Journal of Classification

Many problems entail the analysis of data that are independent and identically distributed random graphs. Useful inference requires flexible probability models for such random graphs; these models should have interpretable location and scale parameters, and support the establishment of confidence regions, maximum likelihood estimates, goodness-of-fit tests, Bayesian inference, and an appropriate analogue of linear model theory. Banks and Carley (1994) develop a simple probability model and sketch some analyses; this paper extends that work so that analysts are able to choose models that reflect application-specific metrics on the set of graphs. The strategy applies to graphs, directed graphs, hypergraphs, and trees, and often extends to objects in countable metric spaces.

The number of trees withn labeled terminal vertices grows too rapidly withn to permit exhaustive searches for Steiner trees or other kinds of optima in cladistics and related areas Often, however, structured constraints are known and may be imposed on the set of trees to be scanned These constraints may be formulated in terms of a consensus among the trees to be searched We calculate the reduction in the number of trees to be enumerated as a function of properties of the imposed consensus

Many common (nonhierarchical) clustering and classification methods are optimization-based methods, in the sense described by Windham (1987) in this Journal. This paper gives some large sample properties for estimates derived by such methods. Under appropriate conditions, such estimates converge with probability one to a limit, and are asymptotically normally distributed around that limiting value. The conditions are satisfied by most of the common examples of optimization-based methods.

It is common practice to perform a principal component analysis (PCA) on a correlation matrix to represent graphically the relations among numerous variables. In such a situation, the variables may be considered as points on the unit hypersphere of an Euclidean space, and PCA provides a sort of best fit of these points within a subspace. Taking into account their particular position, this paper suggests to represent the variables on an optimal three-dimensional unit sphere.