Order-k α-hulls and α-shapes

Krasnoshchekov, D.; Polishchuk, Valentin

doi:10.1016/j.ipl.2013.07.023

Cited by 14 publications

(13 citation statements)

References 32 publications

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“…A depth function based on α shapes can give an idea about the median of each mode in a multimodal distribution [ Krasnoshchekov and Polishchuk , ]. Multimodality can be reflected as disconnected components in the α shape.…”

Section: Methodsmentioning

confidence: 99%

“…Considering that a convex boundary might be ill‐suited to describe the shape of a point cloud (Figure ), we decided to test a nonconvex depth‐measure, based on α shapes [ Edelsbrunner , ]. The two main advantages of such a depth measure are that it does not require the assumption, implicit in most depth functions, that the underlying multivariate distribution is unimodal [ Liu et al ., ; Krasnoshchekov and Polishchuk , ], and that it might give a tighter delimitation of the behavioral parameter space, i.e., less space to be explored. An added advantage is that a depth function based on alpha shapes has potential to detect different modes in a multivariate distribution [ Krasnoshchekov and Polishchuk , ].…”

Section: Introductionmentioning

confidence: 99%

“…This could be advantageous when exploring the behavioral parameter space, since usually little is known a priori regarding its shape. Krasnoshchekov and Polishchuk [] discuss a depth function based on α shapes [ Edelsbrunner , ] that avoids the unimodality assumption.…”

Section: Introductionmentioning

confidence: 99%

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Exploring the hydrological robustness of model-parameter values with alpha shapes

et al. 2013

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[1] Estimation of parameter values in hydrological models has gradually moved from subjective, trial-and-error methods into objective estimation methods. Translation of nature's complexity to bit operations is an uncertain process as a result of data errors, epistemic gaps, computational deficiencies, and other limitations, and relies on calibration to fit model output to observed data. The robustness of the calibrated parameter values to these types of uncertainties is therefore an important concern. In this study, we investigated how the hydrological robustness of the model-parameter values varied within the geometric structure of the behavioral (well-performing) parameter space with a depth function based on shapes and an in-depth posterior performance analysis of the simulations in relation to the observed discharge uncertainty. The shape depth is a nonconvex measure that may provide an accurate and tight delimitation of the geometric structure of the behavioral space for both unimodal and multimodal parameter-value distributions. WASMOD, a parsimonious rainfall-runoff model, was applied to six Honduran and one UK catchment, with differing data quality and hydrological characteristics. Model evaluation was done with two performance measures, the Nash-Sutcliffe efficiency and one based on flow-duration curves. Deep parameter vectors were in general found to be more hydrologically robust than shallow ones in the analyses we performed; model-performance values increased with depth, deviations to the observed data for the high-flow aspects of the hydrograph generally decreased with increasing depth, deep parameter vectors generally transferred in time with maintained high performance values, and the model had a low sensitivity to small changes in the parameter values. The tight delimitation of the behavioral space provided by the shapes depth function showed a potential to improve the efficiency of calibration techniques that require further exploration. For computational reasons only a three-parameter model could be used, which limited the applicability of this depth measure and the conclusions drawn in this paper, especially concerning hydrological robustness at low flows.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exploring the hydrological robustness of model-parameter values with alpha shapes

et al. 2013

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“…α‐shape (Edelsbrunner et al, ) is a classical computational geometry tool for restoring the curve out of a point cloud. Its generalization to k ‐order α‐shape was shown to be extremely effective for datasets featuring samples with outstanding error, extreme values, noise, or even nonrelevant data (Krasnoshchekov & Polishchuk, ). The last cited paper narrates the algorithmic and combinatorial properties of k ‐order α‐shape that enable robust reconstruction of the inner shape of a largely scattered dataset—the type of data to which PKiKP/PcP differential measurements belong.…”

Section: Methodsmentioning

confidence: 99%

Dissimilarity of the Earth's Inner Core Surface Under South America and Northeastern Asia Revealed by Core Reflected Phases

2019

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Resolving topography of the inner core boundary (ICB) and the structure and composition of the nearby region is key to improving our understanding of solidification of the Earth's inner core. Observations of travel times and amplitudes of short-period seismic phases of PKiKP and PcP reflected, respectively, off the inner and outer boundary of the liquid core, provide essential constraints on the properties of this region. We revisit heterogeneities of ICB using a total of more than 1,300 new differential travel times and amplitude ratios of PKiKP and PcP measured at 3.2-35.2°and reflected off the core's boundaries under Northeastern Asia and South America. We observe a statistically significant systematic bias between the measurements collected in the two spots. We carefully examine its origin in terms of contributions by various Earth's shells and find that most of variance in PKiKP-PcP differential travel times measured above the epicentral distance of 16.5°in Northeastern Asia can be accounted for by mantle corrections. We find slight disparity of about 1-3 km between the outer core thickness under Asia and America; the ICB density jump under Northeastern Asia is about 0.3 g/cm 3 , which is three times as small as under South America. The findings are interpretable either as evidence for inner core hemispherical asymmetry, whereby crystallization dominates in the West and melting in the East (not vice versa), or in terms of two disconnected mosaic patches with contrasting properties.

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“…We are interested in the function w k : Del k (X) → R that maps each cell to the minimum radius, r, such that the corresponding intersection of domains contains a point at distance at most r from each one of its k nearest neighbors. The sublevel sets of w k generalize the notion of alpha shapes from k = 1 to orders k ≥ 1 [5,13]. Recently, the stochastic properties of w k have been studied [6] and algorithms for computing the persistence have been presented [7].…”

Section: Introductionmentioning

confidence: 99%

A step in the Delaunay mosaic of order k

2021

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Given a locally finite set $$X \subseteq {{\mathbb {R}}}^d$$ X ⊆ R d and an integer $$k \ge 0$$ k ≥ 0 , we consider the function $${\mathbf{w}_{k}^{}} :{\mathrm{Del}_{k}{({X})}} \rightarrow {{\mathbb {R}}}$$ w k : Del k ( X ) → R on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.

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Order-k α-hulls and α-shapes

Cited by 14 publications

References 32 publications

Exploring the hydrological robustness of model-parameter values with alpha shapes

Exploring the hydrological robustness of model-parameter values with alpha shapes

Dissimilarity of the Earth's Inner Core Surface Under South America and Northeastern Asia Revealed by Core Reflected Phases

A step in the Delaunay mosaic of order k

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