COPS: Cluster optimized proximity scaling

Proximity scaling (i.e., multidimensional scaling and related methods) is a versatile statistical method whose general idea is to reduce the multivariate complexity in a data set by employing suitable proximities between the data points and finding low-dimensional configurations where the fitted distances optimally approximate these proximities. The ultimate goal, however, is often not only to find the optimal configuration but to infer statements about the similarity of objects in the high-dimensional space based on the the similarity in the configuration. Since these two goals are somewhat at odds it can happen that the resulting optimal configuration makes inferring similarities rather difficult. In that case the solution lacks "clusteredness" in the configuration (which we call "c-clusteredness"). We present a version of proximity scaling, coined cluster optimized proximity scaling (COPS), which solves the conundrum by introducing a more clustered appearance into the configuration while adhering to the general idea of multidimensional scaling. In COPS, an arbitrary MDS loss function is parametrized by monotonic transformations and combined with an index that quantifies the c-clusteredness of the solution. This index, the OPTICS cordillera, has intuitively appealing properties with respect to measuring c-clusteredness. This combination of MDS loss and index is called "cluster optimized loss" (coploss) and is minimized to push any configuration towards a more clustered appearance. The effect of the method will be illustrated with various examples: Assessing similarities of countries based on the history of banking crises in the last 200 years, scaling Californian counties with respect to the projected effects of climate change and their social vulnerability, and preprocessing a data set of hand written digits for subsequent classification by nonlinear dimension reduction. (authors' abstract)