Stability of Density-Based Clustering

Alessandro Rinaldo; Aarti Singh; Rebecca Nugent; Larry Wasserman

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arXiv · 1011.2771

Stability of Density-Based Clustering

Alessandro Rinaldo·Aarti Singh·Rebecca Nugent·Larry Wasserman

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Abstract

High density clusters can be characterized by the connected components of a level set $L(\lambda) = \{x:\ p(x)>\lambda\}$ of the underlying probability density function $p$ generating the data, at some appropriate level $\lambda\geq 0$. The complete hierarchical clustering can be characterized by a cluster tree ${\cal T}= \bigcup_{\lambda} L(\lambda)$. In this paper, we study the behavior of a density level set estimate $\widehat L(\lambda)$ and cluster tree estimate $\widehat{\cal{T}}$ based on a kernel density estimator with kernel bandwidth $h$. We define two notions of instability to measure the variability of $\widehat L(\lambda)$ and $\widehat{\cal{T}}$ as a function of $h$, and investigate the theoretical properties of these instability measures.

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Alessandro Rinaldo, Aarti Singh, Rebecca Nugent, Larry Wasserman. 2010-11-11. Stability of Density-Based Clustering. https://arxiv.org/abs/1011.2771

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