Random Point Sets Research Articles

Alignments in two-dimensional random sets of points

Letnpoints in the plane be generated by some specified random mechanism and suppose thatN(∊) of theresulting triads form triangles with largest angle ≧ π – ∊. The main object of the paper is to obtain asymptotic formulae forand Var (N(∊)) when ∊ ↓ 0, and to solve the associated data-analytic problem of testing whether an empirical set ofnpoints should be considered to contain too many such ∊-blunt triads in the situation where the generating mechanism is unknown and where all that can be said about the tolerance ∊ is that it must be allowed to take values anywhere in a given interval (T0,T1) (0 <T0<T1). This problem is solved by the introduction of a plot to be called thepontogramand by the introduction of simulation-based significance tests constructed by random lateral perturbations of the data.

A canonical decomposition of the probability measure of sets of isotropic random points in [formula omitted

The probability measure of X = ( x 0,…, x r ), where x 0,…, x r are independent isotropic random points in R n (1 ≤ r ≤ n − 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ẑ), the probability measure of p, and the probability measure of Y ∗ = (y 0 ∗,…, y r ∗) . Here ω is the r-subspace parallel to the r-flat η determined by X, ẑ is a unit vector in ω ⊥ with ‘initial’ point at the origin [ ω ⊥ is the ( n − r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and y i ∗ = (x i − z) α(p 2) , where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in R n invariant under the group of rotations in R n , while the conditional probability measure of ẑ given ω is uniform on the boundary of the unit ( n − r)-ball in ω ⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of { x 0,…, x r } for 1 ≤ r ≤ n.

On the Average Number of Distinctive Level Curve Points of a Random Field on the Plane

Previous article Next article On the Average Number of Distinctive Level Curve Points of a Random Field on the PlaneV. P. NoskoV. P. Noskohttps://doi.org/10.1137/1124019PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Robert J. Adler and , A. M. Hasofer, Level crossings for random fields, Ann. Probability, 4 (1976), 1–12 MR0405559 0329.60017 CrossrefGoogle Scholar[2] Yu. K. Belyaev, , V. P. Nosko and , A. A. Sveshnikov, Probabilistic characteristics of excursions of a two-dimensional random field, Probability Theory and Mathematical Statistics, Vol. 6, Kiev University, 1972, 24–31, (In Russian.) Google Scholar[3] A. A. Sveshnikov and , V. D. Khabalov, An estimate of the average number of excursions of a two-dimensional random field, Kibernetika, 3 (1973), 94–99, (In Russian.) Google Scholar[4] V. P. Nosko, On the possibility of using the Morse inequalities for the estimation of the number of excursions of a random field in a domain, Theor. Probability Appl., 18 (1973), 821–822 Google Scholar[5] Yu. K. Belyaev, A general formula for the average number of intersections for random processes and fields“Excursions of Random Fields”, MGU, Moscow, 1972, 38–45, (In Russian.) Google Scholar[6] T. L. Malevič, The formula for the mean number of crossings of a surface by random fields, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk, 17 (1973), 15–17, 75, (In Russian.) MR0343353 Google Scholar[7] Yu. K. Belyaev, Random point sets and problems of the level crossing type“Excursions of Random Fields”, MGU, Moscow, 1972, 9–37, (In Russian.) Google Scholar[8] Robert J. Adler, Excursions above a fixed level by n-dimensional random fields, J. Appl. Probability, 13 (1976), 276–289 MR0405566 0368.60048 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails References and Suggested ReadingLevel Sets and Extrema of Random Processes and Fields | 21 July 2008 Cross Ref Volume 24, Issue 1| 1979Theory of Probability & Its Applications History Submitted:12 September 1977Published online:17 July 2006 InformationCopyright © 1979 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1124019Article page range:pp. 181-183ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On the Average Number of Maxima in a Set of Vectors and Applications

A maximal vector of a set ~s one which is not less than any other vector m all components We derive a recurrence relation for computing the average number of maxunal vectors m a set of n vectors m d-space under the assumpUon that all (nl) a relative ordermgs are equally probable. Solving the recurrence shows that the average number of maxmaa is O((ln n) a-~) for fixed d We use this result to construct an algorithm for finding all the maxima that have expected running tmae hnear m n (for sets of vectors drawn under our assumptions) We then use the result to find an upper bound on the expected number of convex hull points m a random point set

Liquid Structure and Self-Diffusion

The method of molecular dynamics has been used to study the structure of a liquid and its relation with the process of self-diffusion. The particles around a given particle are separated into shells of primary, secondary, etc., neighbors by a precise geometrical procedure. Each shell of neighbors of a particle generates a closed convex polyhedron surrounding the polyhedron of the previous shell. The faces of the polyhedra are the bisector planes of lines joining the particle to its neighbors. The primary polyhedron is the smallest closed convex polyhedron around the particle. The primary polyhedra of a random system of points, of a liquid, and of a solid are compared; the average number f̄ of faces are found to be 15.67, 14.45, and 14.26, respectively. The average number p̄ of sides to the polygons forming the faces is obtained through the relation p̄=6—(12/f̄). The pair correlation function g(r) is decomposed into a sum Σig(i)(r), where g(i)(r), arises out of neighbors forming the ith-order polyhedra. The distributions up to i=5 have been calculated. g(1) and g(2) give the first two maxima in g(r). For higher values of i the g(i)(r) do not correspond to maxima in g(r). For the liquid and for the random set of points the average number of faces f̄i of the ith-order polyhedra increases linearly with i; the departure from the linear law is significant only for i=1 and 2 in the case of the liquid. It is found that the shape of the primary polyhedra is related to the direction of displacement, in time τ, of the central particle; and the correlation is maximum when τ=0.5×10−12 sec. The positions of the neighbors are analyzed with respect to this displacement direction and expressed as a distribution g(r, μ; τ), which, in addition to the dependence on the distance, contains the angular correlation between the position of a neighbor and the displacement direction in time τ of the central particle. The nearest neighbors are found to be preferably situated in a direction away from the displacement direction. As the distance increases, the forward direction becomes more favorