Distance decay functions

A distance decay function defines the relation between travel impedance $d_{ij}$ and travel potential $p_{ij}$: how much less attractive a destination becomes as it gets further away. The function must be decreasing, so that nearer or cheaper destinations are weighted no less than farther or more expensive ones.

$$p_{ij} := f(d_{ij})$$

Travel potential can be calculated with the GeoDMS directly from travel impedance tables, see Origin Destination matrix (trip table).

Common forms

Name	Definition	Notes
Inverse	$d \rightarrow d^{-1}$	Diverges as $d \rightarrow 0$, so it needs a floor on short distances
Inverse squared	$d \rightarrow d^{-2}$	Steeper, the classical gravity analogue
Power-b	$d \rightarrow d^{-b}$	General power family, with $b$ estimated from observed flows
Negative exponential	$d \rightarrow e^{-\beta d}$	Better behaved near zero, standard in trip distribution
Cut off	$d \rightarrow \text{if } d < d_{max} \text{ then } 1 \text{ else } 0$	Reduces the potential measure to a cumulative opportunity count
Log logistic	$d \rightarrow 1/(1+e^{a+b \ln d})$	Flat near the origin, then a steep fall, often the best empirical fit

The log logistic form can be rewritten as

$$f(d) = \frac{1}{1+e^{a}d^{b}} = \frac{e^{-a}d^{-b}}{e^{-a}d^{-b}+e^{0}d^{0}}$$

with $d$ in minutes of travel time. For potential accessibility to jobs and employees, $a = -6.899$ and $b = 2.326$ are used, following Geurs & Ritsema van Eck (2001).

Choosing a function

The shape matters more than the family. Three properties are worth checking against what you know about the behaviour being modelled.

• Behaviour near zero. Power functions go to infinity at $d = 0$, which turns a destination in the same zone as the origin into an artefact. Exponential and log logistic forms do not.
• The tail. A heavy tail lets distant destinations keep contributing; a thin one makes the indicator effectively local. This decides how sensitive your result is to the study area boundary.
• The middle. Log logistic functions have a plateau followed by a steep drop, which matches observed behaviour for many trip purposes better than a smooth monotone decline. People are fairly indifferent between 5 and 10 minutes, and much less so between 30 and 45.

Parameters should be estimated from observed trip data for the same purpose and mode where possible. Borrowed parameters are common and often unavoidable, but they carry the travel behaviour of the study they came from. A commuting decay function applied to grocery trips will overstate the value of distant shops.

Always report a sensitivity analysis over the parameters. If the policy conclusion flips between $b = 2.0$ and $b = 2.5$, that is a finding about the study, not a detail.

In the GeoDMS

Inside the interaction model, the decay function is chosen through the impedance options, which offer two forms.

dist_decay(gamma) is the power family, $t_{ij} = d_{ij}^{-\gamma}$. Useful special cases are $\gamma = 1$ for the inverse, $\gamma = 0$ for no decay at all, which turns the interaction model into an unweighted count, and $\gamma = -1$ for a linear increase. Where $d_{ij} \le 0$ and $\gamma \neq 0$ the potential is set to zero rather than infinity, which is the implementation's answer to the near zero problem above; the OrgZone_min and DstZone_min options are the other answer, imposing a floor on the impedance.

dist_logit(alpha,beta,gamma) is the log logistic form, extended with a term in $d$ itself:

$$t_{ij} = \left(1+e^{,alpha + beta \cdot \ln d_{ij} + gamma \cdot d_{ij}}\right)^{-1}$$

At $gamma = 0$ this is exactly the log logistic function in the table above, so the Geurs & Ritsema van Eck parameters translate directly into dist_logit(-6.899, 2.326, 0) with $d$ in minutes. A non-zero $gamma$ adds an exponential tail on top of the power term, which thins the tail faster than the pure log logistic form at long distances.

Note that the parameters are named in Latin characters in the documentation, deliberately, because the surrounding interaction options already use $\alpha$, $\beta$ and $\gamma$ for the elasticities. They are different quantities with the same names.

Decay applies to the impedance used for the interaction, which need not be the impedance used to choose the route. See Travel impedance.

Related

• Accessibility indicators, where decay functions are used
• Spatial Interaction Models, estimating decay parameters from flow data
• Travel impedance, what goes into $d_{ij}$
• Impedance options, the implementation