A Population Background for Nonparametric Density-Based Clustering

Chacón, José E.

doi:10.1214/15-sts526

Cited by 62 publications

(62 citation statements)

References 53 publications

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“…These issues have pushed statisticians to interact with other seemingly unrelated branches of Mathematics, like Computational Geometry, Morse Theory or Algebraic Topology, which have revealed themselves as very useful to develop some modern statistical techniques. For instance, Morse Theory (a branch of Differential Topology) was essential to provide a very precise definition for the ideal population goal of modal clustering in Chacón (); the persistence diagram was borrowed from Computational Geometry to distinguish true from spurious modes in Fasy et al (), and the concept of persistent homology from Algebraic Topology is fundamental for modern Topological Data Analysis (Wasserman, ).…”

Section: Discussionmentioning

confidence: 99%

“…Alternatively, Chacón () showed that this approach can also be formulated in terms of the domains of attraction of the density modes (hence the name of modal clustering): given

x \in {double-struckR}^{d}

, consider the path starting at

x

that follows the direction of steepest ascent (which is determined by the density gradient

\nabla f

); such a curve

γ_{x} : double-struckR \to {double-struckR}^{d}

is the solution to the initial value problem

γ_{x}^{'} false(t false) = \nabla f ((), γ_{x} false(t false))

with

γ_{x} false(0 false) = x

. Under certain regularity conditions, that path always finishes at a critical value of

f

, so that

\nabla f ((), \lim_{t \to \infty} γ_{x} false(t false)) = 0

and, therefore, the set of points

x

whose final destination is a given critical point is called the domain of attraction of that critical point.…”

Section: Modal Clusteringmentioning

confidence: 99%

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The Modal Age of Statistics

Chacón

2020

Int Statistical Rev

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Summary Recently, a number of statistical problems have found an unexpected solution by inspecting them through a ‘modal point of view'. These include classical tasks such as clustering or regression. This has led to a renewed interest in estimation and inference for the mode. This paper offers an extensive survey of the traditional approaches to mode estimation and explores the consequences of applying this modern modal methodology to other, seemingly unrelated, fields.

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Section: Discussionmentioning

confidence: 99%

“…Alternatively, Chacón () showed that this approach can also be formulated in terms of the domains of attraction of the density modes (hence the name of modal clustering): given

x \in {double-struckR}^{d}

, consider the path starting at

x

that follows the direction of steepest ascent (which is determined by the density gradient

\nabla f

); such a curve

γ_{x} : double-struckR \to {double-struckR}^{d}

is the solution to the initial value problem

γ_{x}^{'} false(t false) = \nabla f ((), γ_{x} false(t false))

with

γ_{x} false(0 false) = x

. Under certain regularity conditions, that path always finishes at a critical value of

f

, so that

\nabla f ((), \lim_{t \to \infty} γ_{x} false(t false)) = 0

and, therefore, the set of points

x

whose final destination is a given critical point is called the domain of attraction of that critical point.…”

Section: Modal Clusteringmentioning

confidence: 99%

The Modal Age of Statistics

Chacón

2020

Int Statistical Rev

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“…Proof of Theorem 1. From Theorem 4.1 in [3] it follows that, with probability one, there exists Proof of Lemma 1. From ψ(µ, σ 2 ) = σψ(µ/σ, 1), it suffices to show that ψ(u, 1) ≤ (2/π) 1/2 + (2π) −1/2 u 2 for u ≥ 0.…”

Section: A Proofsmentioning

confidence: 98%

Modal clustering asymptotics with applications to bandwidth selection

Casa¹,

Chacón²,

Menardi³

2020

Electron. J. Statist.

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Density-based clustering relies on the idea of linking groups to some specific features of the probability distribution underlying the data. The reference to a true, yet unknown, population structure allows to frame the clustering problem in a standard inferential setting, where the concept of ideal population clustering is defined as the partition induced by the true density function. The nonparametric formulation of this approach, known as modal clustering, draws a correspondence between the groups and the domains of attraction of the density modes. Operationally, a nonparametric density estimate is required and a proper selection of the amount of smoothing, governing the shape of the density and hence possibly the modal structure, is crucial to identify the final partition. In this work, we address the issue of density estimation for modal clustering from an asymptotic perspective. A natural and easy to interpret metric to measure the distance between density-based partitions is discussed, its asymptotic approximation explored, and employed to study the problem of bandwidth selection for nonparametric modal clustering.

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“…Another density clustering method is mode clustering (Chacón et al, 2015(Chacón et al, , 2013Chacón, 2012;Li et al, 2007;Comaniciu & Meer, 2002;Arias-Castro et al, 2015;Cheng, 1995). The idea is to find modes of the density and then define clusters as the basins of attraction of the modes.…”

Section: Mode Clustering and Morse Theorymentioning

confidence: 99%