We have also the entropy maximization,[2][3] information loss minimization (or cross-entropy)[4] or RAS which consists of factoring the matrix rows to match the specified row totals, then factoring its columns to match the specified column totals; each step usually disturbs the previous step's match, so these steps are repeated in cycles, re-adjusting the rows and columns in turn, until all specified marginal totals are satisfactorily approximated.
IPF has been "re-invented" many times, the earliest by Kruithof in 1937 [6] in relation to telephone traffic ("Kruithof’s double factor method"), Deming and Stephan in 1940[7] for adjusting census crosstabulations, and G.V.
[9] Early proofs of uniqueness and convergence came from Sinkhorn (1964),[10] Bacharach (1965),[11] Bishop (1967),[12] and Fienberg (1970).
Fienberg's proof by differential geometry exploits the method's constant crossproduct ratios, for strictly positive tables.
[14] found necessary and sufficient conditions for general tables having zero entries.
Pukelsheim and Simeone (2009) [15] give further results on convergence and error behavior.
An exhaustive treatment of the algorithm and its mathematical foundations can be found in the book of Bishop et al.
In most cases, IPFP is preferred due to its computational speed, low storage requirements, numerical stability and algebraic simplicity.
Applications of IPFP have grown to include trip distribution models, Fratar or Furness and other applications in transportation planning (Lamond and Stewart), survey weighting, synthesis of cross-classified demographic data, adjusting input–output models in economics, estimating expected quasi-independent contingency tables, biproportional apportionment systems of political representation, and for a preconditioner in linear algebra.
is “indecomposable”, then this process has a unique fixed-point because it is deduced from a program where the function is a convex and continuously derivable function defined on a compact set.
The demonstrations calls some above properties, particularly the Theorem of separable modifications and the composition of biproportions.
Alternatively, we can estimate the row and column factors separately: Choose initial values
set Repeat these steps until successive changes of a and b are sufficiently negligible (indicating the resulting row- and column-sums are close to u and v).
Notes: The vaguely demanded 'similarity' between M and X can be explained as follows: IPFP (and thus RAS) maintains the crossproduct ratios, i.e. since
This property is sometimes called structure conservation and directly leads to the geometrical interpretation of contingency tables and the proof of convergence in the seminal paper of Fienberg (1970).
Direct factor estimation (algorithm 2) is generally the more efficient way to solve IPF: Whereas a form of the classical IPFP needs elementary operations in each iteration step (including a row and a column fitting step), factor estimation needs only operations being at least one order in magnitude faster than classical IPFP.
IPFP can be used to estimate expected quasi-independent (incomplete) contingency tables, with
For fully independent (complete) contingency tables, estimation with IPFP concludes exactly in one cycle.
Similar to the IPF, the NM-method is also an operation of finding a matrix
For instance, the NM-method defines closeness of matrices of the same size differently from the IPF.
is a sample from this population in problems where the IPF is applied as the maximum likelihood estimator.
Macdonald (2023)[20] is at ease with the conclusion by Naszodi (2023)[21] that the IPF is suitable for sampling correction tasks, but not for generation of counterfactuals.
Similarly to Naszodi, Macdonald also questions whether the row and column proportional transformations of the IPF preserve the structure of association within a contingency table that allows us to study social mobility.
Necessary and sufficient conditions for the existence and uniqueness of MLEs are complicated in the general case (see[22]), but sufficient conditions for 2-dimensional tables are simple: If unique MLEs exist, IPFP exhibits linear convergence in the worst case (Fienberg 1970), but exponential convergence has also been observed (Pukelsheim and Simeone 2009).
If unique MLEs do not exist, IPFP converges toward the so-called extended MLEs by design (Haberman 1974), but convergence may be arbitrarily slow and often computationally infeasible.
If all observed values are strictly positive, existence and uniqueness of MLEs and therefore convergence is ensured.
After completing three cycles, each with a row adjustment and a column adjustment, we get a closer approximation: The R package mipfp (currently in version 3.2) provides a multi-dimensional implementation of the traditional iterative proportional fitting procedure.
[23] The package allows the updating of a N-dimensional array with respect to given target marginal distributions (which, in turn can be multi-dimensional).
Python has an equivalent package, ipfn[24][25] that can be installed via pip.
The package supports numpy and pandas input objects.