson (1967, 1971), Hansen (1959) and Bikker (1987). The GTM was debated at length after 1978 by Hua, Ledent, Anselin, Wilson and others; De Vries et al. survey that discussion and are the place to pick up those references. See References for the full list.