Open Access
August 2019 Generalized cluster trees and singular measures
Yen-Chi Chen
Ann. Statist. 47(4): 2174-2203 (August 2019). DOI: 10.1214/18-AOS1744


In this paper we study the $\alpha $-cluster tree ($\alpha $-tree) under both singular and nonsingular measures. The $\alpha $-tree uses probability contents within a set created by the ordering of points to construct a cluster tree so that it is well defined even for singular measures. We first derive the convergence rate for a density level set around critical points, which leads to the convergence rate for estimating an $\alpha $-tree under nonsingular measures. For singular measures, we study how the kernel density estimator (KDE) behaves and prove that the KDE is not uniformly consistent but pointwise consistent after rescaling. We further prove that the estimated $\alpha $-tree fails to converge in the $L_{\infty }$ metric but is still consistent under the integrated distance. We also observe a new type of critical points—the dimensional critical points (DCPs)—of a singular measure. DCPs are points that contribute to cluster tree topology but cannot be defined using density gradient. Building on the analysis of the KDE and DCPs, we prove the topological consistency of an estimated $\alpha $-tree.


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Yen-Chi Chen. "Generalized cluster trees and singular measures." Ann. Statist. 47 (4) 2174 - 2203, August 2019.


Received: 1 November 2016; Revised: 1 February 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082283
MathSciNet: MR3953448
Digital Object Identifier: 10.1214/18-AOS1744

Primary: 62G20
Secondary: 62G05 , 62G07

Keywords: cluster tree , critical points , kernel density estimator , Level set , singular measure , topological data analysis

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 4 • August 2019
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