Asymptotic theory for density ridges

By Yen-Chi Chen, Christopher R. Genovese, Larry A. Wasserman. 2014

88 citations recorded by the source corpus.

The large sample theory of estimators for density modes is well understood. In this paper we consider density ridges, which are a higher-dimensional extension of modes. Modes correspond to zero-dimensional, local high-density regions in point clouds. Density ridges correspond to $s$-dimensional, local high-density regions in point clouds. We establish three main results. First we show that under appropriate regularity conditions, the local variation of the estimated ridge can be approximated by an empirical process. Second, we show that the distribution of the estimated ridge converges to a Gaussian process. Third, we establish that the bootstrap leads to valid confidence sets for density ridges.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f95ceb0f9e8194a932357d7ed3f54b3f3a42ec92

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In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors (2017, methodology): “Examples include sure screening (Fan and Lv, 2008), removing spurious correlation (Fan and Zhou, 2016), testing the equality of two matrices (Cai et al., 2013; Chang et al., 2017), detecting ridges and estimating level sets (Chen et al., 2015, 2016), and many more.”
A tutorial on kernel density estimation and recent advances (2017, methodology): “[22] and [84] both studied the asymptotic theory, and [22] further proposed methods of constructing confidence sets of R.”
Nonparametric modal regression (2014, methodology): “Mode-based clustering (Cheng, 1995; Comaniciu and Meer, 2002; Chen et al., 2014a) is a nonparametric clustering method which uses local density modes to define clusters.”
Topological Data Analysis (2016, methodology): “For example, Chen et al. (2015d) investigated how the properties of galaxies diﬀer depending on the distance from ﬁlaments.”