The mixture method of clustering applied to three-way data

Clustering or classifying individuals into groups such that there is relative homogeneity within the groups and heterogeneity between the groups is a problem which has been considered for many years. Most available clustering techniques are applicable only to a two-way data set, where one of the modes is to be partitioned into groups on the basis of the other mode. Suppose, however, that the data set is three-way. Then what is needed is a multivariate technique which will cluster one of the modes on the basis of both of the other modes simultaneously. It is shown that by appropriate specification of the underlying model, the mixture maximum likelihood approach to clustering can be applied in the context of a three-way table. It is illustrated using a soybean data set which consists of multiattribute measurements on a number of genotypes each grown in several environments. Although the problem is set in the framework of clustering genotypes, the technique is applicable to other types of three-way data sets.

[1]  D. E. Byth,et al.  Two-way pattern analysis of a large data set to evaluate genotypic adaptation , 1976, Heredity.

[2]  J. Carroll,et al.  Synthesized clustering: A method for amalgamating alternative clustering bases with differential weighting of variables , 1984 .

[3]  Douglas M. Hawkins,et al.  Topics in Applied Multivariate Analysis: CLUSTER ANALYSIS , 1982 .

[4]  D. Binder.  Bayesian cluster analysis , 1978 .

[5]  W. Williams,et al.  Numerical analysis of variation patterns in the genus Stylosanthes as an aid to plant introduction and assessment , 1971 .

[6]  W. DeSarbo,et al.  The representation of three-way proximity data by single and multiple tree structure models , 1984 .

[7]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[8]  K. Basford.  The use of multidimensional scaling in analysing multi-attribute genotype response across environments , 1982 .

[9]  B. Morgan.  Three Applications of Methods of Cluster-analysis , 1981 .

[10]  G. J. McLachlan,et al.  9 The classification and mixture maximum likelihood approaches to cluster analysis , 1982, Classification, Pattern Recognition and Reduction of Dimensionality.

[11]  M. Kendall.  A course in multivariate analysis , 1958 .

[12]  J. Kruskal.  Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis , 1964 .

[13]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[14]  J. Kruskal.  Nonmetric multidimensional scaling: A numerical method , 1964 .

[15]  P. Arabie,et al.  Indclus: An individual differences generalization of the adclus model and the mapclus algorithm , 1983 .

[16]  J. Kruskal.  The Relationship between Multidimensional Scaling and Clustering , 1977 .

[17]  J. Ramsay.  Some Statistical Approaches to Multidimensional Scaling Data , 1982 .

[18]  Murray Aitkin,et al.  Statistical Modelling of Data on Teaching Styles , 1981 .

[19]  D. E. Byth,et al.  Genotype × environment interactions and environmental adaptation. I. Pattern analysis — application to soya bean populations , 1974 .

[20]  E. Harner,et al.  Analyses of multivariately determined community matrices using cluster analysis and multidimensional scaling , 1980 .

[21]  J. Wolfe.  A Monte Carlo Study of the Sampling Distribution of the Likelihood Ratio for Mixtures of Multinormal Distributions , 1971 .

[22]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[23]  D. E. Byth,et al.  Genotype × environment interactions and environmental adaptation. II.* Assessment of environmental contributions , 1977 .

[24]  Wei-Chien Chang.  On using Principal Components before Separating a Mixture of Two Multivariate Normal Distributions , 1983 .

[25]  J. Hartigan.  Distribution Problems in Clustering , 1977 .

[26]  Geoffrey J. McLachlan,et al.  Cluster analysis in a randomized complete block design , 1985 .

[27]  Brian Everitt,et al.  Cluster analysis , 1974 .