Mean Absolute Deviations about the Mean, the Cut Norm and Taxicab Correspondence Analysis

Abstract

O ptimization has two faces, minimization of a loss function or maximization of a gain function. We show that the mean absolute deviation about the mean, d , maximizes a gain function based on the power set of the individuals; and nd , where n is the sample size, equals twice the value of the cut-norm o f the deviations about the mean. This property is generalized to double-centered and triple-centered data sets. Furthermore, we show that among the three well known dispersion measures, standard deviation, least absolute deviation and d , d is the most robust based on the relative contribution criterion. More importantly, we show that the computation of each principal dimension of taxicab correspondence analysis (TCA) corresponds to balanced 2-blocks seriation. These ideas are applied on two data sets.