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 \(q_i^r \ge 0\) 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)]".