# **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 $z_i := \sum_{\lim_{r}} Z_r * \frac{s_i * q_i^r}{\sum_{\lim_{j}} s_j * q_j^r}$; 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*).

- all

- When
*q*_{i}^{r}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
*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)

- 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).
- can be done using a incidence proxy
*c*_{i}, aka dasymmetric mapping, in GeoDMS: “c ? Z[r] : 0[valuesunit](valuesunit.html)(Z)".