Comparing partitions

The problem of comparing two different partitions of a finite set of objects reappears continually in the clustering literature. We begin by reviewing a well-known measure of partition correspondence often attributed to Rand (1971), discuss the issue of correcting this index for chance, and note that a recent normalization strategy developed by Morey and Agresti (1984) and adopted by others (e.g., Miligan and Cooper 1985) is based on an incorrect assumption. Then, the general problem of comparing partitions is approached indirectly by assessing the congruence of two proximity matrices using a simple cross-product measure. They are generated from corresponding partitions using various scoring rules. Special cases derivable include traditionally familiar statistics and/or ones tailored to weight certain object pairs differentially. Finally, we propose a measure based on the comparison of object triples having the advantage of a probabilistic interpretation in addition to being corrected for chance (i.e., assuming a constant value under a reasonable null hypothesis) and bounded between ±1.William H.E. Day was Acting Editor for the reviewing of this paper. We are grateful to him, Ove Frank, Charles Lewis, Glenn W. Milligan, Ivo Molenaar, Stanley S. Wasserman, and anonymous referees for helpful suggestions. Lynn Bilger and Tom Sharpe provided competent technical assistance. Partial support of Phipps Arabie's participation in this research was provided by NSF Grant SES 8310866 and ONR Contract N00014-83-K-0733.     

Deep roots of the Messinian salinity crisis

The Messinian salinity crisis--the desiccation of the Mediterranean Sea between 5.96 and 5.33 million years (Myr) ago--was one of the most dramatic events on Earth during the Cenozoic era. It resulted from the closure of marine gateways between the Atlantic Ocean and the Mediterranean Sea, the causes of which remain enigmatic. Here we use the age and composition of volcanic rocks to reconstruct the geodynamic evolution of the westernmost Mediterranean from the Middle Miocene epoch to the Pleistocene epoch (about 12.1-0.65 Myr ago). Our data show that a marked shift in the geochemistry of mantle-derived volcanic rocks, reflecting a change from subduction-related to intraplate-type volcanism, occurred between 6.3 and 4.8 Myr ago, largely synchronous with the Messinian salinity crisis. Using a thermomechanical model, we show that westward roll back of subducted Tethys oceanic lithosphere and associated asthenospheric upwelling provides a plausible mechanism for producing the shift in magma chemistry and the necessary uplift (approximately 1 km) along the African and Iberian continental margins to close the Miocene marine gateways, thereby causing the Messinian salinity crisis.     

Multidimensional scaling in the city-block metric: A combinatorial approach

We present an approach, independent of the common gradient-based necessary conditions for obtaining a (locally) optimal solution, to multidimensional scaling using the city-block distance function, and implementable in either a metric or nonmetric context. The difficulties encountered in relying on a gradient-based strategy are first reviewed: the general weakness in indicating a good solution that is implied by the satisfaction of the necessary condition of a zero gradient, and the possibility of actual nonconvergence of the associated optimization strategy. To avoid the dependence on gradients for guiding the optimization technique, an alternative iterative procedure is proposed that incorporates (a) combinatorial optimization to construct good object orders along the chosen number of dimensions and (b) nonnegative least-squares to re-estimate the coordinates for the objects based on the object orders. The re-estimated coordinates are used to improve upon the given object orders, which may in turn lead to better coordinates, and so on until convergence of the entire process occurs to a (locally) optimal solution. The approach is illustrated through several data sets on the perception of similarity of rectangles and compared to the results obtained with a gradient-based method.     

Bending-related faulting and mantle serpentinization at the Middle America trench

The dehydration of subducting oceanic crust and upper mantle has been inferred both to promote the partial melting leading to arc magmatism and to induce intraslab intermediate-depth earthquakes, at depths of 50-300 km. Yet there is still no consensus about how slab hydration occurs or where and how much chemically bound water is stored within the crust and mantle of the incoming plate. Here we document that bending-related faulting of the incoming plate at the Middle America trench creates a pervasive tectonic fabric that cuts across the crust, penetrating deep into the mantle. Faulting is active across the entire ocean trench slope, promoting hydration of the cold crust and upper mantle surrounding these deep active faults. The along-strike length and depth of penetration of these faults are also similar to the dimensions of the rupture area of intermediate-depth earthquakes.     

Additive two-mode clustering: The error-variance approach revisited

The additive clustering approach is applied to the problem of two-mode clustering and compared with the recent error-variance approach of Eckes and Orlik (1993). Although the schemes of the computational algorithms look very similar in both of the approaches, the additive clustering has been shown to have several advantages. Specifically, two technical limitations of the error-variance approach (see Eckes and Orlik 1993, p. 71) have been overcome in the framework of the additive clustering. The research was supported by the Office of Naval Research under grant number N0014-93-1-0222 to Rutgers University. The authors are indebted both to Fionn Murtagh, who served as Acting Editor, and to anonymous Referees for thoughtful and constructive reviews.