Analytical and statistical properties of local depth functions motivated by clustering applications
Giacomo Francisci, Claudio Agostinelli, Alicia Nieto-Reyes, Anand N. Vidyashankar
AAnalytical and statistical properties of local depthfunctions motivated by clustering applications
Giacomo Francisci , Claudio Agostinelli , Alicia Nieto-Reyes , andAnand N. Vidyashankar Dipartimento di Matematica, Universit`a di Trento, Trento, Italy Departamento de Matem´aticas, Estad´ıstica y Computaci´on, Universidad de Cantabria,Santander, Spain Department of Statistics, George Mason University, Fairfax, Virginia
August 28, 2020
Abstract
Local depth functions (LDFs) are used for describing the local geometric fea-tures of multivariate distributions, especially in multimodal models. In this paper,we undertake a rigorous systematic study of the LDFs and use it to develop a the-oretically validated algorithm for clustering. For this reason, we establish severalanalytical and statistical properties of LDFs. First, we show that, when the under-lying probability distribution is absolutely continuous, under an appropriate scalingthat converge to zero (referred to as extreme localization ), LDFs converge uniformlyto a power of the density and obtain a related rate of convergence result. Second,we establish that the centered and scaled sample LDFs converge in distributionto a centered Gaussian process, uniformly in the space of bounded functions on R p × [0 , ∞ ], as the sample size diverges to infinity. Third, under an extreme lo-calization that depends on the sample size, we determine the correct centering andscaling for the sample LDFs to possess a limiting normal distribution. Fourth, in-voking the above results, we develop a new clustering algorithm that uses the LDFsand their differentiability properties. Fifth, for the last purpose, we establish severalresults concerning the gradient systems related to LDFs. Finally, we illustrate thefinite sample performance of our results using simulations and apply them to twodatasets. a r X i v : . [ m a t h . S T ] A ug ey Words: Local depth, extreme localization, Hoeffding’s decomposition, sample localdepth, uniform central limit theorem, clustering, modes, gradient system, Lyapunov’sstability Theorem
Investigation of data depths is gaining momentum due to its applicability in a varietyof machine learning problems such as non-parametric classification and clustering. Thisconcept, formalized in Liu [1990] and Zuo and Serfling [2000a], dates back to Tukey[1975] and serves to identify a center for multivariate distributions and a multidimensionalcenter-outward order similar to that of a real line. This ordering enables a descriptionof quantiles of multivariate distributions (see Zuo and Serfling [2000b]) and aids in usingdepth functions (DFs) for clustering as described, for instance, in Torrente and Romo[2020], Joernsten [2004] (see also the references therein). The focus of these methods is toimprove k-means based clustering via the use of DFs by identifying the “deepest points”,the depth median of a dataset. The current paper develops the intuitive notion that localdepths possess properties that help in identifying peaks and valleys, and hence cluster-ing based on such identification can improve the quality and stability of the clusteringalgorithm.The notion of local depth, as first described in Agostinelli and Romanazzi [2011] andinvestigated in one-dimension, provides a framework to describe the local multidimen-sional features of multivariate distributions. Section 2 of this paper extends this analysisto multidimensions and provides a detailed study of the analytical properties of localdepth functions (LDFs) and their scaled versions (referred to as τ -approximation). Theseresults form the basic ingredients for a gradient system analysis developed in the latersections. Recent additional work on local depth include that of Dutta et al. [2016] andChandler and Polonik [2020], where the focus is on different types of data (spatial andfunctional, for instance) and construction of supervised classification methods. However,none of these methods apply to the more challenging problem studied here, viz., unsuper-vised classification or clustering. We now turn to a precise description of our contributionsstarting with depth and local depth.In the following, we denote by B ( X ) the Borel subsets of a topological space X and by ( R p ) k the k -fold cartesian product of the p -dimensional Euclidean space R p . Let P be a probability distribution on R p . Then for a set Z G ( x ) ∈ B (( R p ) k ), where x ∈ R p ,2he bounded non-negative function GD ( x, P ) defined by GD ( x, P ) = (cid:90) ( R p ) k I (( x , . . . , x k ) ∈ Z G ( x )) dP ( x ) . . . dP ( x k ) (1.1)is referred to as the depth function at x through Z G ( x ) for P . Depending on the choice of P and Z G ( · ), they can also satisfy the following properties: affine invariance, maximalityat the center, monotonically non-increasing along the rays from the center, and vanishingat infinity. These are sometimes stated as required properties of a “good” depth function(see Liu [1990] and Zuo and Serfling [2000a]). As an example, if P is absolutely continuouswith respect to the Lebesgue measure and is angularly symmetric, the so-called simplicialdepth (see below) satisfies all the properties of a good depth function.As expected, different choices of Z G ( · ) yield different DFs. Specifically, choosing k = 2 and Z G ( · ) to be Z G ( x ) = { ( x , x ) ∈ R p × R p : max i =1 , (cid:107) x − x i (cid:107) ≤ (cid:107) x − x (cid:107)} =: Z L ( x )we obtain the lens depth, LD ( x, P ) [Liu and Modarres, 2011]. Also, for the choice Z G ( · )(with k = 2) given by Z G ( x ) = { ( x , x ) ∈ R p × R p : (cid:107) x − ( x + x ) (cid:107) ≤ (cid:107) x − x (cid:107)} =: Z B ( x )we get the spherical depth, BD ( x, P ) [Elmore et al., 2006]. Finally, choosing k = ( p + 1)and Z G ( x ) = { ( x , . . . , x p +1 ) ∈ ( R p ) ( p +1) : x ∈ (cid:52) [ x , . . . , x p +1 ] } , where (cid:52) [ x , . . . , x p +1 ] is the closed simplex with vertices x , . . . , x p +1 ∈ R p , we get thewell-known simplicial depth, SD ( x, P ) given in Liu [1990]. We denote the correspondingset by Z S ( · ).The local versions of the depths described above, involve an additional nonnega-tive parameter τ , referred to as the localizing parameter , yielding the LDFs, namely, LGD ( · , P, τ ), where G = L, B, S . The corresponding sets now take the form Z Gτ ( x ). Asin (1.1), the LDFs can be expressed as: LGD ( x, P, τ ) = (cid:90) ( R p ) k I (( x , . . . , x k ) ∈ Z Gτ ( x )) dP ( x ) . . . dP ( x k ) . (1.2)For the sake of clarity, we provide the expressions for “the local set” for different choicesof G = L, B, S . Specifically, the local set associated with lens depth is Z Lτ ( x ) = { ( x , x ) ∈ R p × R p : max i =1 , (cid:107) x − x i (cid:107) ≤ (cid:107) x − x (cid:107) ≤ τ } , Z Bτ ( x ) = { ( x , x ) ∈ R p × R p : (cid:107) x − ( x + x ) (cid:107) ≤ (cid:107) x − x (cid:107) ≤ τ } . The local set for the simplicial depth is given by Z Sτ ( x ) = { ( x , . . . , x p +1 ) ∈ ( R p ) ( p +1) : x ∈ (cid:52) [ x , . . . , x p +1 ] , max i,j =1 ,...,p +1 i>j (cid:107) x i − x j (cid:107) ≤ τ } . We observe here that while the definitions of LSD and LLD first appear in Agostinelliand Romanazzi [2008] and Kleindessner and Von Luxburg [2017] respectively, the LBD,as defined here, seems new. Of course, when p = 1 all of the above three local depthscoincide. Finally, when τ = ∞ , the LDFs reduce to the DFs.A useful and important aspect of local depth is its behavior under extreme local-ization , i.e. when τ → + . When P is absolutely continuous, LDFs investigated in thispaper, under appropriate scaling, converge to a power of the probability density of P .Under additional conditions, one obtains convergence to the derivatives of the densitywhich facilitates an enquiry into the modes of the density via a gradient system analysis.This, in turn, allows one to characterize the related stable manifolds paving the way forcluster analysis. Related ideas about clustering appear in Chazal et al. [2013], Chen et al.[2016], and Genovese et al. [2016]. Our methodology differs from the existing literaturein that we take advantage of the local depth notion, specifically the τ -approximation andits properties, developed in Sections 2 and 3 below.Statistical enquiry about local depth requires an investigation into their sample ver-sions, viz., estimators of local depth, which are described in Subsection 2.2. We establishthat, for all choices of G = L, B, S , the estimators are uniformly consistent across thepair ( x, τ ). The key issue is the additional uniformity in τ which requires investigationof the behavior of the classes of sets { Z Gτ ( x ) : ( x, τ ) ∈ R p × [0 , ∞ ] } under the empiricalmeasure. Specifically, we study the behavior of the estimator of LDFs in the space (cid:96) ∞ (Γ),of bounded functions on subsets Γ of R p × [0 , ∞ ], which is not separable. Borrowingtools from empirical process theory, we establish uniform consistency for all choices of G = L, B, S . Turning to the limit distributions of the estimator, we establish a uni-form central limit theorem, in (cid:96) ∞ (Γ). This involves establishing convergence of finitedimensional distributions, admissible Suslin property, and use of Alexander’s Lemma (seeAlexander [1987]) for establishing the finiteness of the bracketing entropy for the abovedescribed VC classes of sets. Both the uniform consistency and the uniform central limitresults rely on the Hoeffding’s decomposition of U-statistics representation of the localdepth which incidentally is a critical component of our analysis.4 natural next question concerns the distributional behavior of sample local depthwhen the localizing parameter τ converges to zero as a function of the sample size. Inthis case, contrary to expectations, the correct centering is not the limiting density. InSubsection 2.2, we identify the correct centering and scaling for the sample local depthto converge, in distribution, to a centered normal distribution.As the title of the manuscript indicates, the primary motivation of the paper is clus-tering in multidimensions. This involves two distinct but connected steps: (i) identifyingthe modes and (ii) identifying the stable manifold generated by them. For (i), we es-tablish sufficient conditions in terms of the eigenvalues of the Hessian matrix. For (ii),which involves defining the clusters, one needs to identify the points whose trajectories(which result from the gradient system) converge to the mode, also referred to as the ω -limit of the path. Thus from a practical perspective, an algorithm can be developedbased on the above two considerations. While this is feasible, what is unclear is that theresulting sets indeed form non-trivial clusters; that is, they have positive Lebesgue mea-sure. We establish that this is the case using tools from dynamical systems. Specifically,since ω -limits are stationary points of the density, to validate the algorithm, it is requiredto show that the trajectories that converge to non-modal points have Lebesgue measurezero. In contrast, those that converge to modal points have positive Lebesgue measure.Lyapunovs stability Theorem plays a significant first step for this analysis. To the bestof our knowledge, the current manuscript seems to be the first to provide such strongtheoretical guarantees for clustering in multidimensional problems.We now provide a brief description of the organization of the rest of the paper. InSubsection 2.1, we establish the analytical properties of LLD while the statistical prop-erties are established in 2.2. In Subsection 2.3, we discuss the modifications requiredfor other local depths, including local half-space depth and local half-region depth. InSubsections 3.1, 3.2 and 3.3, we focus on the mathematical underpinnings behind thealgorithm for clustering which itself is described in Subsection 3.4. In Section 4, we pro-vide several numerical experiments illustrating the finite sample behavior of the proposedmethods and apply them to data examples while, in Section 5, we provide a few conclud-ing remarks. The proofs of all the results are relegated to the appendix. We end thisintroduction with a comment about the notations. When there is no scope for confusion,we suppress P in GD ( x, P ) and LGD ( x, P, τ ) and use the notation GD ( x ) and LGD ( x, τ )for all G = L, S, B . 5
Local depth and extreme localization
We recall that the local lens depth is given by
LLD ( x, τ ) = (cid:90) R p × R p I (( x , x ) ∈ Z τ ( x )) dP ( x ) dP ( x ) , (2.1)where Z τ ( x ) = { ( x , x ) ∈ R p × R p : max i =1 , (cid:107) x − x i (cid:107) ≤ (cid:107) x − x (cid:107) ≤ τ } , and the indexL is suppressed for simplicity. We begin this section by describing properties of the local depth. These results will berequired when investigating the properties of the estimator of the local depth.
Proposition 2.1 (i) For any fixed x ∈ R p , LLD ( x, · ) is a non-decreasing and right-continuous function of τ satisfying lim τ → + LLD ( x, τ ) = P ( { x } ) and lim τ →∞ LLD ( x, τ ) = LD ( x ) . (ii) For τ ∈ [0 , ∞ ] , sup x ∈ R p : (cid:107) x (cid:107)≥ M LLD ( x, τ ) → as M → ∞ .(iii) For τ ∈ [0 , ∞ ] , LLD ( · , τ ) is upper semicontinuous as a function of x . If P is abso-lutely continuous with respect to the Lebesgue measure, then it is jointly continuous in x and τ .(iv) If P is absolutely continuous with respect to the Lebesgue measure, with k -times con-tinuously differentiable density f ( · ) , then LLD ( · , τ ) is k -times continuously differentiablefor each fixed τ ∈ [0 , ∞ ) . (i) of the above proposition was studied for the case p = 1 in Agostinelli and Romanazzi[2011][Proposition 6] and (ii) was investigated for the extreme case, namely τ = ∞ , inLiu and Modarres [2011][Theorem 2]. If P is absolutely continuous with respect to theLebesgue measure on R p with density f ( · ), then Proposition 2.1 (i) shows that LLDconverges to 0 as τ → + . Notice that for x ∈ R p , Z τ ( x ) = ( x, x ) + Z τ (0) and Z τ (0) = τ Z (0). Also, by a change of variable in (2.1), it follows that LLD ( x, τ ) = (cid:90) Z τ ( x ) f ( x ) f ( x ) dx dx = (cid:90) Z τ (0) f ( x + x ) f ( x + x ) dx dx = (cid:90) Z (0) τ p f ( x + τ x ) f ( x + τ x ) dx dx . (2.2)6or τ >
0, let Λ τ := λ ⊗ ( Z τ (0)) denote the Lebesgue measure in R p × R p of Z τ (0).By the translation and dilatation invariance of the Lebesgue measure, it follows that λ ⊗ ( Z ( x )) = Λ and Λ τ = τ p Λ . Our next result is concerned with extreme localization.Specifically, we establish the rate of convergence of the local depth to the square of thedensity as τ approaches 0. In the following, we use a.e. to mean almost everywhere withrespect to the Lebesgue measure on R p and (cid:104)· , ·(cid:105) is the inner product on R p . Also, ∇ f and H f represent the gradient and the Hessian matrix of f ( · ), respectively. Theorem 2.1
Let P be absolutely continuous with respect to the Lebesgue measure on R p , with density f ( · ) . Then the following hold:(i) Let x ∈ R p be fixed, then lim τ → + τ p Λ LLD ( x, τ ) = f ( x ) , a.e. (2.3) Furthermore, if f ( · ) is continuous, then (2.3) holds for all x ∈ R p .(ii) Suppose that f ( · ) is three times continuously differentiable. Then lim τ → + τ (cid:18) τ p LLD ( x, τ ) − Λ f ( x ) (cid:19) = h ( x ) , where h ( x ) := f ( x ) (cid:90) Z (0) x (cid:62) H f ( x ) x dx dx + (cid:90) Z (0) (cid:104)∇ f ( x ) , x (cid:105)(cid:104)∇ f ( x ) , x (cid:105) dx dx . It is worth noticing here that h ( · ) vanishes outside the support of f ( · ), S := { x ∈ R p : f ( x ) > } , while it is, in general, non-zero in the interior of the support of f ( · ). When p = 1 certain simplifications occur. Specifically, Λ = 1. This is summarized in thefollowing corollary. Corollary 2.1
Under the conditions of Theorem 2.1 and for p = 1 , LLD ( x, τ ) = 2 (cid:90) T τ ++ f ( x + x ) f ( x − x ) dx dx , (2.4) where T τ ++ := { ( x , x ) : x ≥ , x ≥ , x + x ≤ τ } . Furthermore, for any x ∈ R , lim τ → + τ LLD ( x, τ ) = f ( x ) a.e. (2.5) If f ( · ) is continuous then (2.5) holds everywhere. One of the motivations of this paper is to perform clustering using LLD. With this aim,we introduce the τ -approximation of f ( · ) as follows:7 efinition 2.1 ( τ -approximation) For any τ > , f τ ( x ) := 1 τ p √ Λ (cid:112) LLD ( x, τ ) . Remark 2.1
From Proposition 2.1 (iii), it follows that when P has a density f ( · ) then, f τ ( · ) is continuous. Let S τ := { x ∈ R p : f τ ( x ) > } be the support of f τ ( · ) . If f ( · ) is k -times continuously differentiable, Proposition 2.1 (iv) implies that f τ ( · ) is k -timescontinuously differentiable in S τ . We note that in the approach to clustering adopted in this paper we will encounterconvergence of level sets (described in Section 3 below). This involves uniform convergenceof f τ ( · ) to f ( · ) and other ideas. To this end, we denote by B (cid:15) ( x ) := { y ∈ R p : (cid:107) x − y (cid:107) ≤ (cid:15) } the closed ball with center x ∈ R p and radius (cid:15) > Theorem 2.2
Let P be absolutely continuous with respect to the Lebesgue measure on R p with density f ( · ) . Then the following hold:(i) If f ( · ) is uniformly continuous and bounded, then lim τ → + sup x ∈ R p | f τ ( x ) − f ( x ) | = 0 . (ii) If f ( · ) is continuous, then for all compact sets K ⊂ R p lim τ → + sup x ∈ K | f τ ( x ) − f ( x ) | = 0 . (2.6) In particular, for all x ∈ R p lim τ,(cid:15) → + sup y ∈ B (cid:15) ( x ) | f τ ( y ) − f ( x ) | = 0 . Remark 2.2
The above theorem implies that the τ -approximation converges uniformlyto the density under extreme localization. We also note that continuity is not enough inTheorem 2.2 (i) (see Appendix B for a counterexample). Let { X , . . . , X n } be independent and identically distributed (i.i.d.) from P on R p ; thenthe estimate of LLD is the empirical distribution of the pair ( X i , X j ) belonging to the set Z τ ( x ); that is, for x ∈ R p and τ ∈ [0 , ∞ ] LLD n ( x, τ ) := 1 (cid:0) n (cid:1) (cid:88) ≤ i LLD n is a non-degenerate U-statistics of order 2 [Korolyuk andBorovskich, 2013] with the kernel K x,τ ( x , x ) := I (( x , x ) ∈ Z τ ( x )) (2.7)whose first projection is J x,τ ( x ) := (cid:90) I (( x , x ) ∈ Z τ ( x )) dP ( x ) = P (( X , X ) ∈ Z τ ( x ) | X = x ) . (2.8)Furthermore, LLD n is an unbiased estimator of LLD , that is, E [ LLD n ( x, τ )] = LLD ( x, τ ).Using Serfling [2009, Lemma A, Section 5.2.1], it follows that V ar [ LLD n ( x, τ )] = 1 (cid:0) n (cid:1) [ a ( x, τ ) + 2( n − b ( x, τ )] , (2.9)where a ( x, τ ) := V ar [ K x,τ ( X , X )] = LLD ( x, τ )(1 − LLD ( x, τ )), and b ( x, τ ) := V ar [ J x,τ ( X )] = (cid:90) ( J x,τ ( x )) dP ( x ) − ( LLD ( x, τ )) . (2.10)From (2.9), it follows that for all n ∈ N V ar [ √ n LLD n ( x, τ )] = 4 n − n − b ( x, τ ) + O (cid:18) n (cid:19) −−−→ n →∞ b ( x, τ ) . The above calculation shows that LLD n is a consistent estimator of LLD . In typicalapplications, the choice of x and τ vary and in exploratory analyses, different choices of x and τ may be investigated. Our next result shows that the LLD n is uniformly consistentover x and τ . In the following, we use the notation a.s. to mean almost surely with respectto P . 9 heorem 2.3 sup x ∈ R p τ ∈ [0 , ∞ ] | LLD n ( x, τ ) − LLD ( x, τ ) | −−−→ n →∞ a.s. We now turn to the uniform central limit theorem for LLD n over a subset Γ of R p × [0 , ∞ ]. Let (cid:96) ∞ (Γ) denote the space of all bounded functions ¯ h ( · ) : Γ → R . Tostudy the convergence in distribution in (cid:96) ∞ (Γ), one needs to address the measurabilityproblems that are encountered due to the non-separability of (cid:96) ∞ (Γ). We address thisusing Theorem 4.9 in Arcones and Gin´e [1993]. In their paper they handle the issue byrequiring F (not defined here) be a “measurable” class, where measurable was describedon Page 1497. Indeed, in our proof, and as also stated in their paper, we address thisissue by establishing that the class of kernels related to the U-statistics is image admissibleSuslin (see Dudley [2014]) and (1.9) of their paper holds. In the following, convergencein distribution in (cid:96) ∞ (Γ) is in the sense of Hoffmann-Jørgensen [Gin´e and Nickl, 2016,Definition 3.7.22]. Theorem 2.4 If Γ ⊂ R p × [0 , ∞ ] is such that b ( x, τ ) > for all ( x, τ ) ∈ Γ , then √ n ( LLD n ( · , · ) − LLD ( · , · )) d −−−→ n →∞ W ( · , · ) in (cid:96) ∞ (Γ) where { W ( x, τ ) } ( x,τ ) ∈ Γ is a centered Gaussian process with covariance function γ : Γ × Γ → R given by γ (( x, τ ) , ( y, ν )) = (cid:90) J x,τ ( x ) J y,ν ( x ) dP ( x ) − LLD ( x, τ ) LLD ( y, ν ) . Remark 2.3 Notice that the variance of W ( x, τ ) is γ (( x, τ ) , ( x, τ )) = b ( x, τ ) , and it ispositive for τ > provided P is absolutely continuous with respect to the Lebesgue measureand x is in its support. In the rest of this subsection, we assume that P is absolutely continuous with respectto the Lebesgue measure with density f ( · ). The plug-in estimator of f τ ( · ) is given by f τ,n ( x ) := 1 τ p √ Λ (cid:112) LLD n ( x, τ ) . (2.11)Our first result summarizes basic properties of f τ,n ( · ) and it is a consequence of Theorems2.2 and 2.3. 10 orollary 2.3 Let P be absolutely continuous with respect to the Lebesgue measure on R p with density f ( · ) . Then the following hold:(i) If f ( · ) is uniformly continuous and bounded, then lim τ → + lim n →∞ sup x ∈ R p | f τ,n ( x ) − f ( x ) | = 0 a.s.(ii) If f ( · ) is continuous, then for all compact sets K ⊂ R p lim τ → + lim n →∞ sup x ∈ K | f τ,n ( x ) − f ( x ) | = 0 a.s. (2.12) In particular, for all x ∈ R p lim τ,(cid:15) → + lim n →∞ sup y ∈ B (cid:15) ( x ) | f τ,n ( y ) − f ( x ) | = 0 a.s. It is well known that extreme localization is an important concept in depth analysis,however, the fluctuations of f τ,n ( · ) are unknown. Our main result in this section charac-terizes the asymptotic variance and establishes a related limit distribution. To this end,for x ∈ R p , we denote the projection of Z τ ( x ) onto the first component of R p × R p by Z τ ( x ) | x := { x ∈ R p : max i =1 , (cid:107) x − x i (cid:107) ≤ (cid:107) x − x (cid:107) ≤ τ } and Λ ∗ := (cid:82) λ ( Z (0) | x ) dx . Theorem 2.5 Let P be absolutely continuous with respect to the Lebesgue measure on R p with continuous density f ( · ) . Let x belong to the support of f ( · ) and { τ n } ∞ n =1 be asequence of positive scalars converging to zero. If √ n τ pn −−−→ n →∞ ∞ , then √ n τ pn ( f τ n ,n ( x ) − f τ n ( x )) d −−−→ n →∞ N (cid:18) , Λ ∗ Λ f ( x ) (cid:19) . Remark 2.4 We notice that the limit distribution in Theorem 2.5 with f τ n ( · ) replaced by f ( · ) cannot hold. In fact, the deterministic term f τ n ( x ) − f ( x ) is, by Theorem 2.1 (ii), oforder O ( τ n ) , while the term f τ n ,n ( x ) − f τ n ( x ) converges to a normal distribution at rate / ( √ nτ pn ) . Since, necessarily, √ n τ pn −−−→ n →∞ ∞ , f τ n ( x ) − f ( x ) is the dominant term. The constant Λ appearing in the limiting variance in Theorem 2.5 can be calculatednumerically by computing the percentage of uniformly distributed random points in thecentered ball in R p × R p with radius √ Z (0) and multiplying the result byits volume (2 p π p )( p !) − . Similarly, the constant Λ ∗ can be calculated by approximatingthe integral with a sum. An alternative form for Theorem 2.5 without the factor f ( x ) inthe variance term is given in the following corollary.11 orollary 2.4 Under the hypothesis of Theorem 2.5, √ n τ pn (cid:18)(cid:113) f τ n ,n ( x ) − (cid:112) f τ n ( x ) (cid:19) d −−−→ n →∞ N (cid:18) , Λ ∗ (cid:19) . An extension of Theorem 2.5 uniformly over S , namely, √ n τ pn ( f τ n ,n ( · ) − f τ n ( · )) d −−−→ n →∞ Λ ∗ Λ W ( · ) in (cid:96) ∞ ( S ) , where { W ( x ) } x ∈ S is a centered Gaussian process with the covariance function γ : S × S → R given by γ ( x, y ) = (cid:112) f ( x ) f ( y ), requires an extension of the results of Arcones and Gin´e[1993] to triangular arrays and it is beyond the scope of the present paper. A result inthis direction is given by Schneemeier [1989], but this is not sufficient in this context sincethe sets { Z τ n ( x ) } ∞ n =1 depend on n and x . In the previous subsections, we established results for the LLD. However, as explainedin the introduction, our results remain valid for other LDFs such as LBD and LSD withappropriate modifications. Also, some of the results hold for a local version of half-spacedepth [Tukey, 1975], denoted by LHD, and a local version of half-region depth [L´opez-Pintado and Romo, 2011], denoted by LRD. We highlight in this section the commonfeatures and the main differences between the other LDFs and LLD. In Section 4, wecompare the performance of LLD with LSD in clustering.We begin by comparing LLD and LSD. An alternative definition of LSD that boundsthe volume of the simplex is introduced and studied in Agostinelli and Romanazzi [2011].However, in this definition, the maximum of the component lengths may remain largeeven if the volume of the simplex converges to 0 (as an example consider the triangle).This prevents the extreme localization analysis for p > Z Sτ ( x ) | x , for x ∈ R p , denote the projection of Z Sτ ( x )onto the first component of ( R p ) ( p +1) .The main difference between LSD and LLD is that the geometry of the sets Z Sτ ( x )and Z Sτ ( x ) | x are different from those of Z τ ( x ) and Z τ ( x ) | x (recall that L is suppressedfor LLD). Specifically, note that Z Sτ ( x ) is a subset of ( R p ) ( p +1) while Z τ ( x ) is a subset of( R p ) . This difference affects the corresponding estimator, since the relevant U-statisticsis now of order ( p + 1) as against two for LLD. The Lebesgue measure of the sets Z τ ( x )and Z τ ( x ) | x , namely, Λ and Λ ∗ , appear in many of the asymptotic results for LLD.This, in the case of LSD, is replaced by the Lebesgue measures of the corresponding sets;12hat is, Λ S = λ ( Z S (0)) and (Λ ∗ S ) = (cid:82) (cid:0) λ ⊗ ( p − (cid:1) ( Z S (0) | x ) dx . This change also affectsthe τ -approximation, since the factor τ for LLD is replaced by τ ( p +1) for LSD. Similarcomments hold for other results as well.On the other hand, except for the difference in the geometry of sets Z Bτ ( x ) and Z τ ( x ) (and hence their projections), both LBD and LLD have similar analytical andprobabilistic properties. Indeed, BD and LD arise as particular instances of a broaderclass of DFs [Yang and Modarres, 2018], which is based on the concept of β -skeletonregion [Kirkpatrick and Radke, 1985]. These DFs are therefore called β -skeleton DFs.For β ≥ 1, the β -skeleton depth is a particular case of (1.1) with k = 2, G = K β and Z K β ( x ) = { ( x , x ) ∈ R p × R p : max ( i,j ) ∈{ (1 , , (2 , } (cid:107) x i + (2 /β − x j − /β x (cid:107) ≤ (cid:107) x − x (cid:107)} . The corresponding local version ( LK β D ) is then obtained by replacing Z Gτ ( x ) with Z K β τ ( x ) = { ( x , x ) ∈ R p × R p : max ( i,j ) ∈{ (1 , , (2 , } (cid:107) x i + (2 /β − x j − /β x (cid:107) ≤ (cid:107) x − x (cid:107) ≤ τ } . Notice that, LK D ( · , · ) = LBD ( · , · ) and LK D ( · , · ) = LLD ( · , · ). In particular, for p = 1, the LDFs LK β D , LLD , LBD , and LSD coincide.Turning to LHD (see Agostinelli and Romanazzi [2011][Definition 12]), notice that,as in LSD, the slabs proposed in there cannot be included in a ball of radius r for any r positive. For this reason, we propose an alternative definition of this local depth. Asbefore, let τ ≥ x ∈ R p . Then the local half-space depth with respect to P is givenby LHD ( x, P, τ ) = inf u ∈ S p − P ( Z Hτ ( x, u )) , (2.13)where Z Hτ ( x, u ) is the hypercube with center x + τ u given by Z Hτ ( x, u ) = (cid:26) y ∈ R p : max i =1 ,...,p (cid:12)(cid:12)(cid:12) x ( i ) + τ u ( i ) − y ( i ) (cid:12)(cid:12)(cid:12) ≤ τ (cid:27) and S p − is the unit sphere in R p .Similar to the notion of local half-space depth is the local half-region depth (LRD)introduced and studied in Agostinelli [2018]. This is given by LRD ( x, P, τ ) = min( P ( Z R − τ ( x )) , P ( Z R + τ ( x )) , where Z R − τ ( x ) = { y ∈ R p : − τ ≤ y ( i ) − x ( i ) ≤ , i = 1 , . . . , p } and Z R + τ ( x ) = { y ∈ R p :0 ≤ y ( i ) − x ( i ) ≤ τ, i = 1 , . . . , p } . In particular, for p = 1, LHD and LRD coincide.We now describe some features of LSD, LBD, LK β D , LHD, and LRD that areshared with LLD allowing one to obtain similar results. Recalling that the superscript13 represents general depth ( G = L, S, B, K β , H, R ), we note that the expression definingLLD, LSD, LBD, and LK β D is an integral of the form (1.2), where Z Gτ ( x ) ⊂ ( R p ) k is aclosed set that satisfies the invariance properties and symmetry conditions Z Gτ ( x ) = ( x, . . . , x ) + Z Gτ (0) (2.14) Z Gτ (0) = τ Z G (0) (2.15)( x, . . . , x ) + ( x , . . . , x k ) ∈ Z Gτ ( x ) ⇐⇒ ( x, . . . , x ) − ( x , . . . , x k ) ∈ Z Gτ ( x ) (2.16)( x, . . . , x ) + ( x , . . . , x k ) ∈ Z Gτ ( x ) ⇐⇒ ( x, . . . , x ) + ( x i , . . . , x i k ) ∈ Z Gτ ( x ) (2.17)for every permutation ( i , . . . , i k ) of (1 , . . . , k ). For LLD, LSD, LBD and LK β D thereare sets L ⊂ R p (lenses, simplices, balls, intersection of balls) such that { L ( x , . . . , x k ) : x , . . . , x k ∈ R p } is a VC-class and ( x , . . . , x k ) ∈ Z Gτ ( x ) if and only if x ∈ L ( x , . . . , x k ).This property proves useful in establishing Theorems 2.3 and 2.4 for LSD, LBD and LK β D . The invariance properties (2.14) and (2.15) are needed to establish Proposition2.1 (i), Corollary 2.2 (i) and the results concerning the density function, for example,Theorems 2.1 and 2.2. The symmetry properties (2.16) and (2.17) are required in Theorem2.1 (ii), and Theorems 3.3 and 3.4 in the next section. A detailed description of themodifications required for the results concerning LLD to hold for LSD is provided inAppendix D. The results for LBD and LK β D are very similar to those of LLD andtherefore omitted.We notice that LHD and LRD also share many properties with LLD . Indeed, LHDis the infimum over u ∈ S p − of integrals of the form (1.2) with k = 1 and Z Gτ ( x ) replacedby Z Hτ ( x, u ). If P is absolutely continuous with respect to the Lebesgue measure, thisinfimum is actually a minimum. Hence, most of the properties and the applications toclustering hold for LHD since (i) we can replace u by the minimizer, (ii) Z Hτ ( x, u ) satisfies(2.14), (2.15), and (2.17), and (iii) the class of hypercubes { Z Hτ ( x, u ) : x ∈ R p , u ∈ S p − , τ ∈ [0 , ∞ ] } forms a VC-class. We refer to Appendix E, for further details. Similarcomments also hold for LRD, as it is the minimum of the probability of two hypercubessatisfying (2.14), (2.15) and (2.17). In particular, it has the same behavior under extremelocalization (see Appendix E). In this section, we describe a new methodology for clustering high-dimensional data. Thebasic idea relies on defining clusters as the stable manifolds generated by the mode, whichare obtained using the limiting trajectory of a gradient system (see Chac´on [2015]). Thefact that such manifolds yield non-trivial meaningful clusters can then be established using14he Lyapunov’s theory in dynamical systems. The development of the algorithm needs aninvestigation of the behavior of the level sets. Chac´on [2015] uses density based methodsfor investigating these clusters in one dimension only. Our methods, which involve theuse of the τ -approximation based on LLD, are for multidimensional distributions.It is a priori unclear why such a procedure would yield meaningful clusters. Indeed,using analytical and statistical results on extreme localization described in Sections 2 and3, it is plausible that the proposal so described will yield meaningful clusters. In the restof this section, we describe step-by-step analytical tools to fill in the gap between localdepths and stable manifolds generated by the modes. Towards this aim, we begin byproviding a precise notion of the key ideas.Let f ( · ) be a twice-continuously differentiable density function. Then for any x ∈ S ,the stable manifold generated by x is C ( x ) := { x ∈ S : lim t →∞ u x ( t ) = x } , (3.1)where u x ( t ) is the solution at time t of the gradient system u (cid:48) ( t ) = ∇ f ( u ( t )) (3.2)with initial value u (0) = x . For any choice of x , it is not required for the stable manifoldso-defined to be non-trivial; i.e. the Lebesgue measure of C ( x ) is non-zero. However, if x is chosen as a local maximum of f ( · ), then, under additional conditions, one can verifythat the resulting manifold has positive Lebesgue measure. In the next subsections, weestablish several results leading up to the verification that using local maxima of f ( · ) and f τ ( · ) yield non-trivial clusters. These results are stated for LLD, but also hold for othernotions of depth such as spherical depth, simplicial depth, and β -skeleton depth. Somedetails of the changes needed for simplicial depth are provided in Appendix D. In this subsection, we collect relevant mathematical background that allows one to developthe algorithm for identification of clusters, also referred to as stable manifolds or basinsof attraction. We begin with the definition of level sets and super level sets. Definition 3.1 For α > , the level sets of f ( · ) and f τ ( · ) are L α = { x ∈ R p : f ( x ) = α } and L ατ = { x ∈ R p : f τ ( x ) = α } , respectively. The super level sets of f ( · ) , f τ ( · ) and f τ,n ( · ) are R α := { x ∈ R p : f ( x ) ≥ α } , R ατ := { x ∈ R p : f τ ( x ) ≥ α } and R ατ,n := { x ∈ R p : f τ,n ( x ) ≥ α } , respectively. f τ ( · ) and f τ,n ( · ) coincide with those induced by f ( · ). We use the notation ˚ A for the interior of aset A . Proposition 3.1 Suppose that f ( · ) is uniformly continuous and bounded. Let { α k } ∞ k =1 be a sequence converging to α > and { τ k } ∞ k =1 a sequence of positive scalars convergingto . Then ˚ R α ⊂ lim inf k →∞ R α k τ k ⊂ lim sup k →∞ R α k τ k ⊂ R α , (3.3) and, if λ ( L α ) = 0 , lim k →∞ R α k τ k = R α a.e. (3.4) If additionally λ ( L ατ k ) = 0 for k ∈ N , then lim k →∞ lim n →∞ R α n τ k ,n = R α a.s. (3.5) Remark 3.1 Proposition 3.1 also holds for continuous f ( · ) if we restrict R α , R ατ and R ατ,n to a compact subset of R p . Since LSD, LBD, LK β D , LHD and LRD satisfy Proposition2.1 (ii), Theorem 2.2 and Theorem 2.3, the proof of Proposition 3.1 shows that the superlevel sets based on all these LDFs satisfy (3.3) , (3.4) and (3.5) . Proposition 3.1 is of independent interest, since a common approach in modal clustering isto define clusters as the connected components of the super level sets R α for some α > α varies,yielding cluster trees.In the rest of this section, we suppose that f ( · ) is twice continuously differentiablewith finite number of stationary points in S . Additionally, we assume that the Hessianmatrix H f has non-zero eigenvalues at its stationary points and that R α is a bounded setfor every α > 0. Since f ( · ) is continuous, R α is compact. We notice that R α is boundedif f ( · ) vanishes at infinity, that is, sup x ∈ R p : (cid:107) x (cid:107)≥ M f ( x ) → M → ∞ , which is satisfied,for example, if S is bounded.Even though well-known, to make the manuscript self-contained, we define the mode.The definition of cluster as basin of attraction of a mode is given in Chac´on [2015]. Definition 3.2 A point m ∈ S is said to be a mode (resp. an antimode) for f ( · ) if it isa stationary point of f ( · ) and H f ( m ) has only negative (resp. positive) eigenvalues, thatis, m is a local maximum (resp. minimum) for f ( · ) . If m , . . . , m k are the modes of f ( · ) ,then the clusters induced by m , . . . , m k are the stable manifolds C ( m ) , . . . , C ( m k ) . u x ( · ) exists and is uniquefor t in some maximal time interval ( a, b ) with a < < b , where a = −∞ or b = ∞ is allowed. To see this, observe that, as f ( · ) is twice continuously differentiable, for ev-ery x ∈ S there exists a convex neighborhood U x of x in which the second order partialderivatives are bounded. By applying Agarwal and Lakshmikantham [1993][Lemma 3.2.1]to ∇ f , it follows that ∇ f is Lipschitz in U x , and therefore ∇ f is locally Lipschitz in S .Now, applying Picard-Lindel¨of Theorem [Teschl, 2012][Theorem 2.2 and 2.5] it followsthat u x ( · ) exists in some time interval, which can be chosen to be maximal in view ofTheorem 2.13 in Teschl [2012].We now show that, using the boundedness of R α , the solution u x ( t ) exists for all t ≥ x ∈ S . Furthermore, the solution starting in R α cannot leave the set. Tothis end, notice that the equilibria of (3.2) are the stationary points of f ( · ). The gradientcomputed at each point gives the direction of most rapid increase of f ( · ). Hence, thetrajectories { u x ( t ) : t ∈ R } for x ∈ S that are not stationary points are curves of steepestascent for f ( · ). More specifically, if u x ( t ) ∈ L α for some x ∈ S and t ∈ R , then anyvector v tangent to L α at u x ( t ) satisfies (cid:104) v, u (cid:48) x ( t ) (cid:105) = 0 (see [Hirsch et al., 1974][Chapter9 § u x ( t ) = x for all t is astationary point of the gradient system (3.2) or the trajectory { u x ( t ) : t ∈ R } crosses L α orthogonally. This also implies that u x ( t ) cannot leave R α for t ≥ 0. Furthermore, thisproperty shows that, for all x ∈ S , the solutions u x ( t ) exists for all t ≥ 0, i.e. the maximaltime interval in which u x is defined is ( a, ∞ ) for some a < 0, where a = −∞ is possible.To see this, for x ∈ S , choose an α > x ∈ R α . Since u x ( t ) cannot leave R α for t ≥ R α is compact, the result follows from Teschl [2012][Corollary 2.15]. Recallingthat our clusters are the stable manifolds generated by modes, we now link modes to thegradient system. This requires the notion of ω -limit which we now define. Definition 3.3 The ω -limit of a point x ∈ S is the set of points y ∈ S such that u x ( t ) goes to y as t → ∞ , in symbols ω ( x ) = { y ∈ S : lim t →∞ u x ( t ) = y } . The following definition is also required (Hirsch et al. [1974][Chapter 9 § W : U → R , W (cid:48) ( u x ( t )) = d W ( u x ( t )) dt . 