ABSTRACT
A method, now called Sindclus, for fitting the Adclus/Indclus models for overlapping clustering of two-way or three-way symmetric proximity data was presented recently by Chaturvedi and Carroll (1992), utilizing a numerical procedure very similar to the Carroll and Chang Candecomp method used for fitting the Indscal model for three-way multidimensional scaling (and for multiway components analysis), Sindclus fits one cluster at a time to a sequence of residual data arrays, doing this by iterating among conditional least-squares estimates of parameters for each of the three ways. A separability property observed by Chaturvedi enables straightforward conditional OLS estimation of the discrete “cluster membership” parameters for the two modes corresponding to objects. Weights for the third mode (e.g., subjects) were fit by OLS regression. While the binary parameters for the two object modes are not constrained to be equal on each iteration, as in Candecomp applied to fitting the symmetric Indscal model—given data symmetric in two modes—the two sets of parameter estimates are generally equal upon convergence, Candclus is a multiway generalization of this model and estimation procedure, allowing binary parameters for any p of the N modes, and continuous parameters for the remaining N − p. A generalized separability property allows straightforward conditional OLS estimation for each mode modeled by binary parameters. Special cases include Candecomp, fit “one dimension at a time,” and a model in which all N modes are modeled via overlapping cluster structures. Hybrid models, in which a given mode can be modeled by a hybrid mixture of continuous (spatial) and discrete (clusterlike) dimensions, can also be fit within this framework. A generalization of Candclus, called Mumclus (multimode clustering) is also formulated and discussed theoretically).
