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The geometry of multivariate object preprocessing

The geometric properties of three common object‐preprocessing transformations (constant sum, or closure; constant length, or normalization; and maximum value, or ratioing) are investigated. An argument is made for using absolute values in the constant sum and maximum value transformations. In general, each transformation distorts the shape and dimensionality of patterns in the data: transformed data lie on (C‐1)‐dimensional surfaces in the original C‐dimensional space. A data set that has been closed by one of these transformations can be reopened if a vector containing the constant sums, constant lengths or maximum values of the original objects was retained. Transformed data sets may be freely interconverted among these three transformations without the loss of information.