17 efinition 3.4 Let u ∈ S be an equilibrium point for (3.2) and U ⊂ S a neighborhood of u . A differentiable function W : U → R is a strict Lyapunov function if (i) W ( u ) = 0 and W ( u ) > for u (cid:54) = u , and (ii) W (cid:48) ( u x ( t )) < when u x ( t ) ∈ U \ { x } . Let V ( · ) := − f ( · ). If m is a mode for f ( · ), there exists a neighborhood U m of m suchthat, for all u ∈ U m \ { m } , V ( u ) − V ( m ) > V ( u ) − V ( m )) (cid:48) = − ( f ( u )) (cid:48) = −(cid:104)∇ f ( u ) , u (cid:48) (cid:105) = − (cid:107)∇ f ( u ) (cid:107) < . Hence, V ( u ) − V ( m ) is a strict Lyapunov function in U m . By the Lyapunov stabilityTheorem (see [Hirsch et al., 1974][Chapter 9 § m is asymptotically stable, that is, there is a neighborhood U ∗ m ⊂ U m of m such that each solution starting from a point x ∈ U ∗ m converges to m , i.e., for all x ∈ U ∗ m , ω ( x ) = { m } .Given a mode m , we now describe the stable manifold generated by m . This isequivalent to describing the properties of the points that have ω -limit m . Indeed, if0 < α < f ( m ) is such that the connected component of m in R α contains no equilibriaother than m , then, since each solutions of (3.2) starting in that component cannot leaveit, by LaSalle’s invariance principle (see [Hirsch et al., 1974][Chapter 9 § V ( · ) − V ( m ), allthe points in that component have m as an ω -limit point. On the other hand, if m is anantimode for f ( · ), there exists a neighborhood U m of m such that for all u ∈ U m \ { m } , V ( m ) − V ( u ) > V ( m ) − V ( u )) (cid:48) > 0. This implies that m is unstable [Hale,1980][Chapter X.1 Theorem 1.2]: for every neighborhood U ∗ m ⊂ U m of m every solution u x starting from a point x ∈ U ∗ m eventually leaves U ∗ m . Furthermore, any ω -limit pointof gradient system (3.2) is an equilibrium point, that is a stationary points of f ( · ) (see[Hirsch et al., 1974][Chapter 9 § x of f ( · ), recall from (3.1) that C ( x ) is the stable mani-fold induced by x , that is, the set of points with ω -limit x . The hypothesis that H f (which is symmetric), at stationary points, has non-zero eigenvalues, implies that the sets C ( x ) have dimension equal to the number of negative eigenvalues of H f ( x ) (see [Teschl,2012][Section 9]).As in Definition 3.2, let m , . . . , m k be the modes of f ( · ). We are now ready toverify that the clusters C ( m ) , . . . , C ( m k ) are non-trivial. We first observe that, by theuniqueness of the limit, C ( m ) , . . . , C ( m k ) are disjoint. Also, if x is not a mode, then C ( x ) has dimension smaller than p and the set S \ ( C ( m ) ∪ · · · ∪ C ( m k )) has Lebesguemeasure 0. However, in practice, the density f ( · ) is generally unknown and it has to be18stimated. A common approach is to use kernel density estimators. However, we replace f ( · ) by f τ ( · ) in (3.2) and consider instead the gradient system u (cid:48) ( t ) = ∇ f τ ( u ( t )) . (3.6)The domain of this new system is S τ , the support of f τ ( · ). We summarize in this subsection the main properties of the gradient system (3.6) as τ → + . Lemma 3.1 For all < τ ≤ τ , we have that S τ ⊂ S τ . Additionally, if f ( · ) iscontinuous, then for all τ > , S ⊂ S τ and lim τ → + S τ = S . The next theorem shows that the gradient and the Hessian matrix of f τ ( · ) converge tothose of f ( · ). Given a function g : D ⊂ R p → R for j = 1 , . . . , p we denote by ∂ j g itspartial derivative with respect to the j -component. Theorem 3.1 Let τ > . If f ( · ) has continuous first order partial derivatives, then, forall j = 1 , . . . , p , ∂ j f τ is continuous in S τ and, for all x ∈ S , lim τ → + ∂ j f τ ( x ) = ∂ j f ( x ) . If also the second order partial derivatives are continuous, then, for all i, j = 1 , . . . , n , ∂ i ∂ j f τ is continuous in S τ and, for all x ∈ S , lim τ → + ∂ i ∂ j f τ ( x ) = ∂ i ∂ j f ( x ) . Remark 3.2 Lemma 3.1 and Theorem 3.1 hold also for LSD, LBD, LK β D , LHD andLRD. The sets { S τ } τ> monotonically decrease to S by Lemma 3.1. Furthermore, because ofTheorem 3.1, f τ ( · ) is twice continuously differentiable in S τ and its gradient and Hessianmatrix converge to those of f ( · ) in S . If it exists, we denote by u x,τ ( t ) the solutionof (3.6) with initial point u x,τ (0) = x . Since f τ ( · ) is continuous, for α > 0, the sets R ατ = { x ∈ R p : f τ ( x ) ≥ α } = f − τ ([ α, ∞ )) are closed. The next lemma shows that theyare also bounded. For a set A ⊂ R p , we define ( A ) + δ := { x ∈ R p : inf y ∈ A (cid:107) x − y (cid:107) ≤ δ } and ( A ) − δ := R p \ ( R p \ A ) + δ = { x ∈ R p : inf y ∈ R p \ A (cid:107) x − y (cid:107) > δ } .19 emma 3.2 For all τ > and α > , ( R α ) − τ ⊂ R ατ ⊂ ( R α ) + τ . In particular, if R α isbounded for α > , then R ατ is also bounded for any τ > . As we have shown for the gradient system (3.2), Lemma 3.2, along with the boundednessof R α for all α > f τ ( · ) is twice continuously differentiable, impliesthat, for all x ∈ S , u x,τ exists and is unique in a maximal time interval ( a, ∞ ), for some −∞ ≤ a < τ → + , the function f τ ( · ) behaves like f ( · ) (see also Theorem 2.2) and Theorem 3.1 shows that the partial derivatives of f τ ( · )up to the second order converge to those of f ( · ). We exploit this in the next theorem toshow that the solutions of the gradient system (3.6) converge for τ → + to those of thegradient system (3.2). Theorem 3.2 Suppose that f ( · ) is continuously differentiable in R p and that for all α > R α is compact. Then, for all t ≥ and x ∈ S lim τ → + u x,τ ( t ) = u x ( t ) . Remark 3.3 Theorem 3.2 holds also for LSD, LBD, LK β D , LHD and LRD using The-orem 3.1 and Lemma 3.2. For LK β D with β > a small modification is required inLemma 3.2; namely, for all τ > and α > , ( R α ) − β τ ⊂ R ατ ⊂ ( R α ) + β τ . Similarly, sincehypercubes in R p have diameter √ p , for LHD and LRD we have that ( R α ) −√ pτ ⊂ R ατ ⊂ ( R α ) + √ pτ . In the next subsection we provide some conditions under which the stationary points(resp. modes, antimodes) of f ( · ) are also stationary points (resp. modes, antimodes) of f τ ( · ) for τ > In this subsection we consider an absolutely continuous distribution with respect to theLebesgue measure, with density f ( · ). The key criteria for the identification of the modesis the notion of symmetry proposed below. Definition 3.5 Given τ > , a density function f ( · ) with support S is said to be τ -centrally symmetric about µ ∈ S if, for all x ∈ R p with (cid:107) x (cid:107) ≤ τ , f ( µ + x ) = f ( µ − x ) . In particular, for p = 1, f ( · ) is τ -centrally symmetric about µ ∈ R if f ( µ − x ) = f ( µ + x )for all x ∈ [0 , τ ]. If f ( · ) has a continuous derivative, a direct computation using Corollary20.1 shows that f (cid:48) τ ( µ ) = 0 (see Appendix C for more details). For example, if f ( · ) is amixture of two normal densities with means µ and µ , variances equal to σ and weightsequal to 1 / 2, that is, f ( x ) = 1 √ π · σ (cid:20) e − ( x − µ σ + e − ( x − µ σ (cid:21) , (3.7)then f ( · ) is τ -centrally symmetric about µ = µ + µ for all τ ≥ µ is astationary point of f τ ( · ). On the other hand, due to the interference of the tail of onecomponent to the other around the means, f ( · ) is not τ -centrally symmetric about them.However, if the two means are far away from each other, the modes of f τ ( · ) are a goodapproximation of those of f ( · ) (see Figure 1 and Propositiion C.1 in Appendix C). We −4 −2 0 2 4 . . . Figure 1: In black, the density (3.7) with µ = − µ = 2 and σ = 1. In red, its sample τ -approximation f τ,n ( · ) for τ = 1 and n = 6000.begin by identifying the stationary points of f τ . Theorem 3.3 Suppose that for some τ > and µ ∈ S , f ( · ) has continuous first orderpartial derivatives in B τ ( µ ) = { x ∈ R p : (cid:107) x − µ (cid:107) ≤ τ } . Then, µ is a stationary point for f τ ( · ) if and only if (cid:90) Z (0) ∇ f ( µ + τ x ) f ( µ + τ x ) dx dx = 0 , (3.8) where the integral of a vector is the vector of the integrals. Furthermore, if f ( · ) is τ -centrally symmetric about µ , then ∇ f τ ( µ ) = 0 . Our next result, which is about the Hessian matrix, gives sufficient conditions for m tobe a mode of f τ ( · ). 21 heorem 3.4 Let τ > . Suppose that f ( · ) is τ -centrally symmetric about a mode (resp.an antimode) m and has continuous second order partial derivatives in B τ ( m ) .If, for all x , x ∈ B τ ( m ) , the matrix G f ( x , x ) := H f ( x ) f ( x ) + ∇ f ( x ) ∇ f ( x ) (cid:62) is negative (resp. positive) definite, then m is also a mode (resp. an antimode) for f τ ( · ) . Notice that G f ( m, m ) = H f ( m ) f ( m ) is negative (resp. positive) definite and therefore thesecond condition of Theorem 3.4 is satisfied by f ( · ), for τ small. Remark 3.4 Theorem 3.3 and Theorem 3.4 hold also for LSD, LBD and LK β D . For LK β D with β > we assume that f ( · ) has continuous derivatives in B β τ ( µ ) , B β τ ( m ) and that it is β τ -centrally symmetric. In this section, we describe the algorithm for the numerical approximation of the clustersinduced by the system (3.6) based on the theoretical developments in Sections 3.2 and3.3. Since the sample τ -approximation is not differentiable in x , we use a finite differenceapproximation that converges to the directional derivative. To this end, for x ∈ R p , τ > n ∈ N , h > v ∈ R p , we use the finite difference approximations of thedirectional derivatives of f τ ( · ) and f n,τ ( · ) along v given by ∇ hv f τ ( x ) = f τ ( x + hv ) − f τ ( x ) h and ∇ hv f τ,n ( x ) = f τ,n ( x + hv ) − f τ,n ( x ) h . Using the proof of Corollary 2.3 and the convergence of partial derivatives in Theorem 3.1,it follows that the finite difference approximations converge to the directional derivative.The next step towards identifying the modes, is finding a local maximum of a func-tion. To this end, we use the steepest ascent or gradient ascent idea; that is, startingfrom a point in the space, the next point is chosen in the direction given by the gradientof the function at that point. This procedure is repeated until convergence to a localmaximum is achieved. In the context of mode hunting, this procedure is often combinedwith kernel density estimators to find the modes of the density underlying some givendata points and to find the clusters associated to these modes [Fukunaga and Hostetler,1975, Menardi, 2016]. In the following, we propose a similar technique, which does notnecessarily require the existence of the gradient and only considers data points as candi-date points toward to which to move in the next step. This yields an efficient procedure, ince the local depth function has a higher potential to differentiate the local features ofthe distribution, especially in the inner part of the data cloud .Turning to the development of the algorithm, starting from a point x ∈ R p , we search,in a given neighborhood of x , the point y that yields the largest directional derivative ∇ hv f τ,n with h = (cid:107) y − x (cid:107) and v = ( y − x ) / (cid:107) y − x (cid:107) . Since, ∇ hv f τ ( x ) τ p (cid:112) Λ = (cid:112) LLD ( x + hv, τ ) − (cid:112) LLD ( x, τ ) h and ∇ hv f τ,n ( x ) τ p (cid:112) Λ = (cid:112) LLD n ( x + hv, τ ) − (cid:112) LLD n ( x, τ ) h , the constant τ p √ Λ does not influence the choice of the point y which maximizes boththe finite differences ∇ hv f τ ( x ) and ∇ hv f τ,n ( x ). This allows one to ignore the constantsin the specification of the algorithm. That is, the finite difference approximation of thedirectional derivative of the square root of the local depth can be used instead, avoidingthe computation of the constant Λ .We now show, in fact, that the constant τ p √ Λ also does not influence the clustersinduced by the system (3.6). Since τ, Λ > 0, if, for x ∈ R p , u x,τ : R → R p is a solutionof the system (3.6) with u x,τ (0) = x , then ˜ u x,τ : R → R p given by ˜ u x,τ ( t ) := u x,τ ( τ p √ Λ t )also satisfies ˜ u x,τ (0) = x and it is a solution of the system˜ u (cid:48) ( t ) = ∇ (cid:16)(cid:112) LLD (˜ u ( t ) , τ ) (cid:17) . (3.9)Moreover, since lim t →∞ u x,τ ( t ) = lim t →∞ ˜ u x,τ ( t ) for all x ∈ R p , the clusters induced by(3.6) and (3.9) are the same. Hence, for x, y ∈ R p with y (cid:54) = x and h = (cid:107) y − x (cid:107) ≤ r smallenough, we consider the finite difference approximation of the directional derivatives of (cid:112) LLD ( x, τ ) and (cid:112) LLD n ( x, τ ) along the direction v = y − x (cid:107) y − x (cid:107) given by d τ ( x ; y ) := (cid:112) LLD ( y, τ ) − (cid:112) LLD ( x, τ ) (cid:107) y − x (cid:107) and d τ,n ( x ; y ) := (cid:112) LLD n ( y, τ ) − (cid:112) LLD n ( x, τ ) (cid:107) y − x (cid:107) . (3.10)Given n data points x , . . . , x n , the localization parameter τ used for the clusteringprocedure is chosen as the quantile of order q , for some 0 ≤ q ≤ 1, of the empiricaldistribution of the (cid:0) n (cid:1) distances (cid:107) x i − x j (cid:107) , i > j , i, j ∈ { , , . . . , n } . We now sum-marize the procedure for computing the clusters in Algorithm 1. It requires as input,the data points { x , . . . , x n } , the quantile q , and two additional parameters, r and s .23dditional points { y , . . . , y o } may also be provided as input. Starting from any point x ∈ { x , . . . , x n } ∪ { y , . . . , y o } , based on the finite difference (3.10), the algorithm movesto another data point y ∈ { x , . . . , x n } (hence, except for the initial step, only data pointsare involved in (3.10)). The parameter r gives a bound on the norm (cid:107) y − x (cid:107) in (3.10)in order to choose only those points that are close to each other. The parameter s isexploited, to avoid trivialities such that there are no points at distance from x smallerthan r or the number of directional derivatives taken into account is too small to have areliable estimate. If there are less than s points at distance smaller than r from x , then theclosest s points are considered instead. The data points toward which each point tendsare returned as output. Typical choices for the parameters q , s and r are q = 0 . s = 30and r = 0 . 05. The parameter r does not significantly influence the output of Algorithm1. For the distributions and data considered in this manuscript, it is fixed to be r = 0 . s plays an important role in Algorithm 1, except whenthere is one true cluster in which case the choice of s does not matter, provided it is nottoo small. However, if the number of true clusters is at least 2, then s cannot be too bigor Algorithm 1 will return only one cluster. If the number of true clusters is large, say,10 or 20, then smaller values of s should be used.The quantity s can also play the role of a smoothing parameter. If q (and hence τ ) issmall with a small sample size n , then the sample local depth can be noisy and have localpeaks with a small basin of attraction that were not present in the original distribution.In this case, the choice of a larger s helps to avoid these local maxima. We begin this section with a one-dimensional example showing the flexibility of the τ -approximation for different values of τ . As described in Section 2, for small values of τ , f τ ( · ) “resembles” the underlying density, while for larger τ it becomes unimodal, as DFsare decreasing from the median of the distribution. We take this univariate distributionto be a mixture of four normal distributions with means − 2, 0, 3, 4, standard deviations0 . 5, 0 . 8, 0 . 5, 0 . . 25, 0 . 5, 0 . 15 and 0 . 1, respectively. The resulting densityis quadrimodal and is depicted in Figure 2 along with its sample τ -approximation for τ = 0 . , , , 4. As can be seen from the Figure 2, for τ = 0 . τ = 1,the clusters corresponding to the modes at x = 3 and x = 4 merge yielding only three24 lgorithm 1: Clustering with local depth Input: { x , . . . , x n } , { y , . . . , y o } (optional), q , s , r Output: Local maxima for input points: { z , . . . , z n + o } Compute the quantile τ of order q of all pairwise distances: (cid:107) x i − x j (cid:107) , i > j , i, j ∈ { , , . . . , n } Compute the local depth of { x , . . . , x n } with the localization parameter τ Store { x , . . . , x n } , { y , . . . , y o } in new variables for i = 1 to n do z ∗ i := x i endfor i = 1 to o do z ∗ i + n := y i end For all points, compute the corresponding local maxima for i = 1 to n + o dorepeat z i := z ∗ i Store the data points (different from z i ) at distance from z i smaller than r or the s closest data points if they are less than s in new variables w , . . . , w l ( l ≥ s ) z ∗ i := argmax j =1 ,...,l d τ,n ( z i ; w j ) until LLD n ( z ∗ i , τ ) > LLD n ( z i , τ ) end τ to 2, we notice that one can still identify two clusters, while,for τ = 4, the τ -approximation has a unimodal shape. −4 −2 0 2 4 . . . x−5 −4 −3 −2 −1 0 1 2 3 4 5Density t = t = t = t = Figure 2: In black the quadrimodal mixture density and in red, green, blue and cyan itssample τ -approximation f τ,n ( · ) for τ = 0 . , , , 4, respectively, and n = 6000.Turning to bivariate examples studied in the literature (see Chac´on [2015]), we con-sider mixtures of bivariate normal distributions with the following characteristics: (i)two-mixture with equal weights (Bimodal) and identity covariance matrix and (ii) themixtures investigated in Wand and Jones [1993] and Chac´on [2009] referred to as (H)Bimodal IV, (K) Trimodal III and true clusters aregiven in Appendix G.1. We apply our algorithm to analyze these models and identifyclusters; these results are displayed in Figure 3 (second row). A comparison of our resultswith the clusters obtained using the kernel density estimator (obtained using the function kms in the R-package ks [Duong, 2018]) are provided in Figure 3 (third row). By a visualinspection of Figure 3, it is clear that a better cluster estimation is performed by LLDthan by KDE. A more detailed analysis of these and other distributions under extremelocalization is provided in Appendix G.2. In this subsection, we provide simulation results of our method for identification of clus-ters. We evaluate the performance in three different ways: (i) true number of clustersidentified by the algorithm, (ii) empirical Hausdorff distance between the “true” clusterand the estimated cluster, and (iii) empirical probability distance (see Chac´on [2015], forinstance). We recall that the symmetric difference between two clusters C, D ⊂ R p is C ∆ D = (( R p \ C ) ∩ D ) ∪ ( C ∩ ( R p \ D )). Let X , . . . , X n be i.i.d. samples from26rueclusters Bimodal −3 −2 −1 0 1 2 3 − − − ll (H) Bimodal IV −3 −2 −1 0 1 2 3 − − − l l −3 −2 −1 0 1 2 3 − − − l lll l LLDestimatedclusters ll ll ll ll l lll l l ll lll lll ll l ll ll l ll ll ll lll l ll l llll ll l ll l ll ll ll ll l ll l ll llllll ll lll ll lll lll l ll lll ll lllll