Iterative proportional fitting

Iterative Proportional Fitting (IPF) are disaggregation methods to obtain doubly constrained matrix cell values \( X_{ij} \) proportional to a given strictly positive interaction measure (aka proxy) \( C_{ij} \) given restrictions on row an column totals \( T_{i.} \) and \( T_{.j} \).

We introduce \( a_i \) and \( b_j \) to express \( X_{ij} \) as \( a_i b_j C_{ij} \).

It follows that:

\( a_i := {{T_{i.}} \over \sum\limits_{j}{b_j C_{ij}}} \) and \( b_j := {{T_{.j}} \over \sum\limits_{i}{a_i C_{ij}}} \)

An IPF is used to solve the Continuous Allocation of the earlier versions of the Land Use Scanner by using \( e^{\beta s_{ij}} \) as proxy.

=Applications of IPF= 
 * Doubly Constrained Gravity Models of Spatial Interaction
 * Trip Distribution in Transport modelling is often modelled as doubly constrained gravity model
 * Continuous Allocation where land units are sources and claims for land use are destinations.

=More Links=
 * http://www.pbl.nl/publicaties/2000/Iteratief_Proportioneel_Fitten__Methodiek_en_toepassing_voor_de_woonruimteverdeling_in_Geografische_Informatiesystemen_voor_de_Vijfde_Nota_Ruimtelijke_Ordening IPF] RIVM/PBL paper (in Dutch)
 * LAND USE SCANNER: An integrated GIS based model for long term projections of land use in urban and rural areas, Maarten Hilferink and Piet Rietveld Article on SpringerLink describing the Continuous Allocation in the Land Use Scanner