A Clustering Technique for Maximizing φ-Divergence, Noncentrality and Discriminating Power

Abstract

The χ ² goodness-of-fit test for the hypothesis H ₀: P = P ₀ (a distribution with density f ₀(x)) is using an m-partition C = {C _l,..., C _m } of the sample space and checks for the hypothetical class probabilities P ₀(C ₁),..., P ₀(C _m ). The asymptotic power performance of this test is characterized by the noncentrality parameter δ _C ²(P ₁, P ₀) = ∑ _{i =1} ^m(P ₁(C _i )−P ₀(C _i ))²/P ₀(C _i ) where P ₁ is a given alternative distribution. In this paper, we show how an optimally efficient partition C, i.e. with a maximum value δ _C ²(P ₁, P ₀) can be obtained (for a given number m of classes). — In fact, this problem can be embedded into the general framework of maximizing a ø-divergence measure I _C (P ₁, P ₀; ø) over all m-partitions C of R ^P (where ø(∙) is a convex function on R ¹). Our algorithm is an adaptation of the well-known k-means clustering technique and uses the support lines of ø. Since ø-divergence measures characterize, quite generally, the performance of tests for distinguishing between two alternatives P ₀ and P ₁ (e.g. in the NeymanPearson or a Bayesian framework) the given methods can be used for obtaining partitions with a maximum discriminating power for the resulting discretized distributions P ₀(C _i ), P1(C _i ), i = 1,..., m. A series of numerical examples is presented.