l l lll l llll l ll ll l ll lll lll l ll ll l ll ll l ll l ll ll l ll l lll l ll llll l lll ll l lll l l l lll l ll ll l llll l ll l lll ll l ll l ll l ll lll ll ll ll ll l lll lll ll l ll lll ll ll llll ll ll lll l ll ll ll l lll ll ll l ll ll lll l l ll l ll ll lll lll l l l ll lll ll lll l ll ll llll l ll lll ll ll ll l l l ll lll l ll ll ll l ll lll ll ll l lllll l ll lll ll lll ll ll ll ll lll l l llll l l llll ll lll l ll l ll lll llll lll l llll lll lll lll ll ll lll ll lll llll l llll llll lll l ll ll ll lll ll lll ll ll ll l ll l ll ll l l lll l l ll l l lll l ll l ll lll l ll lll l l ll lll ll l ll llll lll l ll ll l ll l l lll ll ll l lll l l ll ll ll l lll lll lll ll lll l l ll ll lll ll llll l l llll ll l llll llll l ll ll lll llll ll llll lll ll lll lll ll ll lll lll l ll l lll lll l llll l lll ll l lllllll ll l lll ll ll lll ll l l lll l lll l ll ll ll ll lll ll l ll lll lll ll l lll ll ll ll llll lll l lll ll lll lll llll lll ll l ll ll ll llll l ll lllll ll ll l lll l lll llll lll l l llll ll ll ll l llll l l llll ll l l ll ll l ll ll lll l lllll ll ll ll ll l l lll lll ll llll lll ll l l llll ll llll ll ll l lll l lll ll l l ll l lll ll l lll ll ll lll ll l l l l ll lll llll l ll l lll l ll ll llll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − l l l ll lll ll l ll llll l lll l ll ll l ll ll l lllll ll l ll ll ll ll ll ll llll l lll lll l ll lll l l ll ll lll ll llll l l lllll ll l lllll llll l lll ll ll lll l llll ll llll lll ll lll lll ll ll l lll lll ll ll l llll lll l l llll l llll ll l ll llllll ll l lll ll ll lll ll l l lll l lll l ll ll ll llll lll l ll l ll lll lll lll l lll ll ll ll llll lll l llll ll lll ll ll l llll lll ll l ll ll lll llll l ll llll lll ll lll l ll ll l lll lllll lll l l llll ll ll llll lllll l l llll ll l l ll ll l l ll ll lll l lllll ll ll ll llll l l lll lll ll llll llll ll l l llll ll lllll l ll ll l l lll l l lll ll l l ll l lll ll l lll ll ll lll ll l ll l ll lll llllll l ll lllll l ll ll llll ll ll ll ll lll l lll l l lllll llll ll l ll ll l ll ll ll lll lll l l llll ll l lll ll ll ll ll l lll l llll lll ll lllll l lll l ll l ll ll l ll lllll l l llll ll ll ll lll ll l ll lll lll l ll ll l llll l ll lll ll l lll lll l ll lllll lll ll l lll ll l llll ll ll lll l ll lll 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llll l llll lll l l lllll llll lll ll llllll lll lll ll ll lll ll ll lll l llll llll ll llll lll l lll lllll lll llll lll ll ll ll llll lll l llll ll lll ll ll l llll lll ll l llll lllllll l ll llll lll ll lll lll ll l lll lllll lll ll llll ll ll llll lllll ll llll ll l l ll ll l l ll ll lll llllll ll llll llll l llll lll ll llll llll ll l lllllll lllll l ll ll l l lll ll lll ll l l ll l lll ll l lll ll llllllllll l lllll llllll lll lllll l ll llllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − lllll lll ll Figure 3: Clusters associated with the Bimodal (left), (H) Bimocal IV (middle) and n = 1000 samples from these densities and parameters q = 0 . s = 50 and r = 0 . P . The empirical probability distance between the clus-ters C = { C , . . . , C l } and D = { D , . . . , D s } with l < s is given byˆ d P,η ( C , D ) = 12 n min π ∈P s (cid:32) l (cid:88) i =1 n (cid:88) j =1 I ( X j ∈ C i ∆ D π ( i ) ) + η s (cid:88) i = l +1 n (cid:88) j =1 I ( X j ∈ D π ( i ) ) (cid:33) , where P s is the set of all permutations of { , . . . , l } and η ≥ l = s the second term in the aboveexpression is zero. In our numerical experiments, we choose η = 1. Additional results forother values of η are provided in Appendix G.3.The empirical Hausdorff distance ˆ d H ( C , D ) is given by1 n max (cid:32) max i ∈{ ,...,l } min j ∈{ ,...,s } n (cid:88) k =1 I ( X k ∈ C i ∆ D j ) , max j ∈{ ,...,s } min i ∈{ ,...,l } n (cid:88) k =1 I ( X k ∈ C i ∆ D j ) (cid:33) . In applications, C is taken to be the set of true clusters while D is the estimated clus-ter, produced by the algorithm. If the estimated clusters coincide with the true clusters,then both these distances, viz . the clustering errors , are zero. Thus, small values of thesedistances suggest a good performance. In this manuscript, as explained before, we con-sider the following distributions commonly used in the literature: Bimodal, (H) BimodalIV, (K) Trimodal III and τ . We choose τ so that the corresponding quantiles q are given by 0 . 01, 0 . . q , asdescribed in Subsection 4.3. Clustering errors (Hausdorff distance) (H) Bimodal IV (K) Trimodal III LSD Hclust * LSD * Clustering errors (distance in probability) (H) Bimodal IV (K) Trimodal III LSD Hclust * LSD * * The true number of clusters is given in input. q = 0 . s = 30. q = 0 . s = 30. q = 0 . s = 50. q = 0 . s = 30.Table 1: Mean of the clustering errors based on the Hausdorff distance and the distance inprobability for the densities (H) Bimodal IV, (K) Trimodal III, umber of times the true clusters are detected correctly (H) Bimodal IV (K) Trimodal III (0) 83 (17) (14) 79 (7) (0) 100 (0) LSD (0) 85 (15) (13) 75 (12) (0) 100 (0) Bimodal Mult. Bimodal Mult. QuadrimodalKDE (0) 76 (24) (0) 56 (44) (0) 63 (37)LLD (0) 99 (1) (0) 99 (1) (0) 100 (0) LSD (0) 100 (0) (12) 63 (25) (77) 18 (5)Table 2: Number of times that the procedure identifies the true number of clusters forthe densities (H) Bimodal IV, (K) Trimodal III , In this subsection, we evaluate the performance of our methodology on two datasetstaken from the UCI machine learning repository ( http://archive.ics.uci.edu/ml/ ),namely, the Iris dataset and the Seeds dataset. As is well-known, the Iris dataset consistsof n = 150 observations from three classes (Iris Setosa, Iris Versicolour, and Iris Virginica)with four measurements each (sepal length, sepal width, petal length, and petal width).We compare our results to those based on KDE and Hclust. Our algorithm, based on bothlens and simplicial depth, correctly identifies all three clusters (see Table 3); furthermore,the Hausdorff distance and probability distance from our algorithm are smaller than thoseof the competitors.Seeds dataset consists of n = 210 observations concerning three varieties of wheat;namely, Kama, Rosa and Canadian. High quality visualization of the internal kernelstructure was detected using a soft X-ray technique and seven geometric parameters ofwheat kernels were recorded. They are area, perimeter, compactness, length of kernel,width of kernel, asymmetry coefficient, and length of kernel groove. All of these geometricparameters were continuous and real-valued. Table 3 contains the results of our analysis.The best results are highlighted in bold and correspond to LLD. We notice that both ofour methods, LLD and LSD, correctly identify the true number of clusters.It is worth mentioning here that Hclust was given as input the true number of clusters,three, as required by this methodology. However, the Haudorff distance and probability30istance of our proposed methods are smaller than those of Hclust. KDE, in both theexamples, overestimates the true number of clusters. Clustering errors for Iris data Number of clusters Distance in prob. Hausdorff distanceKDE 7 0.37 0.30LLD LSD Hclust * Clustering errors for Seeds data Number of clusters Distance in prob. Hausdorff distanceKDE 23 0.70 0.33LLD LSD * q = 10 − , s = 20. q = 10 − , s = 20.Table 3: Mean of the clustering errors based on the Hausdorff distance and distance inprobability for the Iris and Seeds data. The true number of clusters is specified as inputfor the hierarchical clustering algorithm. We have studied the properties of local depth in the multivariate setting. Our results in-clude uniform convergence and uniform central limit theorem for the sample local depth.Based on the local depth, we proposed a population and sample approximation of thedensity that converges uniformly and uniformly almost surely to the true density. Wemade use of it in a modal clustering approach (via a gradient system) where the den-sity is replaced by the population approximation. Convergence results show that theapproximated approach provides, in the limit, the same clusters as those given by thetrue density. In particular, we have shown that, for symmetric distributions, our approx-imated approach correctly detects the true modes. Finally, we proposed an algorithm forthe numerical computation of the clusters at sample level.Yet another work on local depth is that of Paindaveine and Van Bever [2013] andis based on the concept of symmetric neighborhoods. It is legitimate to wonder if themethods proposed in the current paper go over to this alternative notion. While we believe31uch an extension is feasible, we have not pursued all the technical details required forsuch analysis and is left for future research. Additionally, it will be interesting to studya smoothed version of the sample approximation and replace the gradient system by thestochastic differential equation based on this approximation, and consequently study theconvergence of the corresponding solutions. References Ratan Prakash Agarwal and V. Lakshmikantham. Uniqueness and nonuniqueness criteriafor ordinary differential equations , volume 6. World Scientific, 1993.Claudio Agostinelli. Local half-region depth for functional data. Journal of MultivariateAnalysis , 163:67–79, 2018.Claudio Agostinelli and Mario Romanazzi. Local depth of multidimensional data. WorkingPaper 3, Ca’ Foscari University of Venice, 2008.Claudio Agostinelli and Mario Romanazzi. Local depth. Journal of Statistical Planningand Inference , 141(2):817–830, 2011.Kenneth S. Alexander. 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Structural properties and convergence results for contoursof sample statistical depth functions. The Annals of Statistics , 28(2):483–499, 2000b. A Proofs We begin this section summarizing relevant properties of Z τ ( x ) = { ( y , y ) ∈ R p × R p :max i =1 , (cid:107) x − y i (cid:107) ≤ (cid:107) y − y (cid:107) ≤ τ } that are required in the proof of several main resultson the paper. In the proofs we will use P ⊗ k for the k -fold product measure. Lemma A.1 For x ∈ R p and τ ∈ [0 , ∞ ] , Z τ ( x ) is a closed set and its interior, exteriorand boundary are given by ˚ Z τ ( x ) = { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) < (cid:107) y − y (cid:107) < τ } Z eτ ( x ) = { ( y , y ) ∈ R p × R p : (cid:107) y − y (cid:107) > τ }∪ { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) > (cid:107) y − y (cid:107) ≤ τ } ∂Z τ ( x ) = { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) ≤ (cid:107) y − y (cid:107) = τ }∪ { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) = (cid:107) y − y (cid:107) < τ } . roof. For the closeness, let ( y ,n , y ,n ) ∈ Z τ ( x ) and suppose that ( y ,n , y ,n ) −−−→ n →∞ ( y , y ). We will show that ( y , y ) ∈ Z τ ( x ). Since max i =1 , (cid:107) x − y i,n (cid:107) ≤ (cid:107) y ,n − y ,n (cid:107) ≤ τ ,for all n ∈ N , it follows that lim n →∞ max i =1 , (cid:107) x − y i,n (cid:107) ≤ lim n →∞ (cid:107) y ,n − y ,n (cid:107) ≤ τ . Usingthe continuity of the norm we get max i =1 , (cid:107) x − y i (cid:107) ≤ (cid:107) y − y (cid:107) ≤ τ , i.e. ( y , y ) ∈ Z τ ( x ).Next, notice that Z eτ ( x ) = ( R p × R p ) \ Z τ ( x ) = ( R p × R p ) \ Z τ ( x )= { ( y , y ) ∈ R p × R p : (cid:107) y − y (cid:107) > τ ∨ max i =1 , (cid:107) x − y i (cid:107) > (cid:107) y − y (cid:107) ≤ τ } = { ( y , y ) ∈ R p × R p : (cid:107) y − y (cid:107) > τ }∪ { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) > (cid:107) y − y (cid:107) ≤ τ } . Now, by the definition of the boundary we have that ∂Z τ ( x ) = Z τ ( x ) \ (( R p × R p ) \ Z τ ( x )) = Z τ ( x ) ∩ Z eτ ( x )where Z eτ ( x ) = { ( y , y ) ∈ R p × R p : (cid:107) y − y (cid:107) ≥ τ }∪ { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) ≥ (cid:107) y − y (cid:107) ≤ τ } . Therefore, ∂Z τ ( x ) = { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) ≤ (cid:107) y − y (cid:107) = τ }∪ { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) = (cid:107) y − y (cid:107) ≤ τ } = { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) ≤ (cid:107) y − y (cid:107) = τ }∪ { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) = (cid:107) y − y (cid:107) < τ } . Finally, the interior of Z τ ( x ) is˚ Z τ ( x ) = Z τ ( x ) \ ∂Z τ ( x )= { ( y , y ) ∈ R p × R p : max i =1 , (cid:107) x − y i (cid:107) < (cid:107) y − y (cid:107) < τ } . The next lemma deals with symmetry properties of the set Z τ ( x ). Its proof follows directlyfrom the definition of Z τ ( x ) and it is omitted.36 emma A.2 For x ∈ R p and τ ∈ [0 , ∞ ] , Z τ ( x ) satisfies the following symmetry condi-tions ( x, x ) + ( x , x ) ∈ Z τ ( x ) ⇐⇒ ( x, x ) − ( x , x ) ∈ Z τ ( x ) (A.1)( x, x ) + ( x , x ) ∈ Z τ ( x ) ⇐⇒ ( x, x ) + ( x , x ) ∈ Z τ ( x ) . (A.2) In particular, (A.1) and (A.2) imply that ( x, x ) + ( x , x ) ∈ Z τ ( x ) ⇐⇒ ( x, x ) − ( x , x ) ∈ Z τ ( x ) . We now proceed with the proofs of the results in the previous sections. Proof of Proposition 2.1. We start by proving (i). For the monotonicity, observethat for ν ≥ τ , Z ν ( x ) ⊃ Z τ ( x ) and therefore LLD ( x, ν ) ≥ LLD ( x, τ ). Now, let { τ n } ∞ n =1 be a sequence of scalars converging to τ from above. Then, since Z τ ( x ) is closed,lim n →∞ I (( x , x ) ∈ Z τ n ( x )) = I (( x , x ) ∈ Z τ ( x )) . Hence, by Lebesgue dominated convergence Theorem (LDCT) we get thatlim n →∞ LLD ( x, τ n ) = (cid:90) lim n →∞ I (( x , x ) ∈ Z τ n ( x )) dP ( x ) dP ( x )= (cid:90) I (( x , x ) ∈ Z τ ( x )) dP ( x ) dP ( x ) = LLD ( x, τ ) . In particular, for τ = 0, it follows thatlim n →∞ LLD ( x, τ n ) = LLD ( x, 0) = P ( { x } ) . Similarly, if { τ n } ∞ n =1 diverges to ∞ , thenlim n →∞ LLD ( x, τ n ) = (cid:90) lim n →∞ I (( x , x ) ∈ Z τ n ( x )) dP ( x ) dP ( x )= (cid:90) I (( x , x ) ∈ Z ( x )) dP ( x ) dP ( x ) = LD ( x ) . For (ii) observe thatsup x ∈ R p : (cid:107) x (cid:107)≥ M LLD ( x, τ ) ≤ (cid:90) sup x ∈ R p : (cid:107) x (cid:107)≥ M I (( x , x ) ∈ Z τ ( x )) dP ( x ) dP ( x ) . Since for ( x , x ) ∈ Z τ ( x ), it follows that (cid:107) x (cid:107) ≤ max i =1 , ( (cid:107) x i (cid:107) + (cid:107) x i − x (cid:107) ) ≤ max i =1 , (cid:107) x i (cid:107) + (cid:107) x − x (cid:107) ≤ i =1 , (cid:107) x i (cid:107) , 37e have thatsup x ∈ R p : (cid:107) x (cid:107)≥ M I (( x , x ) ∈ Z τ ( x )) ≤ sup x ∈ R p : (cid:107) x (cid:107)≥ M I ( (cid:107) x (cid:107) ≤ i =1 , (cid:107) x i (cid:107) ) ≤ I ( M ≤ i =1 , (cid:107) x i (cid:107) ) , where I ( M ≤ i =1 , (cid:107) x i (cid:107) ) converges to 0 as M → ∞ . Now, (ii) follows from LDCT.We now prove (iii). For the continuity of LLD at ( x, τ ) ∈ R p × [0 , ∞ ), we consider for( y, ν ) ∈ R p × [0 , ∞ ) the difference | LLD ( x, τ ) − LLD ( y, ν ) | ≤ (cid:90) | I (( x , x ) ∈ Z τ ( x )) − I (( x , x ) ∈ Z ν ( y )) | dP ( x ) dP ( x ) . The result follows, using LDCT, if we can show that the absolute value of the difference ofthe indicators inside the integral converges pointwise to 0 almost surely as ( y, ν ) convergesto ( x, τ ). This in turn follows from the fact that for ( x , x ) ∈ R p × R p fixed, if ( x , x ) / ∈ ∂Z τ ( x ) (by the absolute continuity P ⊗ ( ∂Z τ ( x )) = 0), then there exists (cid:15) > y ∈ R p with (cid:107) x − y (cid:107) ≤ (cid:15) and all ν ∈ [0 , ∞ ) with | τ − ν | ≤ (cid:15) , I (( x , x ) ∈ Z τ ( x )) = I (( x , x ) ∈ Z ν ( y )).We now establish this as follows. Recall the interior, exterior and boundary of Z τ ( x )as given in Lemma A.1. If ( x , x ) ∈ ˚ Z τ ( x ), then max i =1 , (cid:107) x − x i (cid:107) < (cid:107) x − x (cid:107) < τ .Since, (cid:107) y − x i (cid:107) ≤ (cid:107) y − x (cid:107) + (cid:107) x − x i (cid:107) , for i = 1 , 2, it follows that max i =1 , (cid:107) y − x i (cid:107) < (cid:107) x − x (cid:107) < ν by taking (cid:15) < min ( (cid:107) x − x (cid:107) − max i =1 , (cid:107) x − x i (cid:107) , τ − (cid:107) x − x (cid:107) ); thisyields ( x , x ) ∈ ˚ Z ν ( y ). On the other hand, if ( x , x ) ∈ Z eτ ( x ), then either (cid:107) x − x (cid:107) > τ or max i =1 , (cid:107) x − x i (cid:107) > (cid:107) x − x (cid:107) ≤ τ . In the first case, by taking 0 < (cid:15) < (cid:107) x − x (cid:107) − τ we see that also (cid:107) x − x (cid:107) > ν . In the second case, (cid:107) x − x i (cid:107) > (cid:107) x − x (cid:107) for i = 1 or i = 2, and since (cid:107) y − x i (cid:107) ≥ |(cid:107) x − x i (cid:107) − (cid:107) y − x (cid:107)| , by taking 0 < (cid:15) < (cid:107) x − x i (cid:107) − (cid:107) x − x (cid:107) it follows that (cid:107) y − x i (cid:107) > (cid:107) x − x (cid:107) . In both cases we conclude that ( x , x ) ∈ Z eν ( y ).We now show that LLD ( · , τ ) is upper semicontinuous. Since, for x ∈ R p , Z τ ( x )is a closed set, it follows as above that, if ( x , x ) ∈ Z eτ ( x ), then lim sup y → x I (( x , x ) ∈ Z τ ( y )) = 0. Therefore, lim sup y → x I (( x , x ) ∈ Z τ ( y )) ≤ I (( x , x ) ∈ Z τ ( x )), and byLDCT lim sup y → x LLD ( y, τ ) ≤ (cid:90) lim sup y → x I (( x , x ) ∈ Z τ ( y )) dP ( x ) dP ( x ) ≤ (cid:90) I (( x , x ) ∈ Z τ ( x )) dP ( x ) dP ( x ) = LLD ( x, τ ) . The proof of (iv) is rather routine and, for the skeptic, it is provided in Appendix F.38 roof of Theorem 2.1. A direct proof of (i), when the density f ( · ) is continuous, isobtained from (2.2). Since Z (0) is compact, it follows that for (cid:15) > τ ∈ [0 ,(cid:15) ] sup ( x ,x ) ∈ Z (0) f ( x + τ x ) f ( x + τ x ) < ∞ . Hence, from LDCT it follows thatlim τ → + τ p LLD ( x, τ ) = 1Λ (cid:90) Z (0) lim τ → + f ( x + τ x ) f ( x + τ x ) dx dx = f ( x ) . The general statement follows from Lebesgue differentiation Theorem [Benedetto andCzaja, 2010], since Z τ ( x ) ⊂ B τ ( x ) × B τ ( x ) ⊂ B ⊗ √ τ ( x ), where B ⊗ √ τ ( x ) := { ( w, z ) ∈ R p × R p : (cid:107) w − x (cid:107) + (cid:107) z − x (cid:107) ≤ τ } is the closed ball in R p × R p with center ( x, x ) andradius √ τ , and λ ⊗ ( Z τ ( x )) λ ⊗ ( B ⊗ √ τ ( x )) = λ ⊗ ( Z (0))2 p λ ⊗ ( B ⊗ (0)) = Λ p !2 p π p > . For (ii), notice from (2.2) that1 τ p LLD ( x, τ ) − Λ f ( x ) = (cid:90) Z (0) (cid:2) f ( x + τ x ) f ( x + τ x ) − f ( x ) (cid:3) dx dx . (A.4)Since f ( · ) is twice differentiable, by multivariate Taylor’s expansion, for i = 1 , f ( x + τ x i ) = f ( x ) + τ (cid:104)∇ f ( x ) , x i (cid:105) + τ x (cid:62) i H f ( x ) x i + O ( τ ) . Therefore, f ( x + τ x ) f ( x + τ x ) = f ( x ) + τ f ( x ) (cid:104)∇ f ( x ) , x + x (cid:105) + τ f ( x ) (cid:2) x (cid:62) H f ( x ) x + x (cid:62) H f ( x ) x (cid:3) + τ (cid:104)∇ f ( x ) , x (cid:105)(cid:104)∇ f ( x ) , x (cid:105) + O ( τ ) . (A.5)The continuity of the third order partial derivatives implies that the remainder O ( τ ) isa bounded and continuous function of x and x for τ ∈ [0 , τ ]. By substituting (A.5) in(A.4), we see that1 τ p LLD ( x, τ ) − Λ f ( x ) = τ f ( x ) (cid:90) Z (0) (cid:104)∇ f ( x ) , x + x (cid:105) dx dx + τ f ( x ) (cid:90) Z (0) (cid:2) x (cid:62) H f ( x ) x + x (cid:62) H f ( x ) x (cid:3) dx dx + τ (cid:90) Z (0) (cid:104)∇ f ( x ) , x (cid:105)(cid:104)∇ f ( x ) , x (cid:105) dx dx + O ( τ ) . x , x ) ∈ Z (0) ⇐⇒ ( − x , − x ) ∈ Z (0) , and by this change of variable (cid:90) Z (0) (cid:104)∇ f ( x ) , x + x (cid:105) dx dx = − (cid:90) Z (0) (cid:104)∇ f ( x ) , x + x (cid:105) dx dx . Therefore (cid:90) Z (0) (cid:104)∇ f ( x ) , x + x (cid:105) dx dx = 0 . Lemma A.2 also implies that( x , x ) ∈ Z (0) ⇐⇒ ( x , x ) ∈ Z (0) , and therefore (cid:90) Z (0) x (cid:62) H f ( x ) x dx dx = (cid:90) Z (0) x (cid:62) H f ( x ) x dx dx . We conclude that lim τ → + τ (cid:18) τ p LLD ( x, τ ) − Λ f ( x ) (cid:19) = h ( x ) , where h ( x ) = f ( x ) (cid:90) Z (0) x (cid:62) H f ( x ) x dx dx + (cid:90) Z (0) (cid:104)∇ f ( x ) , x (cid:105)(cid:104)∇ f ( x ) , x (cid:105) dx dx . Proof of Corollary 2.1. From (2.2), we see that LLD ( x, τ ) = (cid:90) Z τ (0) f ( x + x ) f ( x + x ) dx dx . In two dimensions Z τ (0) can be expressed as the union of two triangles T τ − + and T τ + − ;that is, T τ − + := { ( x , x ) ∈ R : x ≤ , x ≥ , x − x ≤ τ } T τ + − := { ( x , x ) ∈ R : x ≥ , x ≤ , x − x ≤ τ } . Now, by a change of variables in the integrals over the triangles it follows that LLD ( x, τ ) = (cid:90) T τ − + f ( x + x ) f ( x + x ) + f ( x − x ) f ( x − x ) dx dx = (cid:90) T τ ++ f ( x − x ) f ( x + x ) + f ( x + x ) f ( x − x ) dx dx = 2 (cid:90) T τ ++ f ( x + x ) f ( x − x ) dx dx . Z (0) has area 1. Proof of Theorem 2.2. We start by proving (i). Observe that (cid:12)(cid:12)(cid:12) √ s − √ t (cid:12)(cid:12)(cid:12) ≤ | s − t | for all s, t ≥ . (A.6)Therefore, for τ > x ∈ R p | f τ ( x ) − f ( x ) | = sup x ∈ R p (cid:12)(cid:12)(cid:12)(cid:12)(cid:114) τ p Λ LLD ( x, τ ) − (cid:112) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup x ∈ R p (cid:12)(cid:12)(cid:12)(cid:12) τ p Λ LLD ( x, τ ) − f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = sup x ∈ R p (cid:112) G τ ( x ) , where G τ ( x ) := (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Z (0) (cid:2) f ( x + τ x ) f ( x + τ x ) − f ( x ) (cid:3) dx dx (cid:12)(cid:12)(cid:12)(cid:12) . Since the square root is a continuous function, by definition of supremum there existsequences { x k } ∞ k =1 such thatsup x ∈ R p (cid:112) G τ ( x ) = lim sup k →∞ (cid:112) G τ ( x k ) ≤ (cid:114) lim sup k →∞ G τ ( x k ) ≤ (cid:114) sup x ∈ R p G τ ( x ) . To complete the proof, it is enough to show thatlim τ → + sup x ∈ R p G τ ( x ) = 0 . (A.7)For this, observe thatsup x ∈ R p G τ ( x ) ≤ (cid:90) Z (0) sup x ∈ R p (cid:12)(cid:12) f ( x + τ x ) f ( x + τ x ) − f ( x ) (cid:12)(cid:12) dx dx . Since f ( · ) is uniformly continuous and bounded, for all ( x , x )lim τ → + sup x ∈ R p (cid:12)(cid:12) f ( x + τ x ) f ( x + τ x ) − f ( x ) (cid:12)(cid:12) = 0 . (A.7) now follows from LDCT, since Z (0) is compact and the supremum is bounded.Since a continuous function is uniformly continuous on a compact set, the proof of thefirst part of (ii) follows from the proof of (i) with R p replaced by K .For the second part of (ii), notice thatsup y ∈ B (cid:15) ( x ) | f τ ( y ) − f ( x ) | ≤ sup y ∈ B (cid:15) ( x ) | f τ ( y ) − f ( y ) | + sup y ∈ B (cid:15) ( x ) | f ( y ) − f ( x ) | . f ( · ). Proof of Corollary 2.2. The result follows from Proposition 2.1, where the probabilitymeasure P ⊗ is replaced by P ⊗ n := ( n ) (cid:80) ≤ i Let F := {K x,τ : x ∈ R p , τ ∈ [0 , ∞ ] } , where K x,τ ( x , x ) = I (( x , x ) ∈ Z τ ( x )) = I ( x ∈ L ( x , x )) I ( (cid:107) x − x (cid:107) ≤ τ )and L ( x , x ) := { y ∈ R p : max i =1 , (cid:107) y − x i (cid:107) ≤ (cid:107) x − x (cid:107)} . We will show thatsup f ∈F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) f ( x , x ) dP ( x ) dP ( x ) − n ( n − n (cid:88) i,j =1 i To prove Theorem 2.4, we will verify the conditions of Theorem4.9 in Arcones and Gin´e [1993]. To this end, first let F := {K x,τ : ( x, τ ) ∈ Γ } , where K x,τ is given in (2.7) and F (cid:48) = {J x,τ : ( x, τ ) ∈ Γ } , where J x,τ is given in (2.8). Notice that,sup f ∈F | f ( · ) | ≤ 1, sup f (cid:48) ∈F (cid:48) | f (cid:48) ( · ) | ≤ F is image admissible Suslin [Dudley, 2014, p.186]. This then shows that F is a measurable class with a bounded envelope and (ii) ofTheorem 4.9 in Arcones and Gin´e [1993] holds. To verify (iii) in Arcones and Gin´e [1993]we appeal to Lemma 4.4 and (4.2) in Alexander [1987] concerning the covering number N ( F , d L ( F ,P ) , · ) of F with respect to the L -distance, d L ( F ,P ) , given by d L ( F ,P ) (( x, τ ) , ( y, ν )) = (cid:18)(cid:90) ( K x,τ ( x , x ) − K y,ν ( x , x )) dP ( x ) dP ( x ) (cid:19) . For this, we observe that F is a VC-subgraph class as in the proof of Theorem 2.3. Thusto complete the proof, we need to verify (i) in Arcones and Gin´e [1993]. To this end, weneed to show: (a) the finite dimensional distributions of √ n ( LLD n ( x, P, τ ) − LLD ( x, p, τ ))converges to a multivariate normal distribution and (b) for each ( x, τ ), the limiting normalrandom variable { W ( x, τ ) } ( x,τ ) ∈ Γ admits a version whose sample paths are all boundedand uniformly continuous with respect to the distance d F (cid:48) ,P on F (cid:48) given by d F (cid:48) ,P (( x, τ ) , ( y, ν )) = (cid:90) ( J x,τ ( x ) − J y,ν ( x )) dP ( x ) − ( LLD ( x, τ ) − LLD ( y, ν )) , where we identify a function J x,τ for ( x, τ ) ∈ Γ with its parameter ( x, τ ). In this sense, d F (cid:48) ,P is a metric on Γ.Since W ( x, τ ) is Gaussian, we can apply Gin´e and Nickl [2016, Theorem 2.3.7] with T = Γ and d = d F (cid:48) ,P . First, note that { W ( x, τ ) } ( x,τ ) ∈ Γ is a sub-Gaussian process relativeto d F (cid:48) ,P . Indeed, using Proposition A.1, for ( x, τ ) , ( y, ν ) ∈ Γ, ( W ( x, τ ) , W ( y, ν )) (cid:62) has abivariate normal distribution with mean (0 , (cid:62) and covariance matrix (cid:32) b ( x, τ ) γ (( x, τ ) , ( y, ν )) γ (( y, ν ) , ( x, τ )) b ( y, ν ) . (cid:33) . It follows that W ( x, τ ) − W ( y, ν ) is normally distributed with mean 0 and variance b ( x, τ ) + b ( y, ν ) − γ (( x, τ ) , ( y, ν )) = d F (cid:48) ,P (( x, τ ) , ( y, ν )). Therefore, for all α ∈ R E [ e α ( W ( x,τ ) − W ( y,ν )) ] = e α d F(cid:48) ,P (( x,τ ) , ( y,ν ))2 { W ( x, τ ) } ( x,τ ) ∈ Γ is sub-Gaussian with respect to d F (cid:48) ,P .We next verify the integrability condition for the metric entropy. To this end, noticethat, for ( x, τ ) , ( y, ν ) ∈ Γ, the L -distance on F (cid:48) , d L ( F (cid:48) ,P ) is given by d L ( F (cid:48) ,P ) (( x, τ ) , ( y, ν )) = (cid:18)(cid:90) ( J x,τ ( x ) − J y,ν ( x )) dP ( x ) (cid:19) . Now using yet another application of Lemma 4.4 of Alexander [1987], it follows that thereare constants C , C > N ( F , d L ( F ,P ) , √ (cid:15) ) ≤ (cid:18) C √ (cid:15) (cid:19) C . By Jensen’s inequality, it follows that d L ( F (cid:48) ,P ) (( x, τ ) , ( y, ν )) ≤ d L ( F ,P ) (( x, τ ) , ( y, ν )) , which in turn, implies that N ( F (cid:48) , d L ( F (cid:48) ,P ) , √ (cid:15) ) ≤ N ( F , d L ( F ,P ) , √ (cid:15) ) ≤ (cid:18) C √ (cid:15) (cid:19) C , where N ( F (cid:48) , d L ( F (cid:48) ,P ) , √ (cid:15) ) is the covering number of F (cid:48) with respect to the distance d L ( F (cid:48) ,P ) . Thus, for any 0 < (cid:15) ≤ N ( F (cid:48) , d F (cid:48) ,P , (cid:15) ) ≤ N ( F (cid:48) , d L ( F (cid:48) ,P ) , (cid:15) ) = N ( F (cid:48) , d L ( F (cid:48) ,P ) , √ (cid:15) ) ≤ (cid:18) C √ (cid:15) (cid:19) C ≤ (cid:18) C (cid:15) (cid:19) C . Hence, it follows that (cid:90) (cid:113) log( N ( F (cid:48) , d F (cid:48) ,P , (cid:15) )) d(cid:15) ≤ (cid:112) C (cid:90) (cid:112) log( C ) − log( (cid:15) ) d(cid:15) ≤ (cid:112) C (cid:32)(cid:112) log( C ) + (cid:90) e − (cid:112) − log( (cid:15) ) d(cid:15) + (cid:90) e − (cid:112) − log( (cid:15) ) d(cid:15) (cid:33) ≤ (cid:112) C (cid:32)(cid:112) log( C ) + (cid:90) e − − log( (cid:15) ) d(cid:15) + 1 − e − (cid:33) = (cid:112) C (cid:16)(cid:112) log( C ) + e − + 1 (cid:17) < ∞ , where we have used that for a, b ≥ √ a + b ≤ √ a + √ b . The proof finally follows fromProposition A.1. given below. 44efore we state the following proposition, we recall that Γ is a subset of R p × [0 , ∞ ] suchthat, for ( x, τ ) ∈ Γ, b ( x, τ ) = V ar [ J x,τ ( X )] > 0. For k ≥ x , τ ) , . . . , ( x k , τ k ) ∈ Γ,we also use the notations LLD n ( x l , τ l ) := ( LLD n ( x , τ ) , . . . , LLD n ( x k , τ k )) (cid:62) , LLD ( x l , τ l ) :=( LLD ( x , τ ) , . . . , LLD ( x k , τ k )) (cid:62) and J x l ,τ l ( X j ) = ( J x ,τ ( X j ) , . . . , J x k ,τ k ( X j )) (cid:62) . Proposition A.1 For ( x , τ ) , . . . , ( x k , τ k ) ∈ Γ , √ n ( LLD n ( x l , τ l ) − LLD ( x l , τ l )) con-verges in distribution to a k -variate normal distribution with mean and covariance matrixwhose ( l , l ) -element is given by γ (( x l , τ l ) , ( x l , τ l )) , where l , l = 1 , . . . , k . Proof of Proposition A.1. Using Hoeffding’s decomposition of U-statistics (Korolyukand Borovskich [2013, Equation 1.1.22]), it follows that LLD n ( x, τ ) − LLD ( x, τ ) = 2 n n (cid:88) j =1 [ J x,τ ( X j ) − LLD ( x, τ )]+ 1 (cid:0) n (cid:1) (cid:88) ≤ i LLD n ( x l , τ l ) − LLD ( x l , τ l ) = 2 n n (cid:88) j =1 (cid:2) J x l ,τ l ( X j ) − LLD ( x l , τ l ) (cid:3) + R n , where R n = R n ( X , . . . , X n ) := ( n ) (cid:80) ≤ i For (i), observe thatsup x ∈ R p | f τ,n ( x ) − f ( x ) | ≤ sup x ∈ R p | f τ,n ( x ) − f τ ( x ) | + sup x ∈ R p | f τ ( x ) − f ( x ) | and, by Theorem 2.2 (i), it is enough to show thatlim n →∞ sup x ∈ R p τ ∈ (0 ,T ] | f τ,n ( x ) − f τ ( x ) | =1 τ p √ Λ lim n →∞ sup x ∈ R p (cid:12)(cid:12)(cid:12)(cid:112) LLD n ( x, τ ) − (cid:112) LLD ( x, τ ) (cid:12)(cid:12)(cid:12) −−−→ n →∞ x ∈ R p (cid:12)(cid:12)(cid:12)(cid:112) LLD n ( x, τ ) − (cid:112) LLD ( x, τ ) (cid:12)(cid:12)(cid:12) ≤ sup x ∈ R p | LLD n ( x, τ ) − LLD ( x, τ ) | , and setting G τ,n ( x ) := | LLD n ( x, τ ) − LLD ( x, τ ) | , notice thatsup x ∈ R p (cid:113) G τ,n ( x ) ≤ (cid:114) sup x ∈ R p G τ,n ( x ) . By Theorem 2.3, sup x ∈ R p G τ,n ( x ) converges to 0 almost surely as n → ∞ , and since thesquare root is a continuous function, (i) follows from the continuous mapping theorem.Since a continuous function on a compact set is uniformly continuous, the proof of thefirst part of (ii) follows from the proof of (i) with R p replaced by K .For the second part of the proof of (ii), notice that for all x ∈ R p sup y ∈ B (cid:15) ( x ) | f τ,n ( y ) − f ( x ) | ≤ sup y ∈ B (cid:15) ( x ) | f τ,n ( y ) − f τ ( y ) | + sup y ∈ B (cid:15) ( x ) | f τ ( y ) − f ( x ) | , where for all τ > (cid:15) > y ∈ B (cid:15) ( x ) | f τ,n ( y ) − f τ ( y ) | −−−→ n →∞ B (cid:15) ( x ), the first part of (ii), (2.12), and Theorem 2.2 (ii), (2.6),it follows that sup y ∈ B (cid:15) ( x ) | f τ ( y ) − f ( x ) | −−−−→ (cid:15),τ → + . Before proving Theorem 2.5, we provide a lemma concerning the order of convergence of b ( x, τ ) to 0, as τ → + . Lemma A.3 If f ( · ) is continuous, then lim τ → + b ( x, τ ) τ p = Λ ∗ f ( x ) , where Λ ∗ = (cid:90) λ ( Z (0) | x ) dx . Proof of Lemma A.3. Let τ > 0. We compute b ( x, τ ) τ p = 1 τ p (cid:90) (cid:32)(cid:90) Z τ ( x ) | x f ( x ) dx (cid:33) f ( x ) dx − (cid:18) LLD ( x, τ ) τ p (cid:19) , (A.11)where, by Theorem 2.1 (i), LLD ( x, τ ) τ p −−−→ τ → + Λ f ( x ) , and LLD ( x, τ ) τ p −−−→ τ → + . 47e now focus on the first term in (A.11). By changing variables twice, we note that1 τ p (cid:90) (cid:32)(cid:90) Z τ ( x ) | x f ( x ) dx (cid:33) f ( x ) dx = (cid:90) (cid:32) τ p (cid:90) Z τ ( x ) | x + τx f ( x ) dx (cid:33) f ( x + τ x ) dx = (cid:90) (cid:32)(cid:90) Z (0) | x f ( x + τ x ) dx (cid:33) f ( x + τ x ) dx . Since f ( · ) is continuous, it follows from LDCT thatlim τ → + (cid:90) Z (0) | x f ( x + τ x ) dx = f ( x ) λ ( Z (0) | x )and lim τ → + (cid:90) (cid:32)(cid:90) Z (0) | x f ( x + τ x ) dx (cid:33) f ( x + τ x ) dx = f ( x ) (cid:90) λ ( Z (0) | x ) dx . Proof of Theorem 2.5. Using Hoeffding’s decomposition of U-statistics ((A.8) with τ replaced by τ n ), it follows that LLD n ( x, τ n ) − LLD ( x, τ n ) = 2 1 n n (cid:88) j =1 [ J x,τ n ( X j ) − LLD ( x, τ n )]+ 1 (cid:0) n (cid:1) (cid:88) ≤ i 0, by Remark 2.3, b ( x, τ n ) > 0. Now,applying Lindeberg-Levy Theorem for triangular arrays [Billingsley, 2012, Theorem 27.2]with r n = n , s n = √ n b ( x, τ n ) and S n = n (cid:88) j =1 [ J x,τ n ( X j ) − LLD ( x, τ n )] , it follows that, √ n n n (cid:88) j =1 [ J x,τ n ( X j ) − LLD ( x, τ n )] /b ( x, τ n ) d −−−→ n →∞ N (0 , , (A.13)provided the Lindeberg condition [Billingsley, 2012, Equation (27.8)]lim n →∞ n b ( x, τ n ) n (cid:88) k =1 (cid:90) A (cid:15),n,k ( J x,τ k ( x ) − LLD ( x, τ k )) f ( x ) dx = 0 (A.14)48olds for all (cid:15) > 0, where A (cid:15),n,k,x := { x ∈ R p : |J x,τ k ( x ) − LLD ( x, τ k ) | ≥ (cid:15) √ n b ( x, τ n ) } . Notice that, due to √ n τ pn −−−→ n →∞ ∞ and Lemma A.3, √ n b ( x, τ n ) −−−→ n →∞ ∞ . Let n ∗ ∈ N be such that, for all n ≥ n ∗ , 1 < (cid:15) √ n b ( x, τ n ). Since |J x,τ k ( x ) − LLD ( x, τ k ) | ≤ x, x ∈ R p and k ∈ N , it follows that A (cid:15),n,k,x = ∅ , for all n ≥ n ∗ and all 1 ≤ k ≤ n . Thus,(A.14) holds true and we obtain (A.13). Finally, let R n = R n ( X , . . . , X n ) := 1 (cid:0) n (cid:1) (cid:88) ≤ i 1) ;equivalently, Z n := √ n τ pn Λ f τ n ( x ) b ( x, τ n ) ( f τ n ,n ( x ) − f τ n ( x )) d −−−→ n →∞ N (0 , . (A.17)To complete the proof, since x ∈ S and τ n > 0, by Remark 2.3 b ( x, τ n ) > 0. By Theorem2.1 (i), f τ n ( x ) f ( x ) = LLD ( x, τ n )Λ τ pn f ( x ) −−−→ n →∞ b ( x, τ n ) τ pn −−−→ n →∞ Λ ∗ f ( x ) > . (A.19)(A.18) and (A.19) imply that Y n := b ( x, τ n ) τ pn f τ n ( x ) · ∗ f ( x ) = b ( x, τ n ) τ pn · ∗ f ( x ) · f ( x ) f τ n ( x ) −−−→ n →∞ . From (A.17) and Slutsky’s Theorem it follows that Y n Z n d −−−→ n →∞ N (0 , , completing the proof. Proof of Corollary 2.4. We will show that f τ n ,n ( x ) f τ n ( x ) = (cid:112) LLD n ( x, τ n ) (cid:112) LLD ( x, τ n ) d −−−→ n →∞ . (A.20)For this, it is enough to verify that LLD n ( x, τ n ) LLD ( x, τ n ) d −−−→ n →∞ . (A.21)Notice that LLD n ( x, τ n ) LLD ( x, τ n ) = τ pn LLD ( x, τ n ) · LLD n ( x, τ n ) − LLD ( x, τ n ) τ pn + 1 , where, by Theorem 2.1 (i), LLD ( x, τ n ) τ pn −−−→ n →∞ Λ f ( x ) > . On the other hand, LLD n ( x, τ n ) − LLD ( x, τ n ) τ pn = b ( x, τ n ) τ pn · √ nτ pn · √ n LLD n ( x, τ n ) − LLD ( x, τ n ) b ( x, τ n ) , where √ n τ pn −−−→ n →∞ ∞ , by Lemma A.3, b ( x, τ n ) τ pn −−−→ n →∞ Λ ∗ f ( x ) > , and by (A.16) √ n LLD n ( x, τ n ) − LLD ( x, τ n ) b ( x, τ n ) d −−−→ n →∞ N (0 , . LLD n ( x, τ n ) − LLD ( x, τ n ) τ pn d −−−→ n →∞ , and, hence (A.21) and (A.20) hold. Now, (A.20) and (A.18) imply that f τ n ,n ( x ) f ( x ) = f τ n ,n ( x ) f τ n ( x ) · f τ n ( x ) f ( x ) d −−−→ n →∞ . (A.22)By Theorem 2.5 Z ∗ n := √ n τ pn (cid:112) f ( x ) ( f τ n ,n ( x ) − f τ n ( x )) d −−−→ n →∞ N (cid:18) , Λ ∗ Λ (cid:19) , where we can write f τ n ,n ( x ) − f τ n ( x ) as f τ n ,n ( x ) − f τ n ( x ) = (cid:18)(cid:113) f τ n ,n ( x ) − (cid:112) f τ n ( x ) (cid:19) (cid:18)(cid:113) f τ n ,n ( x ) + (cid:112) f τ n ( x ) (cid:19) . Also, by (A.18) and (A.22) Y ∗ n := (cid:112) f τ n ,n ( x ) + (cid:112) f τ n ( x ) (cid:112) f ( x ) = (cid:115) f τ n ,n ( x ) f ( x ) + (cid:115) f τ n ( x ) f ( x ) d −−−→ n →∞ , and by another application of Slutsky’s Theorem Z ∗ n Y ∗ n d −−−→ n →∞ N (cid:18) , Λ ∗ (cid:19) , proving the Corollary.Before proving Proposition 3.1, we state a result concerning the super level sets ofthe local depth, which is an immediate consequence of the general theory on DFs (see[Zuo and Serfling, 2000b]). Given a level of localization specified by τ > 0, a commonapproach in cluster analysis exploits LDFs to define clusters as the connected componentsof R ∗ ατ := { x ∈ R p : LLD ( x, τ ) ≥ α } for some α > 0. In practice, R ∗ ατ is obtained through its sample counterpart R ∗ ατ,n := { x ∈ R p : LLD n ( x, P, τ ) ≥ α } . By Proposition 2.1 (ii) and Theorem 2.3, we can apply Zuo and Serfling [2000b, Theorem4.1] to obtain the following result which is required in the proof of Proposition 3.1.51 roposition A.2 For all (cid:15) > , < δ < (cid:15) , α > and sequence α n → α there exists an n such that, for all n ≥ n , R ∗ α + (cid:15)τ ⊂ R ∗ α n + δτ,n ⊂ R ∗ α n τ,n ⊂ R ∗ α n − δτ,n ⊂ R ∗ α − (cid:15)τ . Furthermore, if P ( { x ∈ R p : LLD ( x, τ ) = α } ) = 0 , then R ∗ α n τ,n −−−→ n →∞ R ∗ ατ a.s. Proof of Proposition 3.1. Since α j → α as j → ∞ and using Theorem 2.2 (i), wehave, for all m ∈ N , there exists a k ∈ N such that | α j − α | < m , for all j ≥ k , and (cid:12)(cid:12) f τ j ( x ) − f ( x ) (cid:12)(cid:12) < m , for all j ≥ k and x ∈ R p . It follows, using the definition of limitinferior lim inf k →∞ R α k τ k = ∪ ∞ k =1 ∩ ∞ j = k { x ∈ R p : f τ j ( x ) ≥ α j }⊃ { x ∈ R p : f ( x ) > α + 2 m } = ˚ R α + m ↑ m →∞ ∪ ∞ m =1 ˚ R α + m = ˚ R α and lim sup k →∞ R α k τ k = ∩ ∞ k =1 ∪ ∞ j = k { x ∈ R p : f τ j ( x ) ≥ α j }⊂ { x ∈ R p : f ( x ) ≥ α − m } = R α − m ↓ m →∞ ∩ ∞ m =1 R α − m = R α , establishing (3.3). For the second part, using R α = L α ∪ ˚ R α and (3.3), it follows thatlim inf k →∞ R α k τ k = ˚ R α ∪ (cid:16) lim inf k →∞ R α k τ k ∩ L α (cid:17) andlim sup k →∞ R α k τ k = ˚ R α ∪ (cid:18) lim sup k →∞ R α k τ k ∩ L α (cid:19) , wherelim inf k →∞ R α k τ k ∩ L α ⊂ lim sup k →∞ R α k τ k ∩ L α ⊂ L α are sets of Lebesgue measure 0. Therefore,lim inf k →∞ R α k τ k = lim sup k →∞ R α k τ k = R α except for a set of Lebesgue measure 0 and we obtain (3.4).Next, to verify (3.5), observe that λ ( L α ) = 0 implies P ( { x ∈ R p : f ( x ) = α } ) = (cid:90) { x ∈ R p : f ( x )= α } f ( y ) dy = αλ ( L α ) = 0 , and, therefore, (3.4) holds with α k ≡ α implyinglim k →∞ R ατ k = R α a.s. (A.23)Also note that (using the notation of Proposition A.2) R ατ k = R ∗ α τ pk Λ τ k , R α n τ k = R ∗ α n τ pk Λ τ k and R α n τ k ,n = R ∗ α n τ pk Λ τ k ,n . Finally, since λ ( L ατ k ) = 0 is equivalent to λ ( { x ∈ R p : LLD ( x, τ k ) =52 τ pk Λ } ) = 0, and since P is absolutely continuous with respect to the Lebesgue measure,it follows that P ( { x ∈ R p : LLD ( x, τ k ) = α τ pk Λ } ) = 0. Now, applying PropositionA.2, we obtain lim n →∞ R α n τ k ,n = R ατ k a.s. (A.24)The proof now follows from (A.23) and (A.24). Proof of Lemma 3.1. We first observe that x ∈ S τ if and only if f τ ( x ) > LLD ( x, τ ) > 0. Proposition 2.1 (i) implies that for x ∈ R p , LLD ( x, τ ) ≤ LLD ( x, τ ),from which it follows that S τ ⊂ S τ . Next, suppose that f ( · ) is continuous and let x ∈ S and τ > 0. Since f ( · ) is continuous, S is open and there exists an (cid:15) > B (cid:15) ( x ) is contained in S . Also, since Z τ ( x ) is closed and bounded, it follows that Z (cid:15)τ ( x ) := Z τ ( x ) ∩ (cid:0) B (cid:15) ( x ) × B (cid:15) ( x ) (cid:1) is a compact subset of S × S . Since λ ⊗ ( Z (cid:15)τ ( x )) > 0, itfollows that LLD ( x, τ ) = (cid:90) Z τ ( x ) f ( x ) f ( x ) dx dx ≥ (cid:90) Z (cid:15)τ ( x ) f ( x ) f ( x ) dx dx > . Thus x ∈ S τ and S ⊂ S τ . For the last part, since the sets { S τ } τ> are monotonically de-creasing with τ , we have that lim τ → + S τ = ∩ τ> S τ ⊃ S . For x ∈ S , we have to show that x ∈ ∩ τ> S τ . As before, there exists an (cid:15) > B (cid:15) ( x ) ⊂ S and, for all 0 < τ ≤ (cid:15) , Z τ ( x ) ⊂ B (cid:15) ( x ) × B (cid:15) ( x ) ⊂ S × S . From the compactness of Z τ ( x ) and λ ⊗ ( Z τ ( x )) > < τ ≤ (cid:15) , LLD ( x, τ ) > 0; if τ > (cid:15) , by Proposition 2.1 (i) we havethat LLD ( x, τ ) ≥ LLD ( x, (cid:15) ) > 0. Hence, for all τ > LLD ( x, τ ) > x ∈ ∩ τ> S τ . Proof of Theorem 3.1. Since f ( · ) and its first order partial derivatives (second orderpartial derivatives for the second part) are continuous, by Remark 2.1, f τ ( · ) is continuouslydifferentiable in S τ (two times continuously differentiable for the second part). Its partialderivatives can be computed following the argument in the proof of Proposition 2.1 (iv)(see Appendix F).The j -th component of the first partial derivative of f τ ( · ) is given by ∂ j f τ ( x ) = 12 τ p √ Λ (cid:112) LLD ( x, τ ) ∂ j LLD ( x, τ ) , where, from the proof of Proposition 2.1 (iv), we have that ∂ j LLD ( x, τ ) = (cid:90) Z (0) g j ∗ ( x ; x , x ) dx dx g j ∗ ( x ; x , x ) = τ p [ ∂ j f ( x + τ x ) f ( x + τ x ) + f ( x + τ x ) ∂ j f ( x + τ x )] . Therefore, ∂ j f τ ( x ) = 12 √ Λ (cid:113) τ p LLD ( x, τ ) (cid:18) ∂ j LLD ( x, τ ) τ p (cid:19) −−−→ τ → + f ( x ) (2 Λ ∂ j f ( x ) f ( x )) = ∂ j f ( x ) . The second partial derivative is computed by differentiating ∂ j f τ with respect to the i -thcomponent, namely ∂ i ∂ j f τ ( x ) = 12 τ p √ Λ (cid:34) ∂ i (cid:32) (cid:112) LLD ( x, τ ) (cid:33) ∂ j LLD ( x, τ ) + 1 (cid:112) LLD ( x, τ ) ∂ i ∂ j LLD ( x, τ ) (cid:35) , where ∂ i (cid:32) (cid:112) LLD ( x, τ ) (cid:33) = − LLD ( x, τ )) ∂ i LLD ( x, τ )and, by the proof of Proposition 2.1 (iv), ∂ i ∂ j LLD ( x, τ ) = (cid:90) Z (0) g ij ∗ ( x ; x , x ) dx dx with g ij ∗ ( x ; x , x ) = τ p [ ∂ i ∂ j f ( x + τ x ) f ( x + τ x ) + ∂ j f ( x + τ x ) ∂ i f ( x + τ x )+ ∂ i f ( x + τ x ) ∂ j f ( x + τ x ) + f ( x + τ x ) ∂ i ∂ j f ( x + τ x )] . Hence, ∂ i ∂ j f τ ( x ) = 12 √ Λ − (cid:0) τ p LLD ( x, τ ) (cid:1) (cid:18) ∂ i LLD ( x, τ ) τ p (cid:19) (cid:18) ∂ j LLD ( x, τ ) τ p (cid:19) + 1 (cid:113) τ p LLD ( x, τ ) (cid:18) ∂ i ∂ j LLD ( x, τ ) τ p (cid:19) −−−→ τ → + (cid:20) − f ( x ) (2 ∂ i f ( x ) f ( x )) (2 ∂ j f ( x ) f ( x ))+ 1 f ( x ) (2 ∂ i ∂ j f ( x ) f ( x ) + 2 ∂ i f ( x ) ∂ j f ( x )) (cid:21) = ∂ i ∂ j f ( x ) . roof of Lemma 3.2. Since x ∈ ( R α ) − τ satisfies inf y ∈ R p \ R α (cid:107) x − y (cid:107) > τ , we have that B τ ( x ) ⊂ R α . From Z τ ( x ) ⊂ B τ ( x ) × B τ ( x ) ⊂ R α × R α , it follows that f τ ( x ) = 1 τ p Λ (cid:112) LLD ( x, τ ) ≥ α, and therefore x ∈ R ατ . Next, let x ∈ R ατ . Then there exists ( x , x ) ∈ Z τ ( x ) suchthat f ( x ) f ( x ) ≥ α . In particular, since Z τ ( x ) ⊂ B τ ( x ) × B τ ( x ), there exists a point z ∈ B τ ( x ) with f ( z ) ≥ α . Hence z ∈ R α , and since z ∈ B τ ( x ), (cid:107) x − z (cid:107) ≤ τ , implyingthat x ∈ ( R α ) + τ . Finally, suppose that for α > R α is bounded. Then, there exists r > R α ⊂ B r (0). It follows that, for τ > x ∈ R ατ ⊂ ( R α ) + τ satisfies (cid:107) x (cid:107) ≤ inf y ∈ R α ( (cid:107) y (cid:107) + (cid:107) y − x (cid:107) ) ≤ r + inf y ∈ R α (cid:107) y − x (cid:107) ≤ r + τ . Hence R ατ ⊂ B r + τ (0) isbounded. Proof of Theorem 3.2. Fix x ∈ S and t ≥ 0. Let α > x ∈ R α .Since R α ⊂ S is compact, R p \ S is closed and these two sets are disjoint, we havethat dist( R α , R p \ S ) > 0, where, for A, B ⊂ R p , dist( A, B ) := inf y ∈ A, z ∈ B (cid:107) y − z (cid:107) . Let τ ∗ := dist( R α , R p \ S ) / R α ) + τ ∗ , R p \ S ) = inf y ∈ R p \ S, z ∈ ( R α ) + τ ∗ (cid:107) y − z (cid:107) = inf y ∈ R p \ S inf z ∈ R p : inf w ∈ Rα (cid:107) w − z (cid:107)≤ τ ∗ (cid:107) y − z (cid:107) . By the triangle inequality, we have that for w ∈ R α (cid:107) y − z (cid:107) ≥ (cid:107) y − w (cid:107) − (cid:107) w − z (cid:107) ≥ (cid:107) y − w (cid:107) − τ ∗ . It follows that dist(( R α ) + τ ∗ , R p \ S ) ≥ dist( R α , R p \ S ) − τ ∗ = 2 τ ∗ > . (A.25)Lemma 3.2 implies that for all 0 < τ ≤ τ ∗ , R ατ ⊂ ( R α ) + τ ⊂ ( R α ) + τ ∗ ⊂ S . In the rest ofthe proof, we suppose that 0 < τ ≤ τ ∗ . Also, for all s ≥ u x ( s ) ∈ R α and u x,τ ( s ) ∈ R ατ ;as shown in Subsection 3.1, the solutions of the gradient system (3.2) cannot leave theregions R α , and the same is true for the gradient system (3.6) and R ατ . In particular, forall 0 ≤ s ≤ t , u x ( s ) , u x,τ ( s ) ∈ K , where K := ( R α ) + τ ∗ is a compact subset of S .Now noticing that the integral of a vector is the vector of the integrals of its components,we obtain u x,τ ( t ) − u x ( t ) = (cid:90) t ∇ f τ ( u x,τ ( s )) − ∇ f ( u x ( s )) ds. ∇ f τ ( u x ( s )) inside the integral and taking the Euclideannorm on both sides we see that (cid:107) u x,τ ( t ) − u x ( t ) (cid:107) ≤ (cid:90) t (cid:107)∇ f τ ( u x,τ ( s )) − ∇ f τ ( u x ( s )) (cid:107) ds + (cid:90) t (cid:107)∇ f τ ( u x ( s )) − ∇ f ( u x ( s )) (cid:107) ds. Since ∇ f τ is locally Lipschitz in S , it is Lipschitz in the compact subset K ; that is, thereexists a constant L τ < ∞ such that for all y, z ∈ K (cid:107)∇ f τ ( y ) − ∇ f τ ( z ) (cid:107) ≤ L τ (cid:107) y − z (cid:107) . (A.26)It follows that (cid:107) u x,τ ( t ) − u x ( t ) (cid:107) ≤ L τ (cid:90) t (cid:107) u x,τ ( s ) − u x ( s ) (cid:107) ds + (cid:90) t (cid:107)∇ f τ ( u x ( s )) − ∇ f ( u x ( s )) (cid:107) ds. We now apply Gr¨onwall’s inequality [Hale, 1980][Corollary 6.6] with a = 0, β ( s ) = L τ ,0 ≤ s ≤ t , α = (cid:82) t (cid:107)∇ f τ ( u x ( s )) − ∇ f ( u x ( s )) (cid:107) ds and ϕ ( t ) = (cid:107) u x,τ ( t ) − u x ( t ) (cid:107) , and obtainthat (cid:107) u x,τ ( t ) − u x ( t ) (cid:107) ≤ e L τ (cid:90) t (cid:107)∇ f τ ( u x ( s )) − ∇ f ( u x ( s )) (cid:107) ds. (A.27)To conclude the proof we need to show that this converges to 0 as τ → + . To this end,since ∇ f is also locally Lipschitz in S , there exists a constant L < ∞ such that, for all y, z ∈ K (cid:107)∇ f ( y ) − ∇ f ( z ) (cid:107) ≤ L (cid:107) y − z (cid:107) . (A.28)Also by Theorem 3.1, ∇ f τ converges pointwise to ∇ f , for all y, z ∈ K (cid:107)∇ f τ ( y ) − ∇ f τ ( z ) (cid:107) −−−→ τ → + (cid:107)∇ f ( y ) − ∇ f ( z ) (cid:107) , and by (A.28) for all y, z ∈ K with y (cid:54) = z lim τ → + (cid:107)∇ f τ ( y ) − ∇ f τ ( z ) (cid:107)(cid:107) y − z (cid:107) = (cid:107)∇ f ( y ) − ∇ f ( z ) (cid:107)(cid:107) y − z (cid:107) ≤ L. It follows that { L τ } <τ ≤ τ ∗ in (A.26) can be chosen in such a way that in the limit doesnot blow up. Therefore, from (A.27), it follows thatlim τ → + (cid:107) u x,τ ( t ) − u x ( t ) (cid:107) = 0 , if we can show that (cid:90) t (cid:107)∇ f τ ( u x ( s )) − ∇ f ( u x ( s )) (cid:107) ds −−−→ τ → + . (A.29)56o show this, we first enlarge the compact set K by τ ∗ in such a way that it is stillcontained in S by considering the set ( K ) + τ ∗ ⊂ ( R α ) +2 τ ∗ . As in (A.25), we see thatdist(( R α ) +2 τ ∗ , R p \ S ) ≥ dist( R α , R p \ S ) − τ ∗ = τ ∗ > , and ( K ) + τ ∗ is indeed a compact subset of S . Furthermore, for all y ∈ K , B τ ( y ) ⊂ B τ ∗ ( y ) ⊂ ( K ) + τ ∗ ⊂ S , and in particular, Z τ ( y ) ⊂ B τ ∗ ( y ) × B τ ∗ ( y ) ⊂ S × S . Now,by proceeding as in the proof of Theorem 3.1, we see that for y ∈ K , the j -th partialderivative of f τ ( · ) at y is given by ∂ j f τ ( y ) = 12 τ p Λ (cid:113) τ p Λ LLD ( x, τ ) (cid:90) Z τ ( x ) ∂ j f ( x ) f ( x ) + f ( x ) ∂ j f ( x ) dx dx and | ∂ j f τ ( y ) | ≤ α · α j β < ∞ where α := max z ∈ ( K ) + τ ∗ f ( z ), β := min z ∈ ( K ) + τ ∗ f ( z ) and α j := max z ∈ ( K ) + τ ∗ ∂ j f ( z ) satisfy0 < α , β , α j < ∞ . It follows that, for y ∈ K , (cid:107)∇ f τ ( y ) − ∇ f ( y ) (cid:107) ≤ (cid:107)∇ f τ ( y ) (cid:107) + (cid:107)∇ f ( y ) (cid:107)≤ √ p (cid:18) α β (cid:19) (cid:32) p (cid:88) j =1 (cid:0) α j (cid:1) (cid:33) / + √ p (cid:32) p (cid:88) j =1 (cid:0) α j (cid:1) (cid:33) / < ∞ . Therefore, for all 0 ≤ s ≤ t , (cid:107)∇ f τ ( u x ( s )) − ∇ f ( u x ( s )) (cid:107) is bounded and by Theorem 3.1,for all 0 ≤ s ≤ t , (cid:107)∇ f τ ( u x ( s )) − ∇ f ( u x ( s )) (cid:107) −−−→ τ → + 0. Now, (A.29) follows using LDCT. Proof of Theorem 3.3. As in the proof of Theorem 3.1 we see that for j = 1 , . . . , p∂ j f τ ( x ) = 12 τ p √ Λ (cid:112) LLD ( x, τ ) ∂ j LLD ( x, τ ) , where ∂ j LLD ( x, τ ) = (cid:90) Z τ (0) ∂ j f ( x + x ) f ( x + x ) + f ( x + x ) ∂ j f ( x + x ) dx dx . Also from Lemma A.2, it follows that Z τ (0) satisfies the following symmetry properties( x , x ) ∈ Z τ (0) ⇐⇒ ( x , x ) ∈ Z τ (0) (A.30)and ( x , x ) ∈ Z τ (0) ⇐⇒ ( − x , − x ) ∈ Z τ (0) . (A.31)57rom (A.30) it follows that ∂ j LLD ( x, τ ) = 2 (cid:90) Z τ (0) ∂ j f ( x + x ) f ( x + x ) dx dx which implies that ∂ j f τ ( µ ) = 0 if and only if (cid:90) Z τ (0) ∂ j f ( µ + x ) f ( µ + x ) dx dx = 0 , and hence (3.8) holds. Finally, if f ( · ) is τ -centrally symmetric about µ , then, for all y ∈ R p with (cid:107) y (cid:107) ≤ τ , f ( µ + y ) = f ( µ − y ) and ∂ j f ( µ − y ) = − ∂ j f ( µ + y ). By the changeof variable in (A.31) it follows that, for all 1 ≤ j ≤ p , ∂ j LLD ( µ, τ ) = 2 (cid:90) Z τ (0) ∂ j f ( µ + x ) f ( µ + x ) dx dx = 2 (cid:90) Z τ (0) ∂ j f ( µ − x ) f ( µ − x ) dx dx = − (cid:90) Z τ (0) ∂ j f ( µ + x ) f ( µ + x ) dx dx = 0 , and therefore ∇ f τ ( µ ) = 0. Proof of Theorem 3.4. Since f ( · ) is τ -centrally symmetric about m , as in the proofof Theorem 3.3, we see that ∂ j LLD ( m, τ ) = 0 for j = 1 , . . . , p (A.32)and hence m is a stationary point for f τ ( · ). Now, as in the proof of Theorem 3.1, we seethat (A.32) implies also that for i, j = 1 , . . . , p∂ i ∂ j f τ ( m ) = 12 τ p √ Λ (cid:112) LLD ( m, τ ) ∂ i ∂ j LLD ( m, τ ) , where ∂ i ∂ j LLD ( m, τ ) = 2 (cid:90) Z τ (0) ∂ i ∂ j f ( m + x ) f ( m + x ) + ∂ j f ( m + x ) ∂ i f ( m + x ) dx dx , due to the symmetry properties of Z τ (0) (see (A.30)). Denoting by H f τ the Hessian matrixof f τ ( · ) and noticing that the integral of a matrix is the matrix of the integrals, we getthat H f τ ( m ) = C m,τ (cid:90) Z τ (0) H f ( m + x ) f ( m + x ) + ∇ f ( m + x ) ∇ f ( m + x ) (cid:62) dx dx = C m,τ (cid:90) Z τ (0) G f ( m + x , m + x ) dx dx , C m,τ = 1 τ p √ Λ (cid:112) LLD ( m, τ ) . Since the Hessian is symmetric, there exists an orthogonal matrix Q such that D = Q (cid:62) H f τ ( m ) Q = Q (cid:62) (cid:20) C m,τ (cid:90) Z τ (0) G f ( m + x , m + x ) dx dx (cid:21) Q = C m,τ (cid:90) Z τ (0) Q (cid:62) G f ( m + x , m + x ) Q dx dx is a diagonal matrix. Now, since G f ( m + x , m + x ) is negative (resp. positive) definite,for all y ∈ R p \ { } , y (cid:62) G f ( m + x , m + x ) y < > Q (cid:62) G f ( m + x , m + x ) Q are negative (resp. positive). It follows that thediagonal elements of D (that is the eigenvalues of H f τ ( m )) are negative (resp. positive)and m is a mode (resp. an antimode) for f τ ( · ). B Necessity of conditions in Theorem 2.2 We show in this section that continuity is not sufficient in Theorem 2.2 (i). For p = 1consider the function ˜ f : R → R with support ∪ ∞ n =1 (cid:0) n − n , n + n (cid:1) defined by˜ f ( n + x ) = ˜ f ( n − x ) = n (cid:0) n − x (cid:1) if 0 ≤ x < n n ≤ x ≤ . Notice that, since (cid:80) ∞ n =1 1 n = π , (cid:90) ˜ f ( x ) dx = ∞ (cid:88) n =1 (cid:90) ( n − n ,n + n ) ˜ f ( x ) dx = ∞ (cid:88) n =1 n = π . Let c := π . Then f : R → R defined by f ( x ) = c ˜ f ( x ) is an unbounded, continuousdensity function. The τ -approximation of f ( · ) is given by f τ ( x ) = τ (cid:112) LLD ( x, τ ), where,by Corollary 2.1, LLD ( x, τ ) = 2 (cid:90) T τ ++ f ( x + x ) f ( x − x ) dx dx = 2 (cid:90) τ (cid:20)(cid:90) τ − x f ( x − x ) dx (cid:21) f ( x + x ) dx . f ( · ) is symmetric about n ∈ N and for n ≤ τ ≤ LLD ( n, τ ) = 2 (cid:90) min ( τ, n ) (cid:34)(cid:90) min ( τ − x , n ) n c (cid:18) n − x (cid:19) dx (cid:35) n c (cid:18) n − x (cid:19) dx = 2 c (cid:90) n (cid:34)(cid:90) n n (cid:18) n − x (cid:19) dx (cid:35) n (cid:18) n − x (cid:19) dx = 2 c (cid:90) n n (cid:18) n − x (cid:19) dx = 18 c n . For all 0 < τ ≤ fixed there exists n ∈ N such that n ≤ τ , and thereforesup x ∈ R | f τ ( x ) − f ( x ) | ≥ sup n ∈ N : n ≤ τ | f τ ( n ) − f ( n ) | = sup n ∈ N : n ≤ τ (cid:12)(cid:12)(cid:12)(cid:12) τ √ cn − n (cid:12)(cid:12)(cid:12)(cid:12) ≥ sup n ∈ N : n ≤ τ (cid:12)(cid:12)(cid:12)(cid:12) √ c − n (cid:12)(cid:12)(cid:12)(cid:12) = ∞ . The boundedness assumption in Theorem 2.2 (i) prevents f ( · ) to become arbitrarily largeand allows one to show that the above supremum is bounded. On the other hand, uniformcontinuity ensures that the supremum converges to zero, thus allowing to use LDCT andobtain the statement. C Stationary points for univariate densities If p = 1, from Corollary 2.1, (2.4), we see that f τ ( x ) = 1 τ (cid:112) LLD ( x, τ )where LLD ( x, τ ) = 2 (cid:90) T τ ++ f ( x + x ) f ( x − x ) dx dx and T τ ++ = { ( x , x ) : x ≥ , x ≥ , x + x ≤ τ } . In particular, if f ( · ) has a continuous derivative, it follows that f (cid:48) τ ( x ) = 1 τ (cid:112) LLD ( x, τ ) (cid:90) T τ ++ f (cid:48) ( x + x ) f ( x − x ) + f ( x + x ) f (cid:48) ( x − x ) dx dx . (C.1)Therefore, the sign of f τ ( x ) depends on the sign of f (cid:48) ( · ) in the interval ( x − τ, x + τ ).In particular, if µ ∈ R satisfies f ( µ − x ) = f ( µ + x ) for all x ∈ (0 , τ ), it follows that f (cid:48) ( µ − x ) = − f (cid:48) ( µ + x ), yielding f (cid:48) τ ( µ ) = 0.60he next result, which also holds for asymmetric densities, shows that, for p = 1 and τ small enough, f τ ( · ) has a mode in a neighborhood of a mode of f ( · ). Proposition C.1 Let p = 1 . Suppose that f ( · ) is twice continuously differentiable in aneighborhood of a mode (resp. an antimode) m . Then, there is τ ∗ > such that, for all < τ ≤ τ ∗ , f τ ( · ) has a mode (resp. an antimode) in ( m − τ, m + τ ) . Proof of Proposition C.1. Since m is a mode (resp. an antimode) for f ( · ) there exists (cid:15) > f (cid:48)(cid:48) ( x ) < > 0) for x ∈ [ m − (cid:15), m + (cid:15) ] and (ii) f (cid:48)(cid:48) ( x ) f ( y ) + f (cid:48) ( x ) f (cid:48) ( y ) < > 0) for x, y ∈ [ m − (cid:15), m + (cid:15) ]. In particular, (i) implies that f (cid:48) ( x ) > < 0) for x ∈ [ m − (cid:15), m ) and f (cid:48) ( x ) < > 0) for x ∈ ( m, m + (cid:15) ].Let τ ∗ := (cid:15) . As in the proof of Proposition 2.1 (iv) (see Appendix F), it follows that f τ ( · )is twice continuously differentiable in [ m − τ ∗ , m + τ ∗ ]. In particular, its first derivative isgiven by (C.1). Since T τ ++ ⊂ [0 , τ ] × [0 , τ ], for ( x , x ) ∈ T τ ++ , m − τ + x ∈ [ m − τ, m ] and m − τ − x ∈ [ m − τ, m − τ ], yielding f ( m − τ + x ) > < 0) and f ( m − τ − x ) > < a.e. for ( x , x ) ∈ T τ ++ . Hence f (cid:48) τ ( m − τ ) > < f (cid:48) τ ( m + τ ) < > m τ ∈ ( m − τ, m + τ ) such that f (cid:48) τ ( m τ ) = 0. Finally,by (ii) f (cid:48)(cid:48) τ ( m τ ) = 1 τ (cid:112) LLD ( x, τ ) (cid:90) T τ ++ f (cid:48)(cid:48) ( x + x ) f ( x − x ) + f (cid:48) ( x + x ) f (cid:48) ( x − x )+ f (cid:48) ( x + x ) f (cid:48) ( x − x ) + f ( x + x ) f (cid:48)(cid:48) ( x − x ) dx dx < > 0) and m τ is a mode (resp. an antimode) for f τ ( · ). D Local simplicial depth In this appendix we describe in detail the appropriate changes to be made in the state-ments and proofs about LLD for LSD. Properties of local simplicial depth: We emphasize that for p > 1, LLD and LSDare different, even though they coincide for p = 1. However, as we explain below theresults for the simplicial depth can be obtained from the proof of our results for LLD.In this appendix we suppress S in Z Sτ ( x ), Z S ( x ), Λ S , Λ ∗ S and other quantities related toLSD.Proposition 2.1 remains valid for LSD, with the only difference being thatlim τ → + LSD ( x, P, τ ) = P ( p +1) ( { x } ) and lim τ →∞ LSD ( x, P, τ ) = SD ( x, P ) , Z ( x ) = { ( x , . . . , x p +1 ) ∈ ( R p ) ( p +1) : x ∈ (cid:52) [ x , . . . , x p +1 ] } , and SD ( x, P ) is thesimplicial depth of a point x ∈ R p with respect to P and is given by SD ( x, P ) = (cid:90) Z ( x ) dP ( x ) . . . dP ( x p +1 ) . Next, if P is absolutely continuous with respect to the Lebesgue measure with a density f ( · ), then LSD ( x, τ ) = (cid:90) Z τ ( x ) f ( x ) . . . f ( x p +1 ) dx . . . dx p +1 = (cid:90) Z τ (0) f ( x + x ) . . . f ( x + x p +1 ) dx . . . dx p +1 = (cid:90) Z (0) τ p ( p +1) f ( x + τ x ) . . . f ( x + τ x p +1 ) dx . . . dx p +1 . This then implies that Theorem 2.1 (i) can be rephrased as follows: for almost every point x ∈ R p , lim τ → + τ p ( p +1) Λ LSD ( x, τ ) = f ( p +1) ( x ) . (D.1)where Λ = λ ⊗ p ( Z (0)) is the Lebesgue measure of Z (0). Furthermore, if f ( · ) is contin-uous then (D.1) holds for all x ∈ R p .Proceeding analogously, Theorem 2.1 (ii) becomeslim τ → + τ (cid:18) τ p ( p +1) LSD ( x, τ ) − Λ f ( p +1) ( x ) (cid:19) = h ( x ) , where now h ( · ) is given by h ( x ) = p + 12 f p ( x ) (cid:90) Z (0) x (cid:62) H f ( x ) x dx . . . dx p +1 + p ( p + 1)2 f p − ( x ) (cid:90) Z (0) (cid:104)∇ f ( x ) , x (cid:105)(cid:104)∇ f ( x ) , x (cid:105) dx . . . dx p +1 . Finally, the τ -approximation in Definition 2.1 for LSD becomes f τ ( x ) = 1 τ p ( p +1) √ Λ ( p +1) (cid:112) LSD ( x, τ )and Theorem 2.2 holds as before. Sample local simplicial depth: Turning to the sample version of LSD, again given X , . . . , X n i.i.d. random variables from P , LSD is estimated using the proportion LSD n ( x, τ ) = 1 (cid:0) np +1 (cid:1) (cid:88) ≤ i <...i p +1 ≤ n I (( X i , . . . , X i p +1 ) ∈ Z τ ( x )) , (D.2)62hich is a U-statistics with the kernel K x,τ ( x , . . . , x p +1 ) = I (( x , . . . , x p +1 ) ∈ Z τ ( x ))and first projection J x,τ ( x ) = (cid:90) I (( x , . . . , x p +1 ) ∈ Z τ ( x )) dP ( x ) . . . dP ( x p +1 ) . (D.3)The variance of the first projection, using the same notation as in the LLD case, is givenby b ( x, τ ) = V ar [ J x,τ ( X )] = (cid:90) ( J x,τ ( x )) dP ( x ) − ( LSD ( x, τ )) . (D.4)With this description, Corollary 2.2 (i) can be stated as follows: for any fixed x ∈ R p , LSD n ( x, · ) is a non-decreasing and right continuous function; i.e. ,lim τ → + LSD n ( x, τ ) = 1 (cid:0) np +1 (cid:1) (cid:88) ≤ i < ··· x, τ ) ∈ Γ, then √ n ( LSD n ( · , · ) − LSD ( · , · )) d −−−→ n →∞ ( p + 1) W ( · , · ) in (cid:96) ∞ (Γ) , where { W ( x, τ ) } ( x,τ ) ∈ Γ is a centered Gaussian process with covariance function γ : Γ × Γ → R given by γ (( x, τ ) , ( y, ν )) = (cid:90) J x,τ ( x ) J y,ν ( x ) dP ( x ) − LSD ( x, τ ) LSD ( y, ν ) . The extreme case of this result, that is the case τ = ∞ , was studied in D¨umbgen [1992]and Arcones and Gin´e [1993][Corollary 6.8]. We repeat here that uniformity in τ is notan issue in the other papers, but it is a critical issue in our work.We now turn to the sample τ -approximation, which is needed for clustering. Now,the sample τ -approximation in (2.11) becomes f τ,n ( x ) = 1 τ p ( p +1) √ Λ ( p +1) (cid:112) LSD n ( x, τ ) , viz., √ n τ p n ( f τ n ,n ( x ) − f τ n ( x )) d −−−→ n →∞ N (cid:18) , Λ ∗ Λ f ( x ) (cid:19) , where the sequence { τ n } ∞ n =1 now satisfies τ n −−−→ n →∞ √ n τ ( p + p ) n −−−→ n →∞ ∞ . We noticethat, while the limiting distribution is exactly the same of Theorem 2.5, the quantitiesΛ and Λ ∗ = (cid:82) (cid:0) λ ⊗ ( p − (cid:1) ( Z (0) | x ) dx are now based on Z (0) = Z S (0) and Z (0) | x = Z S (0) | x . It is worth mentioning that √ n τ ( p + p ) n −−−→ n →∞ ∞ is a necessary condition for thelimiting normality to hold. Finally, under these hypotheses, it follows that √ n τ p n (cid:18)(cid:113) f τ n ,n ( x ) − (cid:112) f τ n ( x ) (cid:19) d −−−→ n →∞ N (cid:18) , Λ ∗ (cid:19) . Clustering via local simplicial depth: We describe below how to adapt LSD forthe clustering procedure described in Section 3. Since LSD satisfies Proposition 2.1 (ii)and Theorem 2.3, Proposition A.2 still holds for LSD. This and the fact that f τ ( · ) sat-isfies Theorem 2.2 imply that the super level sets of Definition 3.1 based on LSD satisfyProposition 3.1.Since the algorithm for clustering involves identification of the stationary points andthe modes, first, we need analogous of Theorems 3.3 and 3.4. Specifically, in Theorem 3.3 µ is a stationary point for f τ ( · ) if and only if (cid:90) Z (0) ∇ f ( µ + τ x ) f ( µ + τ x ) . . . f ( µ + τ x p +1 ) dx dx . . . dx p +1 = 0 , and ∇ f τ ( µ ) = 0 if f ( · ) is τ -centrally symmetric about µ . Next, turning to a mode (resp.an antimode) m of f ( · ), Theorem 3.4 can be described as follows: if for all x , x ∈ B τ ( m )the matrix G f ( x , x ) = H f ( x ) f ( x ) + p ∇ f ( x ) ∇ f ( x ) (cid:62) is negative (resp. positive) definite, then m is also a mode (resp. an antimode) for f τ ( · ).Incidentally, it is worth noting here that Lemma 3.1, Theorem 3.1, Lemma 3.2 and The-orem 3.2 all hold true for LSD. Numerical implementation: In this subsection we describe the changes to the numer-ical procedure required for clustering using LSD. First the quantity d τ,n ( · ; · ) in (3.10) isnow replaced by d τ,n ( x ; y ) = ( p +1) (cid:112) LSD n ( y, τ ) − ( p +1) (cid:112) LSD n ( x, τ ) (cid:107) y − x (cid:107) . 