Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions

1985 ◽  
Vol 17 (4) ◽  
pp. 794-809 ◽  
Author(s):  
Charles M. Newman ◽  
Yosef Rinott
Keyword(s):  
Asymptotic Behavior ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Nearest Neighbor ◽  
Correlation Coefficients ◽  
Nearest Neighbors ◽  
Distance Functions ◽  
High Dimensional ◽  
Voronoi Region

Consider a Poisson point process of density 1 in Rd, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e0, · ··, en, in Rd and a variety of distances, we obtain the asymptotic behavior of Ndn, the number of points which have e0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-lp distances related to correlation coefficients.

Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions

1985 ◽  
Vol 17 (04) ◽  
pp. 794-809 ◽  
Author(s):  
Charles M. Newman ◽  
Yosef Rinott
Keyword(s):  
Asymptotic Behavior ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Nearest Neighbor ◽  
Correlation Coefficients ◽  
Nearest Neighbors ◽  
Distance Functions ◽  
High Dimensional ◽  
Voronoi Region

Consider a Poisson point process of density 1 in R d, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e 0, · ··, e n, in Rd and a variety of distances, we obtain the asymptotic behavior of Nd n , the number of points which have e 0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-l p distances related to correlation coefficients.

scholarly journals Tropical Balls and Its Applications to K Nearest Neighbor over the Space of Phylogenetic Trees

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 779
Author(s):  
Ruriko Yoshida
Keyword(s):  
Supervised Learning ◽  
Phylogenetic Trees ◽  
Nearest Neighbor ◽  
Nearest Neighbors ◽  
High Dimensional ◽  
Learning Method ◽  
Dimensional Vector ◽  
K Nearest Neighbor ◽  
K Nearest Neighbors

A tropical ball is a ball defined by the tropical metric over the tropical projective torus. In this paper we show several properties of tropical balls over the tropical projective torus and also over the space of phylogenetic trees with a given set of leaf labels. Then we discuss its application to the K nearest neighbors (KNN) algorithm, a supervised learning method used to classify a high-dimensional vector into given categories by looking at a ball centered at the vector, which contains K vectors in the space.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (03) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k } are the points of a renewal process and {Ak } are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (3) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k} are the points of a renewal process and {Ak} are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

scholarly journals Estimation of the local intensity of a cyclic Poisson process by means of nearest neighbor distances

2002 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
I W. MANGKU
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Poisson Point Process ◽  
Nearest Neighbor ◽  
Single Realization ◽  
Local Intensity ◽  
Strongly Consistent ◽  
Nearest Neighbor Distances

We consider the problem of estimating the local intensity of a cyclic Poisson point process, when we know the period. We suppose that only a single realization of the cyclic Poisson point process is observed within a bounded 'window', and our aim is to estimate consistently the local intensity at a given point. A nearest neighbor estimator of the local intensity is proposed, and we show that our estimator is weakly and strongly consistent, as the window expands.

scholarly journals On the peeling procedure applied to a Poisson point process

2010 ◽  
Vol 42 (3) ◽  
pp. 620-630
Author(s):  
Y. Davydov ◽  
A. Nagaev ◽  
A. Philippe
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Numerical Experiments ◽  
Poisson Point Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Convex Hulls ◽  
Limit Shape ◽  
Empirical Point ◽  
Stable Random Vectors

In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (4) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

scholarly journals Persistence and Equilibria of Branching Populations with Exponential Intensity

2012 ◽  
Vol 49 (1) ◽  
pp. 226-244
Author(s):  
Zakhar Kabluchko
Keyword(s):  
Laplace Transform ◽  
Random Walks ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Local Extinction ◽  
Branching Random Walks ◽  
Cluster Distribution

We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form eλ(du) = e-λudu, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) > 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Πeλχ with intensity eλ. If λφ'(λ) < 0 then the branching population suffers local extinction, meaning that the limiting point process is empty. We characterize point processes on R which are cluster invariant with respect to the cluster distribution χ as mixtures of the point processes Πceλχ over c > 0 and λ ∈ Kst, where Kst = {λ ∈ R: φ(λ) = 0, λφ'(λ) > 0}.

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