Estimate a cross table from a series of row and columntotals

Consider the following problem:

Known:

Requested:

Q_g**h^r, an estimated number of objects with g-type g and h-type h in zone r.
estimated number of objects per h-type as a function (preferably a linear combination) of object numbers per h-type (for estimating a h-type distribution given a g-type distribution of new objects).

We assume that:

For each zone r, the F and G count all objects, thus: $\forall r: \sum\limits_g G^r_g = \sum\limits_h H^r_h$
P_hⁱ is equal for all i in the same r and with the same g, thus depends only on P(h|r**g).
Q_g**h^r has a G_g^r repeated categorical distribution per row g and zone r; thus E[Q_g**h^r] = P(h|r**g) ⋅ G_g^r
Q_g**h^r := f_g ⋅ f_h ⋅ P_g**h such that $\sum\limits_{h} Q^r_{gh} = G^r_g$ and $\sum\limits_{g} Q^r_{gh} = H^r_h$, to be determined by Iterative proportional fitting

Thus:

$\sum\limits_{h} P(h|rg) = 1$, $f_g = G^r_g / \sum\limits_{h} f_h \cdot P_{gh}$, and $f_h = H^r_h / \sum\limits_{g} f_g \cdot P_{gh}$.
P_g**h is to be determined by regression of H_h^r by G_g^r, thus, written in matrix notation: H = G × P + ϵ with ϵ a r × h matrix of independent stochasts with zero expectation.
it follows that: P := (G^T×G)⁻¹ × (G^T×H)

ToDo:

Consider Alternative: Q_bjh is determined by discrete allocation with the given constraints and suitability P_g**h
Consider Alternative: P := (G^T×H) × (H^T×H)⁻¹ which follows from regression of G_g^r by H_h^r.
Consider Alternative: P := (G^T×H)
Consider Effect of possible heteroscedasticity of ϵ with H; i.e. assume var(ϵ) ∼ H, as each element of H is assumed to be the sum over j of the results of G_g^r trials of categorical distributions with conditional probabilities P_g|h.

Estimate a cross table from a series of row and columntotals - ObjectVision/GeoDMS GitHub Wiki