The iterative proportional fitting procedure (IPF or IPFP, also known as biproportional fitting or biproportion in statistics or economics (input-output analysis, etc.), RAS algorithm in economics, raking in survey statistics, and matrix scaling in computer science) is the operation of finding the fitted matrix X {\displaystyle X} which is the closest to an initial matrix Z {\displaystyle Z} but with the row and column totals of a target matrix Y {\displaystyle Y} (which provides the constraints of the problem; the interior of Y {\displaystyle Y} is unknown). The fitted matrix being of the form X = P Z Q {\displaystyle X=PZQ} , where P {\displaystyle P} and Q {\displaystyle Q} are diagonal matrices such that X {\displaystyle X} has the margins (row and column sums) of Y {\displaystyle Y} . Some algorithms can be chosen to perform biproportion. We have also the entropy maximization, information loss minimization (or cross-entropy) 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. However, all algorithms give the same solution. In three- or more-dimensional cases, adjustment steps are applied for the marginals of each dimension in turn, the steps likewise repeated in cycles.
History IPF has been "re-invented" many times, the earliest by Kruithof in 1937 in relation to telephone traffic ("Kruithof’s double factor method"), Deming and Stephan in 1940 for adjusting census crosstabulations, and G.V. Sheleikhovskii for traffic as reported by Bregman. (Deming and Stephan proposed IPFP as an algorithm leading to a minimizer of the Pearson X-squared statistic, which Stephan later reported it does not). Early proofs of uniqueness and convergence came from Sinkhorn (1964), Bacharach (1965), Bishop (1967), and Fienberg (1970). Bishop's proof that IPFP finds the maximum likelihood estimator for any number of dimensions extended a 1959 proof by Brown for 2x2x2... cases. Fienberg's proof by differential geometry exploits the method's constant crossproduct ratios, for strictly positive tables. Csiszár (1975). found necessary and sufficient conditions for general tables having zero entries. Pukelsheim and Simeone (2009) 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. (1975). Idel (2016) gives a more recent survey. Other general algorithms can be modified to yield the same limit as the IPFP, for instance the Newton–Raphson method and the EM algorithm. 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.
Biproportion Biproportion, whatever the algorithm used to solve it, is the following concept: Z {\displaystyle Z} , matrix Y {\displaystyle Y} and matrix X {\displaystyle X} are known real nonnegative matrices of dimension n , m {\displaystyle n,m} ; the interior of Y {\displaystyle Y} is unknown and X {\displaystyle X} is searched such that X {\displaystyle X} has the same margins than Y {\displaystyle Y} , i.e. X s = Y s {\displaystyle Xs=Ys} and s ′ X = s ′ Y {\displaystyle s'X=s'Y} ( s {\displaystyle s} being the sum vector), and such that X {\displaystyle X} is close to Z {\displaystyle Z} following a given criterion, the fitted matrix being of the form X = K ( Z , Y ) = P Z Q {\displaystyle X=K(Z,Y)=PZQ} , where P {\displaystyle P} and Q {\displaystyle Q} are diagonal matrices.
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