Clustering of individuals, segmentation of time series and segmentation of numerical images can all be considered as labeling problems, for each can be described in terms of pairs (x(t),g(t)), t = 1,2,...,n, where x(t) is the observation at instance t and g(t) is the unobservable "label" of instance t. The labels are to be estimated, along with any unspecified distributional parameters. In cluster analysis the values of t are the individuals (cases) observed and the x's are independent. In time series the values of t are time instants and there is temporal correlation. In numerical image segmentation the values of t denote picture elements (pixels) and spatial correlation between neighboring pixels can be utilized.

Signals and time-series often are not homogeneous but rather are generated by mechanisms or processes with various phases. Similarly, images are not homogeneous but contain various objects. "Segmentation" is a process of attempting to recover automatically the phases or objects. A model for representing such signals, time series, and images was discussed in a paper by the present author in the Proceedings of the 30th Conference; some approaches to estimation and segmentation in this model were presented. The present paper summarizes the work on all these types of labeling problems, clustering as well as time series- and image-segmentation.

Key words and phrases: statistical pattern recognition, classification; temporal correlation, spatial correlation; optimization by relaxation method