A distance-based variety of nonlinear multivariate data analysis, including weights for objects and variables

In the distance approach to nonlinear multivariate data analysis the focus is on the optimal representation of the relationships between the objects i n the analysis. In this paper two methods are presented for including weigh ts in distance-based nonlinear multivariate data analysis. In the first met hod weights are assigned to the objects while the second method is concerne d with differential weighting of groups of variables. When each analysis va riable defines a group the latter method becomes a variable weighting metho d Far objects the weights are assumed to be given; for groups of variables they may be given, or estimated. These weighting schemes can also be combin ed and have several important applications. For example, they make it possi ble to perform efficient analyses of large data sets, to use the distance-b ased variety of nonlinear multivariate data analysis as an addition to logl inear 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 efficie nt algorithm is proposed to minimize the corresponding loss function.