n Original Research Paper Chemometrics and Intelligent Laboratory Systems, 19 (1993) 43-51 Elsevier Science Publishers B.V., Amsterdam 43 Linear discriminant hierarchical clustering: a modeling and cross-validable divisive clustering method E. Marengo I/is Druento 115, 10151 Totino (Italy) R. Todeschini Dipartimento di Chimica Fkica ed Elettrochimica, V? Golgi 19, 20133 Milano (Italy) (Received 14 February 1992; accepted 12 October 1992) zyxwvutsrqponmlkjihgfedcbaZYXWV Abstract Marengo, E. and Todeschini, R., 1993. Linear discriminant hierarchical clustering: a modeling and cross-validable divisive clustering method. Chemometrics and Intelligent Laboratory Systems, 19: 43-51. A simple modification of the linear discriminant classification tree (LDCT) (R. Todeschini and E. Marengo, Linear discriminant classification trees, Chemometrics and Intelligent Laboratory Systems, 16 (1992) 25-35) method is.used to deal with clustering problems in the framework of a divisive hierarchical strategy. The method produces a mathematical model which allows the validation of the obtained clusters by using a cross-validation technique; simple parameters can then be estimated to assess the significance of the clusters. The performance of linear discriminant hierarchical clustering has been tested on seven data sets where the class assignments are known. INTRODUCTION Cluster analysis is an unsupervised strategy with the purpose of collecting samples of a train- ing set into a number of different groups called clusters. Within each individual cluster the prop- erties of the objects belonging to that cluster should be more similar to each other than to the Correspondence to: E. Marengo, Via Druento 115, 10151 Torino (Italy). properties of the objects belonging to the other clusters. In particular, two properties of the clus- ters often regarded as desirable are stability and objectivity, but it is difficult to define a single mathematical approach, acceptable to all, which states the general criteria of the acceptability of clustering results. Cluster analysis is a useful and widely used exploratory tool, but little attention has been paid to building clustering methods as modeling tech- niques. Several clustering methods perform hier- 0169-7439/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved