THE GEOMETRY OF MULTIVARIATE OBJECT PREPROCESSING

The geometric properties of three common object-preprocessing transfor mations (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 transf ormations. In general, each transformation distorts the shape and dime nsionality of patterns in the data: transformed data lie on (C-1)-dime nsional surfaces in the original C-dimensional space. A data set that has been closed by one of these transformations can be reopened if a v ector containing the constant sums, constant lengths or maximum values of the original objects was retained. Transformed data sets may be fr eely interconverted among these three transformations without the loss of information.