On boundary estimation

2004 ◽  
Vol 36 (02) ◽  
pp. 340-354 ◽  
Author(s):  
Antonio Cuevas ◽  
Alberto Rodríguez-Casal
Keyword(s):  
Stochastic Geometry ◽  
Simple Structure ◽  
Convergence Rates ◽  
Hausdorff Metric ◽  
Compact Set ◽  
Boundary Estimation ◽  
Geometrical Character ◽  
Computational Identification ◽  
Union Of Balls ◽  
Universal Consistency

We consider the problem of estimating the boundary of a compact set S ⊂ ℝ d from a random sample of points taken from S. We use the Devroye-Wise estimator which is a union of balls centred at the sample points with a common radius (the smoothing parameter in this problem). A universal consistency result, with respect to the Hausdorff metric, is proved and convergence rates are also obtained under broad intuitive conditions of a geometrical character. In particular, a shape condition on S, which we call expandability, plays an important role in our results. The simple structure of the considered estimator presents some practical advantages (for example, the computational identification of the boundary is very easy) and makes this problem quite close to some basic issues in stochastic geometry.

On boundary estimation

2004 ◽  
Vol 36 (2) ◽  
pp. 340-354 ◽  
Author(s):  
Antonio Cuevas ◽  
Alberto Rodríguez-Casal
Keyword(s):  
Stochastic Geometry ◽  
Simple Structure ◽  
Convergence Rates ◽  
Hausdorff Metric ◽  
Compact Set ◽  
Boundary Estimation ◽  
Geometrical Character ◽  
Computational Identification ◽  
Union Of Balls ◽  
Universal Consistency

We consider the problem of estimating the boundary of a compact set S ⊂ ℝd from a random sample of points taken from S. We use the Devroye-Wise estimator which is a union of balls centred at the sample points with a common radius (the smoothing parameter in this problem). A universal consistency result, with respect to the Hausdorff metric, is proved and convergence rates are also obtained under broad intuitive conditions of a geometrical character. In particular, a shape condition on S, which we call expandability, plays an important role in our results. The simple structure of the considered estimator presents some practical advantages (for example, the computational identification of the boundary is very easy) and makes this problem quite close to some basic issues in stochastic geometry.

scholarly journals Convergence rates for estimators of geodesic distances and Frèchet expectations

2018 ◽  
Vol 55 (4) ◽  
pp. 1001-1013
Author(s):  
Catherine Aaron ◽  
Olivier Bodart
Keyword(s):  
Probability Distribution ◽  
Convergence Rate ◽  
Compact Manifold ◽  
Convergence Rates ◽  
Geodesic Distance ◽  
Convergence Result ◽  
Regularity Conditions ◽  
Compact Set ◽  
Geodesic Distances ◽  
General Convergence

Abstract Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

scholarly journals SIFAT KELENGKAPAN DAN KEKOMPAKAN PADA RUANG METRIK HAUSDORFF

2019 ◽  
Vol 3 (2) ◽  
pp. 78
Author(s):  
Dewanti Inesia Putri ◽  
Arta Ekayanti
Keyword(s):  
Metric Space ◽  
Hausdorff Metric ◽  
Compact Set ◽  
Definition Of ◽  
Hausdorff Metric Space

In this paper, will be discuss the definition of the Hausdorff metric space, completeness of the Hausdorff metric space, and compactness of the Hausdorff metric space. By used the theory of the metric space, the compact set was given the definition of the Hausdorff metric space. By used the completeness of the metric space, it is shown that the Hausdorff metric space was complete if the metric space was complete. Furthermore, used the compactness of the metric space was shown the Hausdorff metric space was compact if the metric space was compact

Convergence Rates of a Class of Uncertain Dynamic Systems

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Juan Ignacio Mulero-Martínez
Keyword(s):  
Dynamic Systems ◽  
Uncertain Systems ◽  
Convergence Rates ◽  
Initial Conditions ◽  
Compact Set ◽  
Robust Nonlinear Control ◽  
Riccati Differential Equation ◽  
Uncertain Dynamic Systems ◽  
Complete Treatment ◽  
The Stability

The problem of stabilization of uncertain systems plays a broad and fundamental role in robust control theory. The paper examines a boundedness theorem for a class of uncertain systems characterized as having a decreasing Lyapunov function in a ringlike region. It is a systematic study on stability that embraces both the transient and steady analysis, covering such aspects as the maximum overshoot of the system state, the stability region and the exponential convergence rate. The emphasis throughout is on deriving dominant time constants and explicit time expressions for a state to reach an invariant set. The central theorem provides a complete treatment of the time evolution of trajectories depending on the specific compact set of initial conditions. Toward this end, the comparison lemma along with a particular Riccati differential equation are essential and conclusive. The scope of questions addressed in the paper, the uniformity of their treatment, the novelty of the proposed theorem, and the obtained results make it very useful with respect to other works on the problem of robust nonlinear control.

