M-可分解性、椭圆单峰密度及其在聚类和核密度估计中的应用
M-decomposability, elliptical unimodal densities, and applications to clustering and kernel density estimation
摘要：
Chia和Nakano（2009）引入了一维概率密度的M-可分解性的概念。本文将M-可分解性推广到任意维数。我们证明了所有的椭圆单峰密度都是M-不可分解的。我们还推导出一个不等式，表明它是更好地表示M-可分解的密度通过单峰密度的混合。最后，我们展示了应用M-分解聚类和核密度估计，使用真实和模拟数据。我们的结果表明，M-可分解性可以作为一个非参数的标准来定位模式的概率密度。
英文摘要：
Chia and Nakano (2009) introduced the concept of M-decomposability of probability densities in one-dimension. In this paper, we generalize M-decomposability to any dimension. We prove that all elliptical unimodal densities are M-undecomposable. We also derive an inequality to show that it is better to represent an M-decomposable density via a mixture of unimodal densities. Finally, we demonstrate the application of M-decomposability to clustering and kernel density estimation, using real and simulated data. Our results show that M-decomposability can be used as a non-parametric criterion to locate modes in probability densities.
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