access.weights.gravity

access.weights.gravity(scale, alpha, min_dist=0)

Create a gravity function from a scale \(s\) and \(\alpha\) parameters as well as an optional minimum distance \(x_\text{min}\). The function is of the form \(f(x) = (\text{max}(x, x_\text{min})/s)^\alpha\). Note that there is no overall normalization.

Parameters

scalefloat

Scaling value, normalizing the function input.

alphafloat

Power to which the normalized inputs are raised. Note that it is not implicitly negative (i.e., \(x^\alpha\) instead of \(1/x^\alpha\).

min_distfloat

A ‘standard’ issue with gravity model is the infinite potential at 0 distance or time. This can be rectified crudely by specifying a minimum distance, and setting any input exceeding that minimum to the minimum itself. The default threshold is 0.

Returns

weight_functionfunction

Function returning weight, for input distance or time, x.

Examples

Import the weights:

>>> from access import weights

Create a step function with thresholds at 20, 40, and 60. Travel costs are in minutes here, but the code cannot tell if you mix units!

>>> fn = weights.gravity(scale = 20, alpha = -2, min_dist = 1)

>>> {t : round(fn(t), 2) for t in [0, 1, 2, 20, 40, 60]}
{0: 400.0, 1: 400.0, 2: 100.0, 20: 1.0, 40: 0.25, 60: 0.11}