Iterative proportional fitting

Iterative proportional fitting finds the matrix closest to a given starting matrix whose row and column totals match those of a target matrix. The fitted matrix has the form of the initial matrix scaled by diagonal matrices on each side. The usual procedure scales the rows to match the required row totals, then scales the columns to match the column totals; each step disturbs the match made by the previous one, so the two steps are repeated in cycles until the specified totals are satisfactorily approximated. With three or more dimensions, the margins of each dimension are adjusted in turn. The method goes by many names: biproportional fitting, the RAS algorithm in economics, raking in survey statistics and matrix scaling in computer science. It has been reinvented many times, first by Kruithof in 1937 for telephone traffic and by Deming and Stephan in 1940 for census cross-tabulations. It is applied to survey weighting, trip distribution in transportation planning and input-output models in economics.