Iterative proportional fitting

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...