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.

Related

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