6533b7d2fe1ef96bd125e9a1

RESEARCH PRODUCT

What is the Best Method of Matrix Adjustment? A Formal Answer by a Return to the World of Vectors

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The principle of matrix adjustment methods consists into finding what is the matrix which is the closest to an initial matrix but with respect of the column and row sum totals of a second matrix. In order to help deciding which matrix-adjustment method is the better, the article returns to the simpler problem of vector adjustment then back to matrices. The information-lost minimization (biproportional methods and RAS) leads to a multiplicative form and generalize the linear model. On the other hand, the distance minimization which leads to an additive form tends to distort the data by giving a result asymptotically independent to the initial matrix. The result allows concluding non-ambiguously that biproportional methods and RAS are the best for matrix adjustment as they generalize the linear model and are asymptotically the most respectful of the initial matrix while the do not generate surprising negative terms. Moreover, measuring the gap between the projection and the target cannot help deciding which method is the best because the gap depends on the target matrix, while this gap can be interpreted in terms of structural effect generalizing the shift-share method.