Phipps Arabie

Clustering and Classification

At a moderately advanced level, this book seeks to cover the areas of clustering and related methods of data analysis where major advances are being made. Topics include: hierarchical clustering, variable selection and weighting, additive trees and other network models, relevance of neural network models to clustering, the role of computational complexity in cluster analysis, latent class approaches to cluster analysis, theory and method with applications of a hierarchical classes model in psychology and psychopathology, combinatorial data analysis, clusterwise aggregation of relations, review of the Japanese-language results on clustering, review of the Russian-language results on clustering and multidimensional scaling, practical advances, and significance tests.

Was euclid an unnecessarily sophisticated psychologist

A survey of the current state of multidimensional scaling using the city-block metric is presented. Topics include substantive and theoretical issues, recent algorithmic developments and their implications for seemingly straightforward analyses, isometries with other metrics, links to graph-theoretic models, and future prospects.

Multidimensional scaling in the city-block metric: A combinatorial approach

We present an approach, independent of the common gradient-based necessary conditions for obtaining a (locally) optimal solution, to multidimensional scaling using the city-block distance function, and implementable in either a metric or nonmetric context. The difficulties encountered in relying on a gradient-based strategy are first reviewed: the general weakness in indicating a good solution that is implied by the satisfaction of the necessary condition of a zero gradient, and the possibility of actual nonconvergence of the associated optimization strategy. To avoid the dependence on gradients for guiding the optimization technique, an alternative iterative procedure is proposed that incorporates (a) combinatorial optimization to construct good object orders along the chosen number of dimensions and (b) nonnegative least-squares to re-estimate the coordinates for the objects based on the object orders. The re-estimated coordinates are used to improve upon the given object orders, which may in turn lead to better coordinates, and so on until convergence of the entire process occurs to a (locally) optimal solution. The approach is illustrated through several data sets on the perception of similarity of rectangles and compared to the results obtained with a gradient-based method.

Correspondence analysis and optimal structural representations

Many well-known measures for the comparison of distinct partitions of the same set ofn objects are based on the structure of class overlap presented in the form of a contingency table (e.g., Pearsons chi-square statistic, Rands measure, or Goodman-Kruskalsτb), but they all can be rephrased through the use of a simple cross-product index defined between the corresponding entries from twon ×n proximity matrices that provide particular a priori (numerical) codings of the within- and between-class relationships for each of the partitions. We consider the task of optimally constructing the proximity matrices characterizing the partitions (under suitable restriction) so as to maximize the cross-product measure, or equivalently, the Pearson correlation between their entries. The major result presented states that within the broad classes of matrices that are either symmetric, skew-symmetric, or completely arbitrary, optimal representations are already derivable from what is given by a simple one-dimensional correspondence analysis solution. Besides severely limiting the type of structures that might be of interest to consider for representing the proximity matrices, this result also implies that correspondence analysis beyond one dimension must always be justified from logical bases other than the optimization of a single correlational relationship between the matrices representing the two partitions.