Showing papers by "Henk A.L. Kiers published in 2000"

Towards a standardized notation and terminology in multiway analysis

Abstract: This paper presents a standardized notation and terminology to be used for three- and multiway analyses, especially when these involve (variants of) the CANDECOMP/PARAFAC model and the Tucker model The notation also deals with basic aspects such as symbols for different kinds of products, and terminology for three- and higher-way data The choices for terminology and symbols to be used have to some extent been based on earlier (informal) conventions Simplicity and reduction of the possibility of confusion have also played a role in the choices made Copyright (C) 2000 John Wiley & Sons, Ltd

Three-mode principal components analysis: choosing the numbers of components and sensitivity to local optima.

Abstract: A method that indicates the numbers of components to use in fitting the three-mode principal components analysis (3MPCA) model is proposed This method, called DIFFIT, aims to find an optimal balance between the fit of solutions for the 3MPCA model and the numbers of components The achievement of DIFFIT is compared with that of two other methods, both based on two-way PCAs, by means of a simulation study It was found that DIFFIT performed considerably better than the other methods in indicating the numbers of components The 3MPCA model can be estimated by the TUCKALS3 algorithm, which is an alternating least squares algorithm In a study of how sensitive TUCKALS3 is at hitting local optima, it was found that, if the numbers of components are specified correctly, TUCKALS3 never hits a local optimum The occurrence of local optima increased as the difference between the numbers of underlying components and the numbers of components as estimated by TUCKALS3 increased Rationally initiated TUCKALS3 runs hit local optima less often than randomly initiated runs

Data analysis, classification, and related methods

Abstract: The volume presents new developments in data analysis and classification, and gives a state of the art impression of these scientific fields at the turn of the Millenium. Areas that receive considerable attention in this book are Cluster Analysis, Data Mining, Multidimensional and Symbolic Data Analysis, Decision and Regression Trees. The volume contains a refereed selection of original papers, overview papers, and innovative applications presented at the 7th Conference of the International Federation of Classification Societies (IFCS-2000), with contributions from eminent scientists all over the world. The reader finds introductory material into various areas and kaleidoscopic views of recent technical and methodological developments in widely different areas within data analysis and classification. The presence of a large number of application papers demonstrates the usefulness of the recently developed techniques.

Some procedures for displaying results from three-way methods

Abstract: Three-way Tucker analysis and CANDECOMP/PARAFAC are popular methods for the analysis of three-way data (data pertaining to three sets of entities). To interpret the results from these methods, one can, in addition to inspecting the component matrices and the core array, inspect visual representations of the outcomes. In this paper, first an overview is given of plotting procedures currently in use with three-way methods. Not all of these optimally correspond to the actual approximation of the data furnished by the three-way method at hand. Next it is described how plotting procedures can be designed that do correspond exactly to the low-dimensional description of the data by means of the three-way method at hand, and it is indicated to what extent these correspond to the ones currently in use. Specifically, procedures are described for displaying either one set of entities (e.g. a set of chemical samples) in two- or three-dimensional plots, or a set of combinations of entities (e.g, pertaining to each object at each time point, thus providing 'trajectories' for each object). Furthermore, it is shown how, in these plots, the other entities can be plotted simultaneously (e.g. superimposing the variables on a plot with trajectories for objects), Both procedures are summarized in an appendix. Copyright (C) 2000 John Wiley & Sons, Ltd.

Transforming three-way arrays to multiple orthonormality

Abstract: This paper is concerned with the question to what extent the concept of rowwise or columnwise orthonormality can he generalized to three-way arrays. Whereas transforming a three-way array to multiple orthogonality is immediate, transforming it to multiple orthonormality is far from straightforward. The present paper offers an iterative algorithm for such transformations, and gives a proof of monotonical convergence when only two modes are orthonormalized. Also, it is shown that a variety of three-way arrays do not permit double orthonormalization. This is due to the order of the arrays, and holds regardless of the particular elements of the array. Studying three-way orthonormality has proven useful in exploring the possibilities for simplifying the core, to guide the search for equivalent direct transformations to simplicity; see Murakami et al. (Psychometrika 1998; 63: 255-261) as an example. Also, it appears in various contexts of the mathematical study of three-way analysis. Copyright (C) 2000 John Wiley & Sons, Ltd.

Projections Distinguishing Isolated Groups in Multivariate Data Spaces

Abstract: When multivariate data are available from various different groups of objects, one may wish to discern the most “extreme” groups in the multivariate data space. Four procedures are proposed for finding projections of the high-dimensional data configuration onto a small number of dimensions along which such groups are distinguished best. These procedures involve maximization of either a sum of correlation ratios, or a sum of between-groups variances. It is shown that the use of correlation ratios often leads to trivial dimensions corresponding to directions in the configuration along which the data hardly vary.