Geometric goodness of fit measure to detect patterns in data point clouds

Abstract

In this work, we derive a geometric goodness-of-fit index similar to R2 using geomet- ric data analysis techniques. We build the alpha shape complex from the data-cloud projected onto each variable and estimate the area of the complex and its domain. We create an index that measures the difference of area between the alpha shape and the smallest squared window of observation containing the data. By applying ideas similar to those found in the closest neighbor distribution and empty space distribu- tion functions, we can establish when the characterizing geometric features of the point set emerge. This allows for a more precise application for our index. We pro- vide some examples with anomalous patterns to show how our algorithm performs.