Set Estimation from Reflected Brownian Motion

Cholaquidis, Alejandro; Fraiman, Ricardo; Lugosi, Gábor; Pateiro-López, Beatriz

doi:10.1111/rssb.12149

Cited by 17 publications

(18 citation statements)

References 52 publications

(114 reference statements)

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“…This is the case (for example) for some reflected diffusions and particularly for reflected Brownian motion (RBM). This has been recently proven in corollary 1 in Cholaquidis et al (2016) for RBM without drift (see also Cholaquidis et al (2020) for RBM with drift). RBM with drift is defined as follows: let D be a bounded domain in R d (i.e., a bounded, connected open set), such that ∂D is C 2 .…”

Section: Resultsmentioning

confidence: 52%

“…Given that in general the set S is unknown, the natural idea is to plug into n ∂S an estimator of S. There are different kinds of set estimators, depending on the geometric restrictions imposed on S and the structure of the data (see Devroye and Wise (1980), Cholaquidis et al (2016)) and references therein). One of the most studied in the literature, which is also universally consistent, is the Devroye-Wise estimator (see Devroye and Wise (1980)), given by…”

Section: Devroye-wise Based Approachmentioning

confidence: 99%

“…This result can be applied to many deterministic or random situations, to obtain explicit convergence rates. We focus on two random situations: the case of iid drawn on S (with a density bounded from below), and the case of random trajectories of reflected diffusions on S. In particular, we provide convergence rates when the trajectories are the result of reflected Brownian motion (see Cholaquidis et al (2016Cholaquidis et al ( , 2020). This last setting has several applications in ecology, such as home-range estimation, where the trajectory is obtained by recording the location of an animal (or several animals) living in an area S that is called the home range (see for instance Cholaquidis et al (2016Cholaquidis et al ( , 2020, Baíllo and Chacón (2018) and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…Neither the non-iid case nor the non-convex case, when only a sample inside the set is available and d > 2, have been addressed in the literature. A result for this general setup has also been considered for d = 2 in Theorem 4 in Cholaquidis et al (2016). To fill this gap, we propose to use consistent estimators that are based on Crofton´s formula.…”

Section: Introductionmentioning

confidence: 99%

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Surface and length estimation based on Crofton's formula

Aaron¹,

Cholaquidis²,

Fraiman³

2020

Preprint

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We study the problem of estimating the surface area of the boundary of a sufficiently smooth set when the available information is only a set of points (random or not) that becomes dense (with respect to Hausdorff distance) in the set or the trajectory of a reflected diffusion. We obtain consistency results in this general setup, and we derive rates of convergence for the iid case or when the data corresponds to the trajectory of a reflected Brownian motion. We propose an algorithm based on Crofton's formula, which estimates the number of intersections of random lines with the boundary of the set by counting, in a suitable way (given by the proposed algorithm), the number of intersections with the boundary of two different estimators: the Devroye-Wise estimator and the α-convex hull of the data. As a by-product, our results also cover the convex case, for any dimension.

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Section: Resultsmentioning

confidence: 52%

Section: Devroye-wise Based Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Surface and length estimation based on Crofton's formula

Aaron¹,

Cholaquidis²,

Fraiman³

2020

Preprint

Self Cite

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“…Mind that, in general, it can happen that N x = ∅. In this case, the uniform exterior sphere condition is not satisfied (see, for instance, the examples in [6], Fig. 5 and in [7], page 4).…”

Section: Concept Of Solutionmentioning

confidence: 99%

Well-posedness of a coupled system of Skorohod-like stochastic differential equations

Thieu,

Muntean

2020

Preprint

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We study the existence and uniqueness of solutions to a coupled nonlinear system of Skorohod-like stochastic differential equations with reflecting boundary condition. The setting describes the evacuation dynamics of a mixed crowd composed of both active and passive pedestrians moving through a domain with obstacles, fire and smoke. As main working techniques, we use compactness methods and the Skorohod's representation of solutions to SDEs posed in bounded domains. The challenge is to handle the coupling and the nonlinearities present in the model equations together with the multipleconnectedness of the domain and the pedestrian-obstacle interaction.

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Coupled stochastic systems of Skorokhod type: Well‐posedness of a mathematical model and its applications

Thieu

Muntean

Melnik

2022

Math Methods in App Sciences

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Population dynamics with complex biological interactions, accounting for uncertainty quantification, are critical for many application areas. However, due to the complexity of biological systems, the mathematical formulation of the corresponding problems faces the challenge that the corresponding stochastic processes should, in most cases, be considered in bounded domains. We propose a model based on a coupled system of reflecting Skorokhod‐type stochastic differential equations with jump‐like exit from a boundary. The setting describes the population dynamics of active and passive populations. As main working techniques, we use compactness methods and Skorokhod's representation of solutions to SDEs posed in bounded domains to prove the well‐posedness of the system. This functional setting is a new point of view in the field of modeling and simulation of population dynamics. We provide the details of the model, as well as representative numerical examples, and discuss the applications of a Wilson–Cowan‐type system, modeling the dynamics of two interacting populations of excitatory and inhibitory neurons. Furthermore, the presence of random input current, reflecting factors together with Poisson jumps, increases firing activity in neuronal systems.

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Set Estimation from Reflected Brownian Motion

Cited by 17 publications

References 52 publications

Surface and length estimation based on Crofton's formula

Surface and length estimation based on Crofton's formula

Well-posedness of a coupled system of Skorohod-like stochastic differential equations

Coupled stochastic systems of Skorokhod type: Well‐posedness of a mathematical model and its applications

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