On resampling schemes for polytopes

2019 ◽  
Vol 56 (4) ◽  
pp. 959-980
Author(s):  
Weinan Qi ◽  
Mahmoud Zarepour
Keyword(s):  
Convex Hull ◽  
Asymptotic Theory ◽  
Convex Set ◽  
Real Life ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Underlying Distribution ◽  
Asymptotic Consistency ◽  
Computational Intractability ◽  
Practical Implications

AbstractThe convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. To approximate the distribution of the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately most of the asymptotic results are computationally intractable. To address this computational intractability, we consider consistent bootstrapping schemes for certain cases. Let $S_n=\{X_i\}_{i=1}^{n}$ be a sequence of independent and identically distributed random points uniformly distributed on an unknown convex set in $\mathbb{R}^{d}$ ($d\ge 2$ ). We suggest a bootstrapping scheme that relies on resampling uniformly from the convex hull of $S_n$ . Moreover, the resampling asymptotic consistency of certain functionals of convex hulls is derived under this bootstrapping scheme. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. For $d=2$ , we investigate the asymptotic consistency of the suggested bootstrapping scheme for the area of the symmetric difference and the perimeter difference between the actual convex set and its estimate. In all cases the consistency allows us to rely on the suggested resampling scheme to study the actual distributions, which are not computationally tractable.

The probability that two samples in the plane will have disjoint convex hulls

1978 ◽  
Vol 15 (04) ◽  
pp. 790-802 ◽  
Author(s):  
L. C. G. Rogers
Keyword(s):  
Convex Hull ◽  
Continuous Distribution ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Absolutely Continuous ◽  
Two Samples ◽  
Disjoint Convex ◽  
The Given

Suppose given an absolutely continuous distribution on the plane, and points P i, · · ·, Pj , Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj the convex hull of {P i, · · ·, Pj }, and by Γk the convex hull of (Πt, · · ·, Π k }, and writing pjk for the probability that Gj and Γ k are disjoint, certain properties of the array {pjk ; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p 1n ; n = 1,2, · · ·}, and asymptotic results for {pnn ; n = 1,2, · · ·}. Some examples are considered.

The probability that two samples in the plane will have disjoint convex hulls

1978 ◽  
Vol 15 (4) ◽  
pp. 790-802 ◽  
Author(s):  
L. C. G. Rogers
Keyword(s):  
Convex Hull ◽  
Continuous Distribution ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Absolutely Continuous ◽  
Two Samples ◽  
Disjoint Convex ◽  
The Given

Suppose given an absolutely continuous distribution on the plane, and points Pi, · · ·, Pj, Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj the convex hull of {Pi, · · ·, Pj}, and by Γk the convex hull of (Πt, · · ·, Πk}, and writing pjk for the probability that Gj and Γk are disjoint, certain properties of the array {pjk; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p1n; n = 1,2, · · ·}, and asymptotic results for {pnn; n = 1,2, · · ·}. Some examples are considered.

Standard Asymptotic Theory

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Variance ◽  
Statistical Significance ◽  
Regularity Conditions ◽  
Parameter Estimates ◽  
Asymptotic Results ◽  
Ml Estimation ◽  
True Parameter ◽  
Log Likelihood ◽  
Ml Estimator

This book relies on maximum likelihood (ML) estimation of parameters. Asymptotic theory assumes regularity conditions hold when the ML estimator is consistent. Typically an additional third derivative condition is assumed to ensure that the ML estimator is also asymptotically normally distributed. Standard asymptotic results that then hold are summarized in this chapter; for example, the asymptotic variance of the ML estimator is then given by the Fisher information formula, and the log-likelihood ratio, the Wald and the score statistics for testing the statistical significance of parameter estimates are all asymptotically equivalent. Also, the useful profile log-likelihood then behaves exactly as a standard log-likelihood only in a parameter space of just one dimension. Further, the model can be reparametrized to make it locally orthogonal in the neighbourhood of the true parameter value. The large exponential family of models is briefly reviewed where a unified set of regular conditions can be obtained.

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

An evaluation of public policy student placements: outcomes of a Malta-based study

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Anne Marie Thake
Keyword(s):  
Public Policy ◽  
Tertiary Education ◽  
Real Life ◽  
Added Value ◽  
Academic Disciplines ◽  
Online Questionnaire ◽  
Content Type ◽  
Course Content ◽  
Mandatory Requirement ◽  
Practical Implications

PurposeThe purpose of this paper is to evaluate short-term, unpaid placements offered to students reading for a degree in public policy. They provide added value to their tertiary education experience. Elective placements were offered in 2012 and became a mandatory requirement for students reading for a three-year bachelor of commerce degree in public policy in 2018. To date, no research has been carried out on these placements and this may serve as a model for a post-evaluation assessment.Design/methodology/approachData were collected from students who undertook placements, embedded in the public policy undergraduate programme. A document analysis of selected student and placement provider's reports was carried out to complement the students' responses to an online questionnaire.FindingsPlacements are of value to students as they served as an introduction to the working world. They enable students to establish connections with the course content and carry out research. They were exposed to real-life situations, developing their knowledge, acquiring soft skills and learning new tools, sought after by employers. These placements were valued as a route to graduate employment tailor-made to the degree's requirements. Students were able to embark on a soul-searching, introspective discovery and journey which made them mature and shed light in the direction of future work prospects.Research limitations/implicationsPlacements give students the opportunity to gain insights into real-work environments and are able to link theories learnt in the class-room with real-life situations. Placements have positive implications on students adjusting to their work life easily after graduation. The limitations are that the sample size was small and that the reflective reports which were randomly selected may not have necessarily been representative of the full complement.Practical implicationsThe practical implications are that the placements system and process can easily be implemented and replicated in other academic disciplines and universities as a compulsory component of their studies.Social implicationsPlacements gave students the opportunity to reflect on their learning, develop non-technical skills and enhance their confidence levels. They were also able to network and communicate with different employees.Originality/valuePlacements provided exposure to relevant organisations and personal enrichment in terms of acquiring skills, autonomy and independence. Students with placement experience are also more likely to secure future employment, relevant to their undergraduate degree.

