Ordering and scaling objects in multivariate data under nonlinear transformations of variables

Abstract

An integrated iterative method is presented for the optimal ordering and scaling of objects in multivariate data, where the variables themselves may be transformed in the process of optimizing the objective function. Given an ordering of objects, optimal transformation of variables is guaranteed by the combined use of majorization (a particular (sub)gradient optimization method) and projection methods. The optimal sequencing is a combinatorial task and should not be carried out by applying standard optimization techniques based on gradients, because these are known to result in severe problems of local optima. Instead, a combinatorial data analysis strategy is adopted that amounts to a cyclic application of a number of local operations. A crucial objective for the overall method is the graphical display of the results, which is implemented by spacing the object points optimally over a one-dimensional continuum. An indication is given for how the overall process converges to a (possibly local) optimum. As an illustration, the method is applied to the analysis of a published observational data set. © Springer-Verlag Berlin Heidelberg 2011.