Visual Comparison of Datasets using Mixture Decompositions

Abstract

We describe how a mixture of two densities f 0 and f 1 may be decomposed into a different mixture consisting of three densities. These new densities, f+ , f \Gamma , and f= , summarize differences between f 0 and f 1 : f+ is high in areas of excess of f 1 compared to f 0 ; f \Gamma represents deficiency of f 1 compared to f 0 in the same way; f= represents commonality between f 1 and f 0 . The supports of f+ and f \Gamma are disjoint. This decomposition of the mixture of f 0 and f 1 is similar to the set-theoretic decomposition of the union of two sets A and B into the disjoint sets AnB, BnA, and A " B. Sample points from f 0 and f 1 can be assigned to one of these three densities, allowing the differences between f 0 and f 1 to be visualized in a single plot, a visual hypothesis test of whether f 0 is equal to f 1 . We describe two similar such decompositions and contrast their behavior under the null hypothesis f 0 = f 1 , giving some insight into how such plots may be interpreted. ..