Disaggregation

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

z_i 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 s_i as proxy values. Then $z_i := \sum\limits_{r} Z_r * {{s_i * q_i^r} \over {\sum\limits_{j} s_j * q_j^r}}$; which distributes Z_r proportional to s_i. The pp is guaranteed to match if:
  • all q_i^r 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 q_i^r is discrete, the former can be reformualated to $z_i := Z_{r(i)} * {{s_i} \over {\sum\limits_{j: r(j) = r(i)} s_j}}$ 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 function.

• can be made subject to minimum (zero?) and maximum values for z_i, by transforming and capping the result of scalesum. To comply to the pp, an iterative fitting factor f_r, initially set to 1, can be used. Capping in GeoDMS: min_elem(z, z_max), max_elem(z, z_min), median(z, interval(z_min, z_max)).

• can be combined with disaggregation of other quantities such that each unit i is allocated once (Iterative proportional fitting, Continuous Allocation, or Discrete Allocation).

• can be done by maximizing smoothness of the z_i to adhere to Tobler's first law of geography, aka Smooth Pycnophylactic Interpolation.

intensive quantities

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