In the distance approach to nonlinear multivariate data analysis the focus is on the optimal representation of the relationships between the objects in the analysis. In this paper two methods are presented for including weights in distance-based nonlinear multivariate data analysis. In the first method, weights are assigned to the objects while the second method is concerned with differential weighting of groups of variables. When each analysis variable defines a group the latter method becomes a variable weighting method. For objects the weights are assumed to be given; for groups of variables they may be given, or estimated. These weighting schemes can also be combined and have several important applications. For example, they make it possible to perform efficient analyses of large data sets, to use the distance-based variety of nonlinear multivariate data analysis as an addition to loglinear analysis of multiway contingency tables, and to do stability studies of the solutions by applying the bootstrap on the objects or the variables in the analysis. These and other applications are discussed, and an efficient algorithm is proposed to minimize the corresponding loss function.
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However, the latter approach also carries the computational burden of having to deal with N(N - 1)/2 distances, where N is the number of objects. This burden can become prohibitive This study is funded by The Netherlands Organizationf or Scientific Research (NWO) by grant nr. 030-56403 for the "PIONEER" project "Subject Oriented MultivariateA nalysis"t o the third author. Requests for reprints should be sent to Jacques J.F. Commandeur,D ata Theory Group, Department of Education, Faculty of Social and Behavioral Sciences, Leiden University,P .O. Box 9555, 2300 RB Leiden, THE NETHERLANDS.