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

scholarly journals COMPUTING CONVEX HULLS BY AUTOMATA ITERATION

2009 ◽  
Vol 20 (04) ◽  
pp. 647-667
Author(s):  
FRANÇOIS CANTIN ◽  
AXEL LEGAY ◽  
PIERRE WOLPER
Keyword(s):  
Convex Hull ◽  
Finite Automaton ◽  
Convex Hulls ◽  
The Real ◽  
Starting Point ◽  
Finite Set

This paper considers the problem of computing the real convex hull of a finite set of n-dimensional integer vectors. The starting point is a finite-automaton representation of the initial set of vectors. The proposed method consists in computing a sequence of automata representing approximations of the convex hull and using extrapolation techniques to compute the limit of this sequence. The convex hull can then be directly computed from this limit in the form of an automaton-based representation of the corresponding set of real vectors. The technique is quite general and has been implemented.

scholarly journals Exact Formulae for Variances of Functionals of Convex Hulls

2013 ◽  
Vol 45 (04) ◽  
pp. 917-924
Author(s):  
Christian Buchta
Keyword(s):  
Convex Hull ◽  
Asymptotic Distribution ◽  
Convex Polygon ◽  
Convex Hulls ◽  
Affine Invariance ◽  
Uniform Sample

The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider—in view of affine invariance—n points P 1,…, P n distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number Ñ n of points among P 1,…, P n , which are vertices of the convex hull of (0, 1), P 1,…, P n , and (1, 0). Correspondingly, D̃ n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var Ñ n and var . These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev and Khamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).

Knowledge Transfer Issues in Teaching

2017 ◽  
pp. 58-78 ◽  
Author(s):  
Neeta Baporikar
Keyword(s):  
Knowledge Transfer ◽  
Management Practices ◽  
Real Life ◽  
Research Knowledge ◽  
Professional Schools ◽  
Current State ◽  
Theory To Practice ◽  
Knowledge Management Practices ◽  
Organizational Science ◽  
Practical Implications

Professional schools typically build their raison d'être on the mission of developing knowledge that can be translated into skills that advance the practice of the professions. On the other hand practitioners fail to adopt the findings of research in fields be it medicine, management or engineering. Further, knowledge created is not always in the usage mode, that too in real life practical situation. Action scientists focused on the characteristics and behaviors of researchers to explain this lack of implementation of research knowledge. Identifying the major gaps between scientific knowledge and actual knowledge transfer issues is crucial in today's scenario. Hence, the purpose of this chapter is to identify knowledge transfer issues, discuss the issues and advancements therein, and highlight practical implications of relating theory to practice with focus on management discipline. The issues discussed herein are not only of utmost importance but crucial for understanding, given the current state of management education, organizational science and knowledge management practices.

scholarly journals Myths about HIV and AIDS among serodiscordant couples in Malawi

2017 ◽  
Vol 69 (3) ◽  
pp. 278-293
Author(s):  
Kondwani Wella ◽  
Sheila Webber ◽  
Philippa Levy
Keyword(s):  
Design Methodology ◽  
Real Life ◽  
Hiv And Aids ◽  
Serodiscordant Couples ◽  
Content Type ◽  
Analyse Data ◽  
Information Providers ◽  
Aids Information ◽  
Depth Interviews ◽  
Practical Implications

Purpose The purpose of this paper is to report on research that uncovered myths about HIV and AIDS held by serodiscordant couples in Malawi, and the sources of these myths. The paper reflects on how the myths affect serodiscordant couples’ engagement with HIV and AIDS information. Design/methodology/approach Van Manen’s (1997) approach to analysis of phenomenological data was used to analyse data from in-depth interviews conducted in Malawi with 21 serodiscordant couples and three individuals who had separated from their partners because of serodiscordance. Findings Serodiscordant couples in Malawi believe and hold on to some inaccurate HIV and AIDS information that can be seen as “myths”. Some of these myths are perpetuated by official HIV and AIDS information when it is translated into the local languages. Other myths derive from social norms of the societies where the couples live. Practical implications The findings of this paper have practical implications for how HIV and AIDS information providers should engage with target audiences to understand the origins of the myths they hold. The findings also imply that some myths have technical, religious, moral and cultural bases which need to be addressed before challenging the myth itself. Originality/value Using real-life descriptions of experiences of HIV and AIDS information provided by serodiscordant couples, the authors reveal how myths can affect engagement with the information. The authors make recommendations on how to address myths in ways that contribute to a positive experience of HIV and AIDS information by serodiscordant couples.

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