64s for LLD, τ can be chosen as the q -quantile of the empirical distribution of the (cid:0) np +1 (cid:1) maxima of the form max j,l =1 ,...,p +1 j>l (cid:13)(cid:13) x i j − x i l (cid:13)(cid:13) for all (cid:0) np +1 (cid:1) combinations of indices i , . . . , i p +1 from { , , . . . , n } . Computational complexity: We recall that LLD n is a U-statistics of order 2, while LSD n is a U-statistics of order ( p + 1). This means that the computational complexityof LLD n is of order O ( (cid:0) n (cid:1) ), while the computational complexity of the LSD n is of order O ( (cid:0) np +1 (cid:1) ), which makes a significant difference, especially in high dimensions. For large p and n , an approximation to LSD can be made by considering a large number of sim-plices sampled with replacement amongst all the (cid:0) np +1 (cid:1) simplices that define LSD n ; in oursimulations we sample 10 simplices to reduce the computational cost. E Local half-space depth In this appendix we describe in detail the appropriate changes to be made in the state-ments about LLD that yield the corresponding results for LHD. Additionally, we alsosuppress H in Z Hτ ( x ) and write Z τ ( x ). Properties of local half-space depth: We begin by showing that, for all fixed τ ∈ [0 , ∞ ], LHD ( · , τ ) is upper semicontinuous. This follows from the fact that the infimumof a collection of upper semicontinuous functions is upper semicontinuous and that thefunction x → P ( Z τ ( x, u )) is upper semicontinuous for all u ∈ S p − . In fact, let { x k } ∞ k =1 be a sequence converging to x ∈ R p . Since Z τ ( x, u ) is closed, for all u ∈ S p − lim sup k →∞ P ( Z τ ( x k , u )) = lim sup k →∞ (cid:90) I ( y ∈ Z τ ( x k , u )) dP ( y ) ≤ (cid:90) lim sup k →∞ I ( y ∈ Z τ ( x k , u )) dP ( y ) ≤ (cid:90) I ( y ∈ Z τ ( x, u )) dP ( y ) = P ( Z τ ( x, u )) . If P assigns probability 0 to all hyperplanes [Mass´e, 2004][condition (S)] (for exampleif P is absolutely continuous with respect to the Lebesgue measure), then the function( x, u, τ ) → P ( Z τ ( x, u )) is jointly continuous. In fact, if { τ k } ∞ k =1 ⊂ [0 , ∞ ], { u k } ∞ k =1 ⊂ S p − and { x k } ∞ k =1 are sequences converging to τ ∈ [0 , ∞ ], u ∈ S p − and x ∈ R p , respectively,then | P ( Z τ k ( x k , u k )) − P ( Z τ ( x, u )) | ≤ (cid:90) | I ( y ∈ Z τ k ( x k , u k )) − I ( y ∈ Z τ ( x, u )) | dP ( y ) −−−→ k →∞ ∂Z τ ( x, u ) has probability 0, and if y / ∈ ∂Z τ ( x, u ), thenlim k →∞ I ( y ∈ Z τ k ( x k , u k )) = I ( y ∈ Z τ ( x, u )). In particular, since the infimum of a65ollection of continuous functions over a compact set is a continuous function, LHD isjointly continuous in x and τ . Hence Proposition 2.1 (iii) holds. The compactness of S p − also implies that the infimum in (2.13) is a minimum. The other statements in Proposition2.1 remain true, the only change being that in part (i) we require P to assign probability0 to all hyperplanes to obtainlim τ → + LHD ( x, P, τ ) = P ( { x } ) = 0 and lim τ →∞ LHD ( x, P, τ ) = HD ( x, P ) , where HD ( x, P ) is the half-space depth of x ∈ R p with respect to P given by HD ( x, P ) = inf u ∈ S p − P ( Z ( x, u )) , with Z ( x, u ) = { y ∈ R p : (cid:104) u, x (cid:105) ≤ (cid:104) u, y (cid:105)} . If P is absolutely continuous with respect to the Lebesgue measure with a density f ( · ),due to Z τ ( x, u ) = x + Z τ (0 , u ) and Z τ (0 , u ) = τ Z (0 , u ), by a change of variable we seethat LHD ( x, τ ) = min u ∈ S p − (cid:90) Z τ ( x,u )) f ( y ) dy = τ p min u ∈ S p − (cid:90) Z (0 ,u )) f ( x + τ y ) dy. This allows one to obtain in Theorem 2.1lim τ → + τ p LHD ( x, τ ) = f ( x ) , and for twice continuously differentiable f ( · )lim τ → + τ (cid:18) τ p LHD ( x, τ ) − f ( x ) (cid:19) = min u ∈ S p − (cid:90) Z (0 ,u ) (cid:104)∇ f ( x ) , y (cid:105) dy. Theorem 2.2 remains valid for the τ -approximation based on LHD given by f τ ( x ) = 1 τ p LHD ( x, τ ) . Sample local half-space depth: Given X , . . . , X n i.i.d. random variables from P ,LHD is estimated using the proportion LHD n ( x, τ ) = inf u ∈ S p − n n (cid:88) i =1 I ( X i ∈ Z τ ( x, u )) . With this definition, Corollary 2.2 remains valid, but part (i) becomeslim τ → + LHD n ( x, τ ) = 1 n n (cid:88) i =1 I ( X i = x )and lim τ →∞ LHD n ( x, τ ) = inf u ∈ S p − n n (cid:88) i =1 I ( X i ∈ Z ( x, u )) . R p are VC [Despres, 2017], we also obtainTheorem 2.3 and Corollary 2.3. Asymptotic normality results for the half-space depthare more involved [Arcones et al., 2006, Mass´e, 2004] and are beyond the scope of thispaper. The properties derived in this and in the previous sections allow to obtain all theresults derived for clustering in Subsections 3.1 and 3.2 for LHD. However, Z τ ( x, u ) isnot symmetric in the sense of (2.16) (with k = 1). Therefore, the results concerning themodes and stationary points of f ( · ) in Subsection 3.3 do not hold for LHD. F Proof of Proposition 2.1 (iv) We provide in this section a detailed proof of Proposition 2.1 (iv). We first observe thatif τ = 0 then LLD ( x, τ ) = 0 for all x ∈ R p and the statement is trivial. Let τ > k ≥ 1. We will show that, for all 0 ≤ l ≤ k , the partial derivatives of LLD up to order l exist and are given by the integral over Z (0) of a ( k − l )-times continuously differentiablefunction. To this end, notice that Z (0) is bounded by definition and closed by LemmaA.1, and hence compact. For l = 0, by (2.2), we have that LLD ( x, τ ) = (cid:90) Z (0) g ∗ ( x ; x , x ) dx dx , where for x, x , x ∈ R p g ∗ ( x ; x , x ) := τ p g ( x + τ x , x + τ x )and for y , y ∈ R p g ( y , y ) := f ( y ) f ( y ) . Since f ( · ) is k -times continuously differentiable, g ∗ ( x ; x , x ) is k -times continuously dif-ferentiable with respect to x , x , and x .Given a function g : D ⊂ R p → R for j = 1 , . . . , p we denote by ∂ j g its partialderivative with respect to the j -component. Suppose by induction that the partial deriva-tives of the local depth up to order l − ≤ l ≤ k ) exist and for some choice of indices1 ≤ i , . . . , i l − ≤ p are given by ∂ i l − . . . ∂ i LLD ( x, τ ) = (cid:90) Z (0) g i l − ...i ∗ ( x ; x , x ) dx dx , where g i l − ...i ∗ ( x ; x , x ) = τ p g i l − ...i ( x + τ x , x + τ x ) is ( k − ( l − x , x and x . In particular, since Z (0) and B ( x ) are67ompact, for all 1 ≤ i l ≤ p , z ∈ B ( x ) and ( x , x ) ∈ Z (0), ∂ i l g i l − ...i ∗ ( z ; x , x ) ≤ C i l ...i x := sup ( y ,y ) ∈ Z (0) sup y ∈ B ( x ) τ p ∂ i l g i l − ...i ( y + τ y , y + τ y ) < ∞ . (F.1)Let { h n } ∞ n =1 be a sequence of scalars with 0 < h n ≤ h n −−−→ n →∞ 0. For x ∈ R p and h > i -th partial finite difference of a function g ∗ : R p → R by ∂ hi g ∗ ( x ) = g ∗ ( x + he i ) − g ∗ ( x ) h , where { e i : i = 1 , . . . , p } is the standard basis of R p . Clearly, ∂ h n i l g i l − ...i ∗ ( x ; x , x ) −−−→ n →∞ ∂ i l g i l − ...i ∗ ( x ; x , x ) = τ p ∂ i l g i l − ...i ( x + τ x , x + τ x ). Now, for y , y ∈ R p , define g i l ...i ( y , y ) := ∂ i l g i l − ...i ( y , y )and for x, x , x ∈ R p let g i l ...i ∗ ( x ; x , x ) := τ p g i l ...i ( x + τ x , x + τ x ) .g i l ...i ∗ is ( k − l )-times continuously differentiable with respect to x , x and x and by themean value theorem there are scalars 0 ≤ c n ≤ ∂ i l g i l − ...i ∗ ( x + c n h n e i ; x , x ) = ∂ h n i l g i l − ...i ∗ ( x ; x , x ) . Since x + c n h n e i ∈ B ( x ) for all n ∈ N , the sequence of functions { ∂ h n i l g i l − ...i ∗ ( x + c n h n e i ; · , · ) } ∞ n =1 is dominated on Z (0) by the constant C i l ...i x (see (F.1)). By LDCTit follows that ∂ i l . . . ∂ i LLD ( x, τ ) = lim n →∞ ∂ h n i l ∂ i l − . . . ∂ i LLD ( x, τ )= lim n →∞ (cid:90) Z (0) ∂ h n i l g i l − ...i ∗ ( x ; x , x ) dx dx = (cid:90) Z (0) ∂ i l g i l − ...i ∗ ( x ; x , x ) dx dx = (cid:90) Z (0) g i l ...i ∗ ( x ; x , x ) dx dx , where g i l ...i ∗ is ( k − l )-continuously differentiable with respect to x , x and x .In particular, for l = k this means that for all 1 ≤ i , . . . , i k ≤ p the partial derivatives ∂ i k . . . ∂ i LLD ( x, τ ) exist for all x ∈ R p and are continuous since they are given by theintegral over the compact set Z (0) of the continuous function g i k ...i ∗ . Hence the statement.We finally observe that the functions g i k ...i ∗ can be computed explicitly as it is done, forinstance, in the proof of Theorem 3.1 for k = 1 and k = 2.68 Additional simulations G.1 True clusters In this subsection we provide the analytical expression for the distributions Bimodal, (H)Bimodal IV, (K) Trimodal III, − , 0) and (2 , − , − , − , − 2) and (2 , µ = (1 , − (cid:62) , µ = ( − , (cid:62) and covariancesΣ = 49 (cid:32) (cid:33) and Σ = 49 (cid:32) (cid:33) . (iv) The (K) Trimodal III density is a mixture of three normal distributions with weights w = w = and w = ; means µ = ( − , (cid:62) , µ = (1 , · √ ) (cid:62) and µ = (1 , − · √ ) (cid:62) ;and covariances Σ = (cid:32) 925 710 · · 925 49100 (cid:33) and Σ = Σ = (cid:32) (cid:33) . (v) The (L) Quadrimodal density is a mixture of four normal distributions with weights w = w = and w = w = ; means µ = ( − , (cid:62) , µ = ( − , − (cid:62) , µ = (1 , − (cid:62) and µ = (1 , (cid:62) ; and covariancesΣ = (cid:32) 49 25 · · 49 49 (cid:33) , Σ = (cid:32) 49 35 · · 49 49 (cid:33) , Σ = (cid:32) − · − · 49 49 (cid:33) and Σ = (cid:32) − · − · 49 49 (cid:33) . (vi) The w = and w = w = w = w = w = ; means µ = µ = (0 , (cid:62) , µ = ( − , (cid:62) , µ = ( − , − (cid:62) , µ = (1 , − (cid:62) and µ = (1 , (cid:62) ; and covariancesΣ = (cid:32) (cid:33) , Σ = Σ = Σ = Σ = Σ = (cid:32) (cid:33) . − , , , , 0) and (2 , , , , 0) and (viii) the Mult.Quadrimodal density is a mixture of four normal distributions with means ( − , , , , − , − , , , , − , , , 0) and (2 , , , , f ( · ), f ( · ), f ( · ) and f ( · ) (respectively), where f ( x ) = 0 . (cid:0) − . − (cid:107) x (cid:107) ) (cid:1) (cid:18) . − x (1) (cid:107) x (cid:107) (cid:19) + 0 . (cid:0) − . − . (cid:107) x (cid:107) ) (cid:1) (cid:18) . x (1) (cid:107) x (cid:107) (cid:19) ,f ( x ) = (cid:18) . (cid:18) . x (1) (cid:107) x (cid:107) (cid:19)(cid:19)(cid:30) (cid:0) − (cid:107) x (cid:107) ) (cid:1) + (cid:18) . (cid:18) . x (1) (cid:107) x (cid:107) (cid:19)(cid:19)(cid:30) (cid:0) − . (cid:107) x (cid:107) ) (cid:1) ,f ( x ) = 0 . (cid:18) − − . (cid:107) x (cid:107) ) (cid:19) (cid:18) . − x (1) (cid:107) x (cid:107) (cid:19) + 0 . 15 exp (cid:18) − − . (cid:107) x (cid:107) ) (cid:19) (cid:18) . x (1) (cid:107) x (cid:107) (cid:19) + 0 . 55 exp (cid:18) − − . (cid:107) x (cid:107) ) (cid:19) (cid:18) . x (1) (cid:107) x (cid:107) (cid:19) ,f ( x ) = (cid:18) (cid:18) (cid:18) x (1) (cid:107) x (cid:107) (cid:19)(cid:19)(cid:19)(cid:30) (cid:0) − (cid:107) x (cid:107) ) (cid:1) . Figure 4 shows the functions f ( · ), f ( · ), f ( · ) and f ( · ). The true clusters associatedwith these densities are shown in Figure 5. Although the density Circular 4 Cauchy hasa circular structure, it does not have clusters of a circular form, which makes it easier toidentify the true clusters in the simulations.70 ircular 2 Circular 3Circular 2 Cauchy Circular 4 Cauchy Figure 4: Plots of the functions f ( · ), f ( · ), f ( · ) and f ( · ) proportional to the Circular 2,Circular 2 Cauchy, Circular 3 and Circular 4 Cauchy densities, respectively.71 − − − ll Circular 2 −3 −2 −1 0 1 2 3 − − − ll l Circular 3 −3 −2 −1 0 1 2 3 − − − l l Circular 2 Cauchy −3 −2 −1 0 1 2 3 − − − l l ll Circular 4 Cauchy Figure 5: True clusters associated with the Circular 2, Circular 2 Cauchy, Circular 3 andCircular 4 Cauchy densities. The modes are plotted in red.72 .2 Illustrative examples In this subsection, we provide additional illustrations of clustering (as in Figure 3, Section4), by considering the distributions (K) Trimodal III, Quadrimodal, and (L) Quadrimodal.For this, we compare the second and the third row in Figure 6 with the true clusters inthe first row. Based on this comparison, we observe that the cluster estimates based onthe proposed LLD method (second row) are better than those based on KDE (third row).Also, Figures 7 through 12 provide further illustrations of clustering for different choicesof s and q , similar to the second rows of Figure 3 and Figure 6. These may be regardedas the bivariate analogue of Figure 2 for the densities Bimodal, (H) Bimodal IV, (K)Trimodal III, Quadrimodal, (L) Quadrimodal and r = 0 . 05. However, thechoice of q affects the estimated clusters and a recommendation on its choice is given inSection 3. G.3 Numerical experiments In this subsection, we provide additional simulation results complementing those in Sec-tion 4 of the main paper. Tables 4 through 7 contain results for all the distributions inAppendix G.1 for several choices of q and s beyond what is described in the main paper.Since the parameter r does not affect the output of Algorithm 1, we leave it fixed at r = 0 . 05. The expressions LLD- q - s and LSD- q - s refer to LLD and LSD with parameters q and s . To investigate the effect of sample size, in Tables 8 through 11 n is set at 500,instead of 1000. In the second part of Tables 4 through 6 and 8 through 10, the first rowrefers to the case η = 0 and the second row to the case η = 1. From these two values, itis possible to compute the distance in probability for all values of η . In all the tables thebest results are in bold face. For the probability distance, the best results for the case η = 1 are in bold face.From the tables, we can observe that generally the best results are obtained forour proposed LLD and LSD cluster methodologies. In particular, for the distributionsMult. Bimodal and Mult. Quadrimodal, the best results are always obtained by LLD,see Tables 6, 7, 10 and 11. For the distributions (H) Bimodal IV and Circular 2, LSDoutperforms all the other procedures with the LLD being the second best, as can be seenfrom Tables 4, 5, 7, 8, 9 and 11. From the tables, it also clear that the best clusteringresults for the distributions (K) Trimodal III −3 −2 −1 0 1 2 3 − − − l ll Quadrimodal −3 −2 −1 0 1 2 3 − − − l lll (L) Quadrimodal −3 −2 −1 0 1 2 3 − − − l lll LLDestimatedclusters ll ll ll lll l lll l l lllll llll ll l ll ll l ll ll ll lll lll l l llll ll l lll ll ll ll ll l lll l llll lll ll lllll l lll l ll l ll ll l ll lllll l l llll ll llll lll ll l ll lll lll l ll ll l llll l ll lll ll l lll lll l ll lllll lll ll l lll ll l llll ll ll lll l ll lll l ll ll lll ll l l lll ll lll ll ll ll ll l lll ll ll ll l llll ll llll llll ll ll llll l ll ll ll l lll ll lll ll ll lll ll ll l ll ll lll lll ll lll ll l lllll l ll l l l ll lll ll lll ll ll lll l llllll l ll ll lll ll l lll ll ll l lllll l lllll ll ll ll ll ll ll ll lll ll llll l l lll l ll lll l ll l ll llll ll lll lll l lll llll l ll lll ll ll lll ll lll llll l lllll llll lll l ll ll ll lll ll lll ll lll ll l ll l ll ll l l lll l l ll l l lll l lll ll lll lll lll ll ll lll ll l ll llll l lll l ll ll l ll ll l lllll ll l ll ll ll ll ll ll llll l lll lll l ll lll l l ll ll lll ll llll l l lllll ll l lllll llll l lll ll ll lll l llll llllll lll ll lll lll ll ll l lll lll ll ll l llll lll l l llll l llll ll l ll llllll ll l lll ll ll lll ll ll lll l lll l ll ll ll llll lll l ll l ll lll lll lll l lll ll ll ll llll lll l llll ll lll ll ll l llll lll ll l ll ll lllllll l ll llll lll ll lll l ll ll l lll lllll lll l l llll ll ll llll lllll l l llll ll l l ll ll l l ll ll lll l lllll ll ll ll llll l l lll lll ll llll llll ll l l llll ll lllll l ll ll l l lll l l lll ll l l ll l lll ll l lll ll ll lll lll ll l lllll llllll l ll lllll l ll ll llll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − l ll ll ll ll ll l lll l l ll lll lll ll l ll ll l ll ll ll lll l ll l llll ll l ll l ll ll ll ll l ll l ll llllll ll lll ll lll lll l ll lll ll lllll l l lll l llll l ll ll l ll lll lll l ll ll l ll ll l ll l ll ll l ll l lll l ll llll l lll ll l lll l l l lll l ll ll l llll l ll l lll ll l ll l ll l ll lll ll ll ll l l lll lll ll l ll lll lll llll ll ll lll l ll ll ll lll ll ll l ll ll lll l l ll l ll ll lll lll l l l ll lll ll lll l ll ll llll l ll lll ll ll ll l l l ll lll l ll ll ll l ll lll lll l lllll l ll lll ll lll ll ll ll ll lll l l llll l l llll ll lll l ll l ll ll llll lll l llll lll lll lll ll ll lll ll lll llll l llll llll lll l ll ll ll lll ll lll ll ll ll l ll l ll ll l l lll l l ll l l l ll l ll ll l ll lll ll ll lll ll l ll llll lll l ll ll l ll l lll ll ll l ll l l ll ll l l lll lll llll lll l l ll ll lll ll llll l l llll ll l llll llll l ll ll lll llll ll llll lll ll lll lll ll ll lll lll l ll l lll lll l llll l lll ll l lllllll ll l lll ll 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−3 −2 −1 0 1 2 3 − − − l l ll ll ll ll lll l lll l l lllll llll ll l ll ll l ll ll ll lll lll l l llll ll l ll l ll ll ll ll l lll lllll lll ll lll ll l lll l ll l ll lll ll lllll l l llll ll llll lll ll l ll lll lll l ll ll l ll ll l ll lll ll l lll lll l ll lllll lll ll l lll ll l llll ll ll lll l ll lll l ll ll lll ll l l lll ll lll ll ll ll ll l lll ll ll ll l llll ll llll llll ll ll llll l ll ll ll l lll ll lll ll ll lll ll ll l ll ll lll lll ll lll ll l lllll l ll l l l ll lll ll lll ll ll lll l llllll l ll ll ll l ll l lll ll ll l lllll l lllll ll ll ll ll ll ll ll lll l l llll l l lll l ll lll l ll l ll llll ll lll lll l lll llll lll lll ll ll lll ll lll llll l lllll lll llll l ll ll ll lll ll lll ll lll ll l lll ll ll l llll l l ll l l lll l lll ll lll l ll lll ll ll lll ll lll llll l llll ll lll llll l lllll ll l ll lll l ll ll ll l ll ll lll lll l ll ll ll l ll l l ll l ll lllll l lllll ll l lllll llll l ll l ll ll lll l llll ll llll lll l l lll lll ll ll l lll lll ll ll l 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−3 −2 −1 0 1 2 3 − − − ll ll KDEestimatedclusters ll ll ll lll l lll l l lllll llll ll l ll ll l ll ll ll lll lll l l llll ll l lll ll ll ll ll l lll l llll lll ll lllll l lll l ll l ll ll l ll lllll l l llll ll llll lll ll l ll lll lll l ll ll l llll l ll lll ll l lll lll l ll lllll lll ll l lll ll l llll ll ll lll l ll lll l ll ll lll ll l l lll ll lll ll ll ll ll l lll ll ll ll l llll ll llll llll ll ll llll l ll ll ll l lll ll lll ll ll lll ll ll l ll ll lll lll ll lll ll l lllll l ll l l l ll lll ll lll ll ll lll l llllll l ll ll lll ll l lll ll ll l lllll l lllll ll ll ll ll ll ll ll lll ll llll l l lll l ll lll l ll l ll llll ll lll lll l lll llll l ll lll ll ll lll ll lll llll l lllll llll lll l ll ll ll lll ll lll ll lll ll l ll l ll ll l l lll l l ll l l lll l lll ll lll lll lll ll ll lll ll l ll llll l lll l ll ll l ll ll l lllll ll l ll ll ll ll ll ll llll l lll lll l ll lll l l ll ll lll ll llll l l lllll ll l lllll llll l lll ll ll lll l llll llllll lll ll lll lll ll ll l 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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − ll ll l Figure 6: Clusters associated with the (K) Trimodal III (left), Quadrimodal (middle) and(L) Quadrimodal (right) densities. True clusters (first row). 