Presence in areal units

Introduction

Maybe the most fundamental quantity in all studies of segregation is how we quantify the presence of a category in areal units. All other measures are then based on the measure of presence. Several indicators exist, and one needs to be aware of their meaning, their qualities and their shortcomings. Per se, if one's goal is to identify areal units with high levels of segregation everything else being equal, one should use the representation defined below. Other measures can have their use, but are biaised and thus cannot be used for an assessment of segregation.

Measures

---

concentration (distribution, classes=None)

The concentration measures the proportion of individuals from category $\alpha$ in the areal unit $t$.

$$\frac{n_\alpha(t)}{N_\alpha}$$

The concentration has the nice property to be composition-invariant, that is it does not depend on the relative proportion of category $\alpha$ in the geographical zone as a whole.

However, it strongly depends on the total population of the areal unit we are studying: more population areal units will mechanically lead to higher values of the concentration. Segregation measures based on the concentration (such as the dissimilarity index) will thefore be dominated by the values in highly population areal units.

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

• concentration dictionary

  Returns a dictionary of dictionaries with
  concentration[areal_unit][category] = value

---

proportion (distribution, classes=None)

Sometimes, however, we prefer to know the proportion of people of a given category in an unit. In our notations, it is defined as

$$\frac{n_\alpha(t)}{n(t)}$$

Although the values of the proportion are easier to interpret (`x% of the individuals living in this areal units live in this neighbourhood'), they are not a good indicator of segregation. They strongly depend on the relative proportion of individuals of the category in the geographical area being studied.

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

• proportion dictionary

  Returns a dictionary of dictionaries with
  proportion[areal_unit][category] = value

---

representation (distribution, classes=None)

The representation solves the problems linked to both measures of concentration and of representation. The idea behind the measure of representation is that segregation is a departure from the situation where all categories would be spatially distributed at random. The properties of such a `random', unsegregated city are however well known, and the distribution of categories in each areal unit is given by a binomial distribution. The representation is thus defined as the number $n_\alpha(t)$ divided by its expected value in an unsegregated city

$$r_\alpha(t) = \frac{n_\alpha(t) / n(t)}{N_\alpha / N} $$

In the perfectly unsegregated city (number of individuals equal to the mean of the binomial distribution), $r_\alpha = 1$. Of course, there always is a possibility for any situation that it has been obtained by chance. Given the binomial distribution, however, we can compute how likely it is that the representation $r_\alpha(t)$ we measure has been obtained by chance. To do that, we first compute the variance:

$$\mathrm{Var}\left[r_\alpha(t)\right] = \sigma_\alpha(t)^2 = \frac{1}{N_\alpha} \left[\frac{N}{n(t)} - 1\right]$$

we would like to know the areal units that depart from the random situation with 99% confidence. Therefore, we will say that

• $\alpha$ is overrepresented in $t$ iff $r_\alpha(t) > 1 + 2.57\,*\sigma_\alpha(t)$
• $\alpha$ is underrepresented in $t$ iff $r_\alpha(t) < 1 + 2.57\,\sigma_\alpha(t)$

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

• representation dictionary

  Returns a dictionary of dictionaries with representation[areal_unit][category] = ($r_\alpha(t)$, $\mathrm{Var}\left[r_\alpha(t)\right]$)

Examples

Let us look at how to compute the concentration, proportion and representation 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}}
>>> co = mb.concentration(city)
>>> print co['A'][0]
1.0

>>> pr = mb.proportion(city)
>>> print pr['A'][0]
0.303

>>> rep = mb.representation(city)
>>> print rep['A'][0]
(1.55, 0.054)