Some Remarks on Fractals Generated by a Sequence of Finite Systems of Contractions

2001 ◽  
Vol 8 (4) ◽  
pp. 733-752
Author(s):  
Giorgio Follo
Keyword(s):  
Metric Space ◽  
Radon Measure ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Compact Set ◽  
Regular Measure ◽  
Real Numbers ◽  
Finite Systems ◽  
Bounded Set ◽  
Image Measure

Abstract We generalize some results shown by J. E. Hutchinson in [Indiana Univ. Math. J. 30: 713–747, 1981]. Let be finite systems of contractions on a complete metric space; then, under some conditions on (𝔉𝑛), there exists a unique non-empty compact set 𝐾 such that the sequence defined by ((𝔉1 ○ 𝔉2 ○ ⋯ ○ 𝔉𝑛)(𝐶)) converges to 𝐾 in the Hausdorff metric for every non-empty closed and bounded set 𝐶. If the metric space is also separable and for every there are real numbers strictly between 0 and 1, satisfying the condition , then there exists a unique probability Radon measure μ 𝐾 such that the sequence weakly converges to μ 𝐾 for every probability Borel regular measure ν with bounded support (where by we denote the image measure of ν under a contraction 𝑓). Moreover, 𝐾 is the support of μ 𝐾.

On the estimation of a star-shaped set

2001 ◽  
Vol 33 (4) ◽  
pp. 717-726 ◽  
Author(s):  
Amparo Baíllo ◽  
Antonio Cuevas
Keyword(s):  
Image Analysis ◽  
Random Sample ◽  
Hausdorff Metric ◽  
Point Of View ◽  
Estimation Problem ◽  
Mild Conditions ◽  
Statistical Image Analysis ◽  
Sample Points ◽  
Union Of Balls ◽  
Topological Boundary

The estimation of a star-shaped set S from a random sample of points X1,…,Xn ∊ S is considered. We show that S can be consistently approximated (with respect to both the Hausdorff metric and the ‘distance in measure’ between sets) by an estimator ŝn defined as a union of balls centered at the sample points with a common radius which can be chosen in such a way that ŝn is also star-shaped. We also prove that, under some mild conditions, the topological boundary of the estimator ŝn converges, in the Hausdorff sense, to that of S; this has a particular interest when the proposed estimation problem is considered from the point of view of statistical image analysis.

On the estimation of a star-shaped set

2001 ◽  
Vol 33 (04) ◽  
pp. 717-726 ◽  
Author(s):  
Amparo Baíllo ◽  
Antonio Cuevas
Keyword(s):  
Image Analysis ◽  
Random Sample ◽  
Hausdorff Metric ◽  
Point Of View ◽  
Estimation Problem ◽  
Mild Conditions ◽  
Statistical Image Analysis ◽  
Sample Points ◽  
Union Of Balls ◽  
Topological Boundary

The estimation of a star-shaped set S from a random sample of points X 1,…,X n ∊ S is considered. We show that S can be consistently approximated (with respect to both the Hausdorff metric and the ‘distance in measure’ between sets) by an estimator ŝ n defined as a union of balls centered at the sample points with a common radius which can be chosen in such a way that ŝ n is also star-shaped. We also prove that, under some mild conditions, the topological boundary of the estimator ŝ n converges, in the Hausdorff sense, to that of S; this has a particular interest when the proposed estimation problem is considered from the point of view of statistical image analysis.

Stochastic Geometry Analysis of Cellular Networks

2018 ◽  
Author(s):  
Bartłomiej Błaszczyszyn ◽  
Martin Haenggi ◽  
Paul Keeler ◽  
Sayandev Mukherjee
Keyword(s):  
Cellular Networks ◽  
Stochastic Geometry ◽  
Geometry Analysis

A Comparison of Simple Structure Rotation Criteria in Temporal Exploratory Factor Analysis for Event-Related Potential Data

Methodology ◽  
2019 ◽  
Vol 15 (Supplement 1) ◽  
pp. 43-60 ◽  
Author(s):  
Florian Scharf ◽  
Steffen Nestler
Keyword(s):  
Factor Analysis ◽  
Exploratory Factor Analysis ◽  
Simple Structure ◽  
Ground Truth ◽  
Event Related Potential ◽  
Suitable Alternative ◽  
Factor Rotation ◽  
Rotation Methods ◽  
Simple Structure Rotation ◽  
Temporal Overlap

Abstract. It is challenging to apply exploratory factor analysis (EFA) to event-related potential (ERP) data because such data are characterized by substantial temporal overlap (i.e., large cross-loadings) between the factors, and, because researchers are typically interested in the results of subsequent analyses (e.g., experimental condition effects on the level of the factor scores). In this context, relatively small deviations in the estimated factor solution from the unknown ground truth may result in substantially biased estimates of condition effects (rotation bias). Thus, in order to apply EFA to ERP data researchers need rotation methods that are able to both recover perfect simple structure where it exists and to tolerate substantial cross-loadings between the factors where appropriate. We had two aims in the present paper. First, to extend previous research, we wanted to better understand the behavior of the rotation bias for typical ERP data. To this end, we compared the performance of a variety of factor rotation methods under conditions of varying amounts of temporal overlap between the factors. Second, we wanted to investigate whether the recently proposed component loss rotation is better able to decrease the bias than traditional simple structure rotation. The results showed that no single rotation method was generally superior across all conditions. Component loss rotation showed the best all-round performance across the investigated conditions. We conclude that Component loss rotation is a suitable alternative to simple structure rotation. We discuss this result in the light of recently proposed sparse factor analysis approaches.

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