基于聚类应用的局部深度函数的分析和统计性质
Analytical and statistical properties of local depth functions motivated by clustering applications
摘要：
局部广义深度（$LGD$）函数用于描述多元分布中的局部几何特征和模式。在本文中，我们对$LGD$进行了严格的系统研究，并建立了几个分析和统计特性。首先，我们表明，当潜在的概率分布是绝对连续的密度$f(\cdot)$，缩放版本的$LGD$（称为$τ$近似）收敛，一致和$L^d(\mathbb{R}^p)$的$f(\cdot)$时，$τ$收敛到零。其次，我们建立，作为样本大小发散到无穷大的中心和缩放的样本$LGD$收敛于分布的中心高斯过程均匀的有界函数的空间中的$\mathcal{H}_G$，一类功能产生$LGD$。第三，利用$τ$ -近似（$S τA$）和梯度系统分析，我们开发了一种新的聚类算法。该算法的有效性需要与$S τA$有关的梯度系统的一致有限差分近似的几个结果。为此，我们建立了\emph{Bernstein}型不等式的中心和缩放样本$LGD$之间的偏差，这也是独立的利益。最后，调用上述结果，我们建立了一致性的聚类算法。最后给出了该方法在模式估计和上水平集估计中的应用。有限样本性能的方法进行了评估，使用数值实验和数据分析。
英文摘要：
Local general depth ($LGD$) functions are used for describing the local geometric features and mode(s) in multivariate distributions. In this paper, we undertake a rigorous systematic study of $LGD$ and establish several analytical and statistical properties. First, we show that, when the underlying probability distribution is absolutely continuous with density $f(\cdot)$, the scaled version of $LGD$ (referred to as $τ$-approximation) converges, uniformly and in $L^d(\mathbb{R}^p)$ to $f(\cdot)$ when $τ$ converges to zero. Second, we establish that, as the sample size diverges to infinity the centered and scaled sample $LGD$ converge in distribution to a centered Gaussian process uniformly in the space of bounded functions on $\mathcal{H}_G$, a class of functions yielding $LGD$. Third, using the sample version of the $τ$-approximation ($S τA$) and the gradient system analysis, we develop a new clustering algorithm. The validity of this algorithm requires several results concerning the uniform finite difference approximation of the gradient system associated with $S τA$. For this reason, we establish \emph{Bernstein}-type inequality for deviations between the centered and scaled sample $LGD$, which is also of independent interest. Finally, invoking the above results, we establish consistency of the clustering algorithm. Applications of the proposed methods to mode estimation and upper level set estimation are also provided. Finite sample performance of the methodology are evaluated using numerical experiments and data analysis.
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