Abstract
ABSTRACTThis article provides a framework for assessing and quantifying “clusteredness” of a data representation. Clusteredness is a global univariate property defined as a layout diverging from equidistance of points to the closest neighboring point set. The OPTICS algorithm encodes the global clusteredness as a pair of clusteredness-representative distances and an algorithmic ordering. We use this to construct an index for quantification of clusteredness, coined the OPTICS Cordillera, as the norm of subsequent differences over the pair. We provide lower and upper bounds and a normalization for the index. We show the index captures important aspects of clusteredness such as cluster compactness, cluster separation, and number of clusters simultaneously. The index can be used as a goodness-of-clusteredness statistic, as a function over a grid or to compare different representations. For illustration, we apply our suggestion to dimensionality reduced 2D representations of Californian counties with respect to 48 climate change related variables. Online supplementary material is available (including an R package, the data and additional mathematical details).
Highlights
Introduction and MotivationRepresentation of a data matrix in Rm is an integral part of exploratory data analysis
We suggest a specific instance of the Cordillera utilizing the OPTICS (Ordering Points To Identify The Clustering Structure; Ankerst et al 1999) algorithm for obtaining the algorithmic ordering and reachabilities, which fits neatly into the distance-density based framework and has the properties of making only weak assumptions about the object arrangement in the representation
We find the OPTICS Cordillera and the plot of the clusteredness-representative ordering-reachability pair for the representations in the left column
Summary
Representation of a data matrix in Rm is an integral part of exploratory data analysis. While the approach of visually interpreting clusteredness data representations is common, it appears to be highly subjective. We suggest a specific instance of the Cordillera utilizing the OPTICS (Ordering Points To Identify The Clustering Structure; Ankerst et al 1999) algorithm for obtaining the algorithmic ordering and reachabilities, which fits neatly into the distance-density based framework and has the properties of making only weak assumptions about the object arrangement in the representation. We call this instance the OPTICS Cordillera, and we give results on its behavior.
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