Measuring Mass Concentrations and Estimating Density Contour Clusters-An Excess Mass Approach

By Wolfgang Polonik. 1995

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Citations in the corpus

Security analysis of online centroid anomaly detection (2010, methodology): “Hence several direct methods have been proposed for anomaly detection, e.g., oneclass SVM (Schölkopf et al., 2001), support vector data description (SVDD) (Tax and Duin, 1999a,b), and density level set estimation (Polonik, 1995; Tsybakov, 1997; Steinwart et al., 2005).”
PERFORMANCE GUARANTEES FOR INDIVIDUALIZED TREATMENT RULES (2011, background): “3) is similar to the margin condition in classification [18, 24, 32]; in classification this assumption is often used to obtain sharp upper bounds on the excess 0–1 risk in terms of other surrogate risks [2].”
Topological Data Analysis (2016, background): “The excess mass functional is studied in Polonik (1995); M¨uller & Sawitzki (1991).”
Asymptotics and optimal bandwidth selection for highest density region estimation (2010, background): “nment [R Development Core Team (2008)] is made available on the Internet. The HDR estimation problem has an established literature. Contributions include Hartigan (1987), Mu¨ller and Sawitzki (1991), Polonik (1995), Hyndman (1996), Tsybakov (1997), Ba´ıllo, Cuesta-Albertos and Cuevas (2001), Ba´ıllo”
On Statistical Properties of Sets Fulfilling Rolling-Type Conditions (2011, methodology): “2) in order to apply the results in [25] for classes of sets defined in terms of reach properties.”
Theory of classification : a survey of some recent advances (2005, background): “For regression, Birgé and Massart [36] and Lee, Bartlett and Williamson [134] proved exponential inequalities The fact that this phenomenon also occurs in the context of discriminant analysis and classification, under conditions on the noise (sometimes called margin) has been pointed out by Mammen a”
Optimal rates for plug-in estimators of density level sets (2006, background): “The assumption under which the underlying density has γ-exponent at level λ was first introduced by Polonik (1995) .”
A Classification Framework for Anomaly Detection (2005, background): “[17, 21]). Some other applications including estimation of non-linear functionals of densities, density estimation, regression analysis and spectral analysis are briefly described in [ 19 ].”
Distribution‐free prediction bands for non‐parametric regression (2014, methodology): “Assumption 2 is related to the notion of the ‘γ-exponent’ condition that was introduced by Polonik (1995) and widely used in the density level set literature (Tsybakov, 1997; Rigollet and Vert, 2009).”
Smooth Discrimination Analysis (1999, background): “Taking into account (13) and (14), we obtain that, if E holds, Vn;m pnj 1 2nPni=1f(h1 h1;0)(Xi) E(h1 h1;0)(Xi)gj d(1 )=2 4 (Ĝn;m; GK) +pnj 1 2mPmi=1f(h2 h2;0)(Yi) E(h2 h2;0)(Yi)gj d(1 )=2 4 (Ĝn;m; GK) sup G2G:d4(G;GK) c1n 2 2+ pnj 1 2n Pni=1f(h1 h1;0)(Xi) E(h1 h1;0)(Xi)gj d(1 )=2 4 (G;GK) + sup G2G:”
Distribution Free Prediction Sets (2013, methodology): “To control the regularity of p at level t (α), a common assumption is the γ -exponent condition, which was first introduced by Polonik (1995) and has been used by many others (see, e.g., Tsybakov 1997 and Rigollet and Vert 2009).”
A nonparametric approach to the estimation of lengths and surface areas (2007, background): “In a way, the length and surface area estimation problem can be seen as a further, more difficult, stage in set estimation theory, after the early developments concerned with the estimation of volume (associated with the L1 (measure) distance; see [8]), “visual” shape (associated with the Hausdorff ”
Set estimation: another bridge between statistics and geometry (2009, methodology): “Asymptotic results for the estimator restricted to much more general classes B are given by Polonik (1995) using the empirical process theory.”
Total Error in a Plug-in Estimator of Level Sets (2003, unknown): “The statement Lebff = cg = 0 just means that f is not at at level c, an unavoidable condition in nonparametric estimation of level sets ff > cg (see, for example, Tsybakov 1997, Polonik 1995, 1997).”
Estimating the Support of a High-Dimensional Distribution (2001, unknown): “…1980), estimating manufacturing yields (Stoneking, 1999), econometrics and generalized non-linear principal curves (Tsybakov, 1997; Korostelev and Tsybakov, 1993), regression and spectral analysis (Polonik, 1997), tests for multimodality and clustering (Polonik, 1995b) and others (M¨uller, 1992).”
A Unifying Review of Deep and Shallow Anomaly Detection (2020, background): “We can formally express this objective as the problem of density level set estimation [214]–[217], which is equivalent to minimum volume set estimation [218]–[220] for the special case of density-based sets.”
Asymptotic normality of plug-in level set estimates. (2009, methodology): “Relevant published work (not yet mentioned above) include Hartigan (1987), Polonik (1995), Cavalier (1997), Tsybakov (1997), Walther (1997), Baíllo, Cuevas and Justel (2000), Baíllo, Cuestas-Albertos and Cuevas (2001), Cuevas, Febrero and Fraiman (2000), Baíllo (2003), Tsybakov (2004), Steinwart,…”
PLUG‐IN ESTIMATION OF GENERAL LEVEL SETS (2006, background): “The estimation of such density level sets is useful in mode estimation (Müller & Sawitzki, 1991; Polonik, 1995), clustering (Cuevas, Febrero & Fraiman, 2000; Cuevas, Febrero & Fraiman, 2001) or detection of abnormal behavior in a system (Devroye & Wise, 1980; Baı́llo, Cuevas & Justel, 2000;…”