Disaggegation is the process of estimating a quantity zi for a finer grained domain i, given quantity Zr for a coarser domain r and an incidence relation qir that indicates the fraction of unit i that belongs to region r such that qir ≥ 0 and $\forall i: \sum\limits_{r} q_i^r = 1$. In most cases, qir is discrete, thus either 0 or 1 and one can define r(i) such that qir(i) = 1.

The reverse of disaggregation is aggregation.

Quantitative modelling of attribute values can often be considered as some sort of (combination of ) disaggregation of known aggregates, restrictions and other proxy values.

zi can be an extensive (additive) quantity of i or an intensive quantity (such as discrete class values or density measures).

extensive quantities

  • should adhere to the pycnophylactic principle (further: pp), i.e. $\forall r: \sum\limits_{i} z_i * q_i^r = Z_r$
  • can be done using si as proxy values. Then $z_i := \sum_{\lim_{r}} Z_r * \frac{s_i * q_i^r}{\sum_{\lim_{j}} s_j * q_j^r}$; which distributes Zr proportional to si. The pp is guaranteed to match if:
    • all qir are discrete (thus each i relates to a single aggregate) and
    • for each r: ${\sum\limits_{j} s_j * q_j^r} > 0 \vee {Z_r = 0}$ (thus each nonzero aggregate relates to at least one i ).
  • When qir is discrete, the former can be reformualated to $z_i := Z_{r(i)} * \frac{s_i}{\sum\limits_{j: r(j) = r(i)} s_j}$ which can be done with the GeoDMS function scalesum(s, r, Z).
  • can be smoothed out by convolution when disaggregating to proxies with approximate locations, such as point-related data, by using the potential.
  • can be made subject to minimum (zero?) and maximum values for zi, by transforming and capping the result of scalesum. To comply to the pp, an iterative fitting factor fr, initially set to 1, can be used. Capping in GeoDMS: min_elem(z, z_max), max_elem (z, z_min), median(z, interval)

intensive quantities

  • can be done using homogeneous distribution (choropleth mapping), which can be done in with the GeoDMS function lookup(r, Z).
  • can be done using a incidence proxy ci, aka dasymmetric mapping, in GeoDMS: “c ? Z[r] : 0[valuesunit](valuesunit.html)(Z)".