Interaction

Introduction

Having found a way to quantify the presence of categories in the various areal units, and identified neighbourhoods, we would like to quantify the extent to which individuals from various categories are exposed to one another. In other words, the extent to which they potentially interact with one another.

Measure

exposure (distribution, classes=None)

The following measure of exposure is inspired by a measure proposed by Marcon & Puech to study the co-localisation of firms. In our case, it reads

$$E_{\alpha \beta} = \frac{1}{N} \sum_{t=1}^T n(t)\, r_\alpha(t)\, r_\beta(t)$$

Although it is not straightforward when written in this form, the exposure is the average representation of $\beta$ witnessed by the $\alpha$, or conversely the average representation of $\alpha$ as witnessed by the $\beta$. It is independent of the city's overall inequality levels.

In the unsegregated configuration, the exposure is equal on average to

$$\mathrm{E} \left [E_{\alpha \beta} \right] = 1$$

In other words, the diferent categories are indifferent to one another. The variance of the exposure in the unsegregated configuration, $\mathrm{Var}(E_{\alpha \beta})$, can also be computed analytically.

Parameters

distribution dictionary

Takes a dictionary of dictionaries with
distribution[areal_unit][category] = number
classes dictionary, optional

Takes a dictionary of lists with classes[class] = [cat1, cat2, ...]
If not specified, the algorithm will use the categories found in distribution

Output

exposure

Returns a dictionary of dictionaries with
exposure[alpha][beta] = ($\mathrm{E}_{\alpha \beta}$, $\mathrm{Var}(E_{\alpha \beta})$)

Examples

Let us look at how to compute the exposure for the categories 0, 1 and 2 in a fictional region with two areal units A and B.

>>> import marble as mb
>>> city = {"A":{0: 10, 1:0, 2:23},
          "B":{0: 0, 1:10, 2:8}}
>>> exp = mb.exposure(city)
>>> exp[0][1] # Value of the exposure between cat. 0 and 1
(0, 0.01)

Now, imagine that for some reason, you have found that the original categories in fact split in two classes: first, that is composed of the category 0, and second, that is composed of categories 1 and 2.

>>> import marble as mb
>>> city = {"A":{0: 10, 1:0, 2:23},
          "B":{0: 0, 1:10, 2:8}}
>>> classes = {'first': [0], 
             'second': [1,2]}
>>> exp = mb.exposure(city, classes)
>>> exp['first']['second']
(0.867, 0.002)