Emergent classes

Using the exposure as a measure of interaction between classes, we can aggregate the different categories into classes based on their mutual interaction/repulsion.

Measures

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cluster_categories (distribution, exposure)

At each step of the aggregation, we look for the pair $(\beta, \delta)$ of categories that has the highest exposure (renormalised by the maximum possible value). We aggregate the pair in a new category $\gamma$ whose exposure with the other categories $\alpha$ is given by

$$ E_{\alpha, \gamma} = \frac{1}{N_\beta + N_\delta} \left( N_\beta\, E_{\alpha, \beta} + N_\delta\, E_{\alpha, \delta} \right) $$

The function returns a linkage matrix that encodes the hierarchical tree. At the $i$th iteration of the algorithm, $L[i,0]$ and $L[i,1]$ are aggregated to form the $n+i$th cluster. The exposure between $L[i,1]$ and $L[i,0]$ is given by $L[i,3]$, the variance in the corresponding unsegregated city is given by $L[i,4]$.

Parameters

• distribution dictionary

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

• exposure dictionary

  Takes a dictionary dictionary of dictionaries, the result of the exposure function with
  exposure[$\alpha$][$\beta$] = ($E_{\alpha \beta}$, $\mathrm{Var}(E_{\alpha \beta})$)

Output

• linkage matrix

  Returns a list of lists. See above for the description of the linkage matrix's structure.

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uncover_classes (distribution, exposure, ci_factor=10)

The classes are uncovered using the spatial repartition of individuals from different categories, using their relative exposure.

We only aggregate the pair $(\beta$,$\delta)$ in the same class if the two categories attract each other, that is if the exposure

$$E_{\beta, \delta} > 1 + 10\; \sigma_{\beta, \delta}$$

($99\%$ CI according to the Chebyshev inequality). The aggregation procedure may therefore stop before all categories are aggregated in one unique class, and output the repartition of the original categories into classes.

Parameters

• distribution dictionary

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

• exposure dictionary

  Takes a dictionary dictionary of dictionaries, the result of the exposure function with
  exposure[$\alpha$][$\beta$] = ($E_{\alpha \beta}$, $\mathrm{Var}(E_{\alpha \beta})$)

• ci_factor float

  Number of standard deviations over which we consider to have a $99\%$ confidence interval on the exposure value. The default value $10$, is the upper bound given by Chebyshev's inequality.

Output

• classes nested lists

  Returns a list of lists. Each list corresponds to a class with
  classes[$i$] = [categories in class $i$]