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−3 −2 −1 0 1 2 3 − − − l l Figure 7: Local depth clustering of n = 1000 samples from the Bimodal density. Thepredicted local maxima are plotted in red. The parameters are r = 0 . s = 10 , , 50 ineach column (from left to right) and q = 0 . , . , . , . 50 in each row (from the topdown). 75 ll lll ll l ll llll l lll l ll ll l ll ll l lllll ll l ll ll ll ll ll ll llll l lll lll l ll lll l l ll ll lll ll llll l l lllll ll l lllll llll l lll ll ll lll l llll ll llll lll ll lll lll ll ll l lll lll ll ll l llll lll l l llll l llll ll l ll llllll ll l lll ll ll lll ll l l lll l lll l ll ll ll llll lll l ll l ll lll lll lll l lll ll ll ll llll lll l llll ll lll ll ll l llll lll ll l ll ll lll llll l ll llll lll ll lll l ll ll l lll lllll lll l l llll ll ll llll lllll l l llll ll l l ll ll l l ll ll lll l lllll ll ll ll llll l l lll lll ll llll llll ll l l llll ll lllll l ll ll l l lll l l lll ll l l ll l lll ll l lll ll ll lll ll l ll l ll lll llllll l ll lllll l ll ll llll ll ll ll ll lll l lll l l lllll llll ll l ll ll l ll ll ll lll lll l l ll ll ll l lll ll ll ll ll l lll l llll lll ll lllll l lll l ll l ll ll l ll lllll l l llll ll ll ll lll ll l ll lll lll l ll ll l llll l ll lll ll l lll lll l ll lllll lll ll l lll ll l llll ll ll lll l ll lll l ll ll lll ll l l lll ll lll ll ll ll ll l lll ll ll ll l llll ll llll llll ll ll llll l ll ll ll l lll ll lll llll lll ll ll l ll ll lll lll ll lll ll l lllll l ll l l l ll lll ll lll ll ll lll l llll ll l ll ll lll ll l lll ll l l l llll l l lllll ll ll ll ll ll ll ll lll ll llll l l lll l ll lll l ll l ll llll ll lll lll l lll llll l ll lll ll ll lll ll lll llll l lllll lll ll ll l ll ll ll lll ll lll ll lll lll lll ll ll l llll l l ll l l lll l lll ll lll lll ll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − l l l ll ll ll lll l lll l l lllll llll ll l ll ll l ll ll ll lll lll l l llll ll l lll ll ll ll ll l lll l llll lll ll lllll l lll l ll l ll ll l ll lllll l l llll ll ll ll lll ll l ll lll lll l ll ll l llll l ll lll ll l lll lll l ll lllll lll ll l lll ll l llll ll ll lll l ll lll l ll ll lll ll l l lll ll lll ll ll ll ll l lll ll ll ll l llll ll llll llll ll ll llll l ll ll ll l lll ll lll ll ll lll ll ll l ll ll lll lll ll lll ll l lllll l ll l l l ll lll ll lll ll ll lll l llll ll l ll ll ll l ll l lll ll ll l lllll l lllll ll ll ll ll ll ll ll lll ll llll l l lll l ll lll l ll l ll llll ll lll lll l lll llll l ll lll ll ll lll ll lll llll l lllll llll lll l ll ll ll lll ll lll ll lll ll l ll l ll ll l l lll l l ll l l lll l lll ll lll l ll lll ll ll lll ll l ll llll l lll l ll ll l ll ll l lllll ll l ll ll ll ll ll ll llll l lll lll l ll lll l l ll ll lll ll llll l l lllll ll l lllll llll l lll ll ll lll l llll llllll lll ll lll lll ll ll l lll lll ll ll l 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−3 −2 −1 0 1 2 3 − − − l Figure 9: Local depth clustering of n = 1000 samples from the (K) Trimodal III density.The predicted local maxima are plotted in red. 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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − l Figure 10: Local depth clustering of n = 1000 samples from the Quadrimodal density. Thepredicted local maxima are plotted in red. The parameters are r = 0 . s = 10 , , 50 ineach column (from left to right) and q = 0 . , . , . , . 50 in each row (from the topdown). 78 l ll ll lll l lll l l lllll llll ll l ll ll l ll ll ll lll lll l l llll ll l ll l ll ll ll ll l lll lllll lll ll lll ll l lll l ll l ll lll ll lllll l l llll ll llll lll ll l ll lll lll l ll ll l ll ll l ll lll ll l ll l lll l ll lllll lll ll l lll ll l llll ll ll lll l ll lll l ll ll ll l ll l l lll ll lll ll ll ll ll l lll ll ll ll l llll ll llll llll ll ll llll l ll ll ll l lll ll lll ll ll lll ll ll l ll ll lll lll ll lll ll l lllll l ll l l l ll lll ll lll ll ll lll l llllll l ll ll ll l ll l lll ll ll l llll l l lllll ll ll ll ll ll ll ll lll l l llll l l lll l ll lll l ll l ll llll ll lll lll l lll llll lll lll ll ll lll ll lll llll l lllll lll llll l ll ll ll lll ll lll ll lll ll l lll ll ll l llll l l ll l l lll l lll ll lll l ll lll ll ll lll ll lll llll l llll ll lll llll l lllll ll l ll lll l ll ll ll l ll ll lll lll l ll ll ll l ll l l ll l ll lllll l lllll ll l lllll llll l ll l ll ll lll l llll ll llll lll l l lll lll ll ll l lll lll ll ll l llll l ll l l lllll llll ll l ll llllll ll l lll ll ll lll ll ll lll l llll llll ll llll lll l lll lllll lll llll lll ll ll ll llll ll l l llll ll ll l ll ll l llll lll lll ll ll lll llll l ll llll l ll ll lll lll ll l lll lllll lll l l llll ll ll llll lllll ll ll ll ll l l ll ll l l ll ll lll l ll lll ll ll ll llll l llll lll ll llll llll ll l l llll ll lllll l llll l l lll l l lll ll l l ll l lll ll l lll ll ll llll lll l l lll ll l lllll l ll ll lll l ll ll ll ll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − l Figure 11: Local depth clustering of n = 1000 samples from the (L) Quadrimodal density.The predicted local maxima are plotted in red. 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llll l lll ll ll lll l llll llllll lll ll lll lll ll ll l lll lll llll l llll lll l l lllll llll lll ll llllll lll lll ll ll lll ll ll lll l llll ll ll ll llll lll l lll lllll lll llll lll ll ll ll llll lll l llll ll lll ll ll l llll lll ll l llll lllllll l ll llll lll ll lll lll ll l lll lllll lll ll llll ll ll llll lllll ll llll ll l l ll ll l l ll ll lll llllll ll llll llll l llll lll ll llll llll ll l lllllll lllll l ll ll l l lll l l lll ll l l ll l lll ll l lll ll llllllll ll l lllll llllll lll lllll l ll llllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − l ll ll ll lll l lll l l ll lll llll ll l ll ll l ll ll ll lll l ll l l llll ll l ll l ll ll ll ll l ll l ll llllll ll lll ll l lll lll l ll lll ll lllll l l llll ll llll l ll ll l ll lll lll l ll ll l ll ll l ll l ll ll l ll l lll l ll llll l lll ll l lll l l l lll l ll ll l ll lll l ll l ll ll ll l ll l l ll l ll lll ll ll ll ll l lll ll ll ll l ll ll ll ll ll llll ll ll l lll l ll ll ll l lll ll ll l ll ll lll l l ll l ll ll lll lll l l l ll lll ll lll l ll ll lll ll l ll lll ll ll ll l l l ll lll l ll ll ll l ll l lll ll ll l lllll l ll lll ll ll ll ll ll ll ll lll l l llll l l llll ll lll l ll l ll llll ll lll lll l llll lll lll lll ll ll lll ll lll llll l l llll llll lll l ll ll ll lll ll lll ll lll ll l lll ll ll l llll ll ll l lllll lll ll lll lll lll llll lllll l ll llll l lll l ll ll l ll lll lllll ll l ll ll ll ll ll ll llll l llllll l ll llll l ll ll lll ll llll lllllll ll l lllll llll l lll ll ll lll l llll llllll lll ll lll lll ll ll l lll lll llll l llll lll l l lllll llll lll ll llllll lll lll ll ll lll ll ll lll l llll ll ll ll llll lll l lll lllll lll llll lll ll ll ll llll lll l llll ll lll ll ll l llll lll ll l llll lllllll l ll llll lll ll lll lll ll l lll lllll lll ll llll ll ll llll lllll ll llll ll l l ll ll l l ll ll lll llllll ll llll llll l llll lll ll llll llll ll l lllllll lllll l ll ll l l lll l l lll ll l l ll l lll ll l lll ll llllllll ll l lllll llllll lll lllll l ll llllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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−3 −2 −1 0 1 2 3 − − − l Figure 12: Local depth clustering of n = 1000 samples from the r = 0 . s = 10 , , q = 0 . , . , . , . 50 in each row (from the topdown). 80re provided by LLD, LSD, and KDE. Turning to the true number of clusters, the bestresults for (K) Trimodal III are obtained using LLD while the best results for Bimodaland Circular 3 are obtained using LSD. Finally, we notice that the merging algorithm inChazal et al. [2013] may be used to improve the results of KDE, LLD and LSD for thecircular densities in Tables 5, 7, 9 and 11. As explained previously in Subsection 4.2, onecan improve the performance of LSD using smaller values of q . Clustering errors (Hausdorff distance) (H) Bimodal IV (K) TrimodalIII (L) Quadri-modal LLD-0.05-50 0.22 (0.17) 0.17 (0.12) 0.16 (0.10) LLD-0.1-30 0.05 (0.11) LLD-0.1-50 0.02 (0.08) 0.12 (0.17) 0.26 (0.16) LSD-0 . LSD-0 . LSD-0 . . * Clustering errors (distance in probability) (H) Bimodal IV (K) TrimodalIII (L) Quadri-modal LLD-0.1-30 0.09 (0.19)0.13 (0.28) 0.04 (0.04) LLD-0.1-50 0.04 (0.12)0.05 (0.18) 0.04 (0.03) LSD-0 . . LSD-0 . . * n = 1000 samples for the densities (H) BimodalIV, (K) Trimodal III, (L) Quadrimodal and Clustering errors (Hausdorff distance) Circular 2 Circular 2Cauchy Circular 3 Circular 4CauchyKDE 0.57 (0.07) 0.54 (0.08) 0.32 (0.01) 0.08 (0.10)LLD-0.05-30 0.61 (0.05) 0.56 (0.07) 0.33 (0.01) 0.02 (0.03)LLD-0.05-50 0.59 (0.05) 0.52 (0.07) 0.34 (0.02) 0.02 (0.01)LLD-0.1-30 0.52 (0.06) 0.45 (0.08) 0.32 (0.02) LLD-0.1-50 0.49 (0.07) 0.43 (0.10) LLD-0.15-30 0.43 (0.12) 0.35 (0.20) 0.32 (0.03) 0.02 (0.01)LLD-0.15-50 0.39 (0.16) . LSD-0 . LSD-0 . . . . * Clustering errors (distance in probability) Circular 2 Circular 2Cauchy Circular 3 Circular 4Cauchy82DE 0.31 (0.04)0.62 (0.08) 0.32 (0.05)0.63 (0.10) 0.29 (0.03)0.58 (0.06) 0.17 (0.25)0.21 (0.30)LLD-0.05-30 0.34 (0.03)0.66 (0.07) 0.32 (0.05)0.65 (0.09) 0.32 (0.03)0.59 (0.07) 0.04 (0.09)0.04 (0.11)LLD-0.05-50 0.33 (0.04)0.64 (0.07) 0.33 (0.07)0.62 (0.09) 0.32 (0.05)0.57 (0.08) 0.03 (0.04)0.03 (0.05)LLD-0.1-30 0.31 (0.06)0.59 (0.10) 0.32 (0.12)0.55 (0.15) 0.32 (0.06)0.55 (0.11) 0.02 (0.01) LLD-0.1-50 0.32 (0.10)0.58 (0.12) 0.33 (0.15)0.53 (0.18) 0.34 (0.08)0.54 (0.14) 0.02 (0.01) LLD-0.15-30 0.30 (0.13)0.51 (0.18) 0.27 (0.20)0.40 (0.26) 0.35 (0.07)0.55 (0.11) 0.02 (0.01) LLD-0.15-50 0.30 (0.18)0.47 (0.23) 0.24 (0.20)0.35 (0.26) 0.38 (0.09)0.54 (0.14) 0.03 (0.01)0.03 (0.01)LLD-0.2-30 0.25 (0.18)0.40 (0.25) 0.19 (0.19)0.25 (0.22) 0.41 (0.09)0.55 (0.10) 0.04 (0.03)0.04 (0.05)LLD-0.2-50 0.20 (0.18)0.30 (0.26) 0.14 (0.15)0.20 (0.17) 0.46 (0.09)0.51 (0.10) 0.04 (0.03)0.05 (0.06)LSD-0 . LSD-0 . LSD-0 . . . . * n = 1000 samples for the densities Circular 2, Circular2 Cauchy, Circular 3 and Circular 4 Cauchy. In parentheses the standard deviation.83 lustering errors (Hausdorff distance) Bimodal Quadrimodal Mult. Bimodal Mult. Quadri-modalKDE 0.09 (0.15) 0.05 (0.08) 0.20 (0.21) 0.09 (0.09)LLD-0.05-30 0.25 (0.19) 0.04 (0.06) 0.05 (0.11) 0.03 (0.04)LLD-0.05-50 0.16 (0.18) 0.02 (0.04) LLD-0.1-30 0.01 (0.04) LSD-0 . . . . * Clustering errors (distance in probability) Bimodal Quadrimodal Mult. Bimodal Mult. Quadri-modalKDE 0.03 (0.04)0.05 (0.09) 0.08 (0.15)0.11 (0.20) 0.05 (0.06)0.08 (0.12) 0.25 (0.29)0.31 (0.36)LLD-0.05-30 0.06 (0.06)0.12 (0.12) 0.08 (0.16)0.11 (0.21) 0.024 (0.06)0.04 (0.10) 0.08 (0.19)0.09 (0.22)LLD-0.05-50 0.05 (0.05)0.09 (0.11) 0.05 (0.14)0.06 (0.17) 0.01 (0.01) LLD-0.1-30 0.01 (0.02)0.01 (0.03) 0.02 (0.06)0.02 (0.07) 0.02 (0.06)0.04 (0.10) 0.08 (0.19)0.09 (0.22)LLD-0.1-50 0.01 (0.01)0.01 (0.02) 0.01 (0.00) LSD-0 . . . . * n = 1000 samples for the densities Bimodal, Quadrimodal,Mult. Bimodal and Mult. Quadrimodal. In parentheses the standard deviation. Number of times the true clusters are detected correctly (H) Bimodal IV (K) TrimodalIII (L) Quadri-modal (29) 45 (26) (0) 100 (0) LLD-0.1-30 (0) 83 (17) (14) 79 (7) (69) 29 (2) (0) 100 (0) LLD-0.1-50 (0) 93 (7) (20) 78 (2) (82) 16 (2) (0) 100 (0) LSD-0 . (0) 100 (0) LSD-0 . (0) 100 (0) LSD-0 . (0) 100 (0) (51) 49 (0) (100) 0 (0) (89) 11 (0)LSD-0 . (0) 100 (0) (61) 39 (0) (100) 0 (0) (94) 6 (0)Circular 2 Circular 2Cauchy Circular 3 Circular 4CauchyKDE (0) 0 (100) (0) 0 (100) (0) 0 (100) 0) 68 (32)LLD-0.05-30 (0) 0 (100) (0) 0 (100) (0) 0 (100) (0) 97 (3)LLD-0.05-50 (0) 0 (100) (0) 0 (100) (0) 0 (100) (0) 99 (1)LLD-0.1-30 (0) 0 (100) (0) 2 (98) (0) 0 (100) (0) 100 (0) LLD-0.1-50 (0) 0 (100) (0) 5 (95) (0) 1 (99) (0) 100 (0) LLD-0.15-30 (0) 7 (93) (0) 27 (73) (0) 0 (100) (0) 100 (0) LLD-0.15-50 (0) 15 (85) (0) 37 (63) (0) 1 (99) (0) 100 (0) LLD-0.2-30 (0) 24 (76) (26) 49 (25) (0) 2 (98) (4) 96 (0)LLD-0.2-50 (0) 42 (58) (46) 41 (13) (7) 47 (46) (7) 93 (0)LSD-0 . (0) 100 (0) LSD-0 . (0) 100 (0) LSD-0 . . (0) 69 (31) (44) 43 (13) (4) 51 (45) (6) 94 (0)LSD-0 . . (0) 99 (1) (0) 100 (0) LLD-0.1-30 (0) 98 (2) (0) 99 (1) (0) 85 (15) (0) 93 (7)LLD-0.1-50 (0) 99 (1) (0) 100 (0) (0) 99 (1) (0) 100 (0) LSD-0 . (0) 100 (0) (13) 48 (39) (93) 6 (1)LSD-0 . (0) 100 (0) (32) 49 (19) (99) 1 (0)LSD-0 . (0) 100 (0) (0) 100 (0) (12) 63 (25) (77) 18 (5)LSD-0 . (0) 100 (0) (0) 100 (0) (29) 66 (5) (97) 3 (0)Table 7: Number of times over 100 replications with n = 1000 samples that the procedureidentifies the true number of clusters for the densities (H) Bimodal IV, (K) Trimodal III,(L) Quadrimodal, Clustering errors (Hausdorff distance) (H) Bimodal IV (K) TrimodalIII (L) Quadri-modal LLD-0.1-50 0.03 (0.08) 0.20 (0.20) 0.38 (0.14) 0.07 (0.02)LSD-0 . LSD-0 . . . * Clustering errors (distance in probability) (H) Bimodal IV (K) TrimodalIII (L) Quadri-modal LLD-0.1-50 0.01 (0.03)0.02 (0.06) 0.07 (0.06)0.09 (0.07) 0.10 (0.03)0.18 (0.04) 0.07 (0.01)0.07 (0.02)LSD-0 . LSD-0 . . . * n = 500 samples for the densities (H) BimodalIV, (K) Trimodal III, (L) Quadrimodal and Clustering errors (Hausdorff distance) Circular 2 Circular 2Cauchy Circular 3 Circular 4CauchyKDE 0.55 (0.06) 0.52 (0.08) 0.32 (0.02) 0.10 (0.10)LLD-0.05-30 0.58 (0.05) 0.53 (0.07) 0.34 (0.02) 0.05 (0.06)LLD-0.05-50 0.53 (0.06) 0.49 (0.06) 0.33 (0.04) LLD-0.1-30 0.51 (0.06) 0.46 (0.09) 0.32 (0.04) LLD-0.1-50 0.49 (0.06) 0.41 (0.15) LLD-0.15-30 0.45 (0.10) 0.37 (0.17) 0.32 (0.04) 0.04 (0.04)LLD-0.15-50 0.40 (0.15) . LSD-0 . LSD-0 . . . . * Clustering errors (distance in probability) Circular 2 Circular 2Cauchy Circular 3 Circular 4CauchyKDE 0.31 (0.04)0.62 (0.08) 0.32 (0.06)0.61 (0.10) 0.29 (0.03)0.56 (0.07) 0.18 (0.23)0.23 (0.28)LLD-0.05-30 0.34 (0.04)0.65 (0.07) 0.33 (0.07)0.62 (0.09) 0.34 (0.05)0.59 (0.09) 0.09 (0.17)0.11 (0.21)LLD-0.05-50 0.33 (0.07)0.59 (0.10) 0.34 (0.10)0.58 (0.10) 0.37 (0.09)0.56 (0.13) 0.05 (0.07)0.05 (0.09)LLD-0.1-30 0.32 (0.06)0.59 (0.10) 0.34 (0.14)0.54 (0.15) 0.35 (0.08)0.54 (0.13) 0.03 (0.02) LLD-0.1-50 0.34 (0.10)0.57 (0.12) 0.32 (0.17)0.48 (0.20) 0.39 (0.12)0.52 (0.18) 0.04 (0.02)0.04 (0.03)LLD-0.15-30 0.32 (0.12)0.54 (0.16) 0.31 (0.19)0.45 (0.24) 0.38 (0.09)0.53 (0.14) 0.05 (0.03)0.05 (0.04)LLD-0.15-50 0.30 (0.16)0.47 (0.22) 0.25 (0.19)0.34 (0.25) 0.40 (0.09)0.46 (0.12) 0.05 (0.03)0.06 (0.04)LLD-0.2-30 0.29 (0.17)0.46 (0.23) 0.23 (0.19)0.30 (0.20) 0.44 (0.09)0.51 (0.10) 0.09 (0.05)0.12 (0.10)LLD-0.2-50 0.26 (0.20)0.37 (0.27) 0.19 (0.17)0.25 (0.16) 0.37 (0.07)0.50 (0.08) 0.10 (0.06)0.16 (0.12)LSD-0 . . . . . . * n = 500 samples for the densities Circular 2, Circular 2Cauchy, Circular 3 and Circular 4 Cauchy. In parentheses the standard deviation. Clustering errors (Hausdorff distance) Bimodal Quadrimodal Mult. Bimodal Mult. Quadri-modalKDE 0.13 (0.17) 0.04 (0.06) 0.45 (0.01) 0.20 (0.06)LLD-0.05-30 0.26 (0.17) 0.05 (0.06) 0.03 (0.06) LLD-0.05-50 0.12 (0.16) 0.02 (0.04) . . . . * Clustering errors (distance in probability) Bimodal Quadrimodal Mult. Bimodal Mult. Quadri-modalKDE 0.04 (0.05)0.07 (0.10) 0.06 (0.11)0.08 (0.15) 0.08 (0.07)0.13 (0.12) 0.50 (0.22)0.67 (0.27)LLD-0.05-30 0.08 (0.06)0.14 (0.11) 0.09 (0.17)0.12 (0.21) 0.02 (0.02) . . . . * n = 500 samples for the densities Bimodal, Quad-rimodal, Mult. Bimodal and Mult. Quadrimodal. In parentheses the standard deviation. Number of times the true clusters are detected correctly (H) Bimodal IV (K) TrimodalIII (L) Quadri-modal (47) 35 (18) (0) 29 (71)LLD-0.05-30 (0) 15 (85) (0) 34 (66) (24) 32 (44) (0) 99 (1)LLD-0.05-50 (0) 45 (55) (17) 50 (33) (79) 16 (5) (0) 100 (0) LLD-0.1-30 (0) 82 (18) (14) 78 (8) (88) 11 (1) (0) 100 (0) LLD-0.1-50 (0) 92 (8) (37) 63 (0) (99) 1 (0) (1) 99 (0)LSD-0 . (0) 100 (0) LSD-0 . . (0) 100 (0) (67) 33 (0) (100) 0 (0) (96) 4 (0)LSD-0 . (0) 100 (0) (84) 16 (0) (100) 0 (0) (99) 1 (0)Circular 2 Circular 2Cauchy Circular 3 Circular 4CauchyKDE (0) 0 (100) (0) 0 (100) (0) 0 (100) (0) 64 (36)90LD-0.05-30 (0) 0 (100) (0) 0 (100) (0) 0 (100) (1) 86 (13)LLD-0.05-50 (0) 0 (100) (0) 0 (100) (0) 1 (99) (1) 97 (2)LLD-0.1-30 (0) 0 (100) (0) 3 (97) (0) 2 (98) (1) 99 (0) LLD-0.1-50 (0) 0 (100) (0) 13 (87) (0) 11 (89) (1) 99 (0) LLD-0.15-30 (0) 3 (97) (0) 21 (79) (0) 7 (93) (3) 97 (0)LLD-0.15-50 (0) 15 (85) (0) 43 (57) (8) 47 (45) (4) 96 (0)LLD-0.2-30 (0) 15 (85) (35) 38 (27) (10) 38 (52) (32) 68 (0)LLD-0.2-50 (0) 35 (65) (53) 36 (11) (71) 24 (5) (44) 56 (0)LSD-0 . (1) 99 (0) LSD-0 . (1) 99 (0) LSD-0 . (48) 43 (9) (9) 35 (56) (43) 57 (0)LSD-0 . . (22) 51 (27) (92) 8 (0)LSD-0 . (12) 65 (23) (97) 3 (0) (94) 6 (0) (93) 7 (0)Bimodal Quadrimodal Mult. Bimodal Mult. Quadri-modalKDE (0) 67 (33) (0) 86 (14) (0) 5 (95) (0) 13 (87)LLD-0.05-30 (0) 29 (71) (0) 78 (22) (0) 95 (5) (0) 97 (3) LLD-0.05-50 (0) 65 (35) (0) 94 (6) (0) 99 (1) (6) 94 (0)LLD-0.1-30 (0) 87 (13) (0) 100 (0) (0) 94 (6) (6) 94 (0)LLD-0.1-50 (0) 96 (4) (0) 100 (0) (0) 99 (1) (10) 90 (0)LSD-0 . (0) 100 (0) (61) 36 (3) (100) 0 (0)LSD-0 . (0) 100 (0) (85) 15 (0) (100) 0 (0)LSD-0 . (0) 100 (0) (0) 100 (0) (56) 42 (2) (96) 4 (0)LSD-0 . (0) 100 (0) (1) 99 (0) (73) 27 (0) (100) 0 (0)Table 11: Number of times over 100 replications with nn