Ball divergence is a non-parametric two-sample statistical test method in metric spaces. It measures the difference between two population probability distributions by integrating the difference over all balls in the space.[1] Therefore, its value is zero if and only if the two probability measures are the same. Similar to common non-parametric test methods, ball divergence calculates the p-value through permutation tests.

Background

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Distinguishing between two unknown samples in multivariate data is an important and challenging task. Previously, a more common non-parametric two-sample test method was the energy distance test.[2] However, the effectiveness of the energy distance test relies on the assumption of moment conditions, making it less effective for extremely imbalanced data (where one sample size is disproportionately larger than the other). To address this issue, Chen, Dou, and Qiao proposed a non-parametric multivariate test method using ensemble subsampling nearest neighbors (ESS-NN) for imbalanced data.[3] This method effectively handles imbalanced data and increases the test's power by fixing the size of the smaller group while increasing the size of the larger group.

Additionally, Gretton et al. introduced the maximum mean discrepancy (MMD) for the two-sample problem.[4] Both methods require additional parameter settings, such as the number of groups 𝑘 in ESS-NN and the kernel function in MMD. Ball divergence addresses the two-sample test problem for extremely imbalanced samples without introducing other parameters.

Definition

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Let's start with the population ball divergence. Suppose that we have a metric space ( ), where norm   introduces a metric   for two point   in space   by  . Besides, we use   to show a closed ball with the center   and radius  . Then, the population ball divergence of Borel probability measures   is

 

For convenience, we can decompose the Ball Divergence into two parts:   and   Thus  

Next, we will introduce the sample ball divergence. Let   denote whether point   locates in the ball  . Given two independent samples   form   and   form  

  where   means the proportion of samples from the probability measure   located in the ball   and   means the proportion of samples from the probability measure   located in the ball  . Meanwhile,   and   means the proportion of samples from the probability measure   and   located in the ball  . The sample versions of   and   are as follows

  Finally, we can give the sample ball divergence

 

Properties

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1. Given two Borel probability measures   and   on a finite dimensional Banach space  , then   where the equality holds if and only if  .

2. Suppose   and   are two Borel probability measures in a separable Banach space  . Denote their support   and  , if   or  , then we have   where the equality holds if and only if  .

3.Consistency: We have

  where   for some  .

Define  , and then let   where

  The function   has spectral decomposition:   where   and   are the eigenvalues and eigenfunctions of  . For  ,   are i.i.d.  , and  

4.Asymptotic distribution under the null hypothesis: Suppose that both   and   in such a way that  . Under the null hypothesis, we have  

5. Distribution under the alternative hypothesis: let   Suppose that both   and   in such a way that  . Under the alternative hypothesis, we have  

6. The test based on   is consistent against any general alternative  . More specifically,   and   More importantly,   can also be expressed as   which is independent of  .

References

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  1. ^ Pan, Wenliang; Tian, Yuan; Wang, Xueqin; Zhang, Heping (2018-06-01). "Ball Divergence: Nonparametric two sample test". The Annals of Statistics. 46 (3): 1109–1137. doi:10.1214/17-AOS1579. ISSN 0090-5364. PMC 6192286. PMID 30344356.
  2. ^ Székely, Gábor J.; Rizzo, Maria L. (August 2013). "Energy statistics: A class of statistics based on distances". Journal of Statistical Planning and Inference. 143 (8): 1249–1272. doi:10.1016/j.jspi.2013.03.018. ISSN 0378-3758.
  3. ^ Chen, Lisha; Dou, Winston Wei; Qiao, Zhihua (December 2013). "Ensemble Subsampling for Imbalanced Multivariate Two-Sample Tests". Journal of the American Statistical Association. 108 (504): 1308–1323. doi:10.1080/01621459.2013.800763. ISSN 0162-1459.
  4. ^ Gretton, Arthur; Borgwardt, Karsten M.; Rasch, Malte; Schölkopf, Bernhard; Smola, Alexander J. (2007-09-07), "A Kernel Method for the Two-Sample-Problem", Advances in Neural Information Processing Systems 19, The MIT Press, pp. 513–520, doi:10.7551/mitpress/7503.003.0069, hdl:1885/37327, ISBN 978-0-262-25691-9, retrieved 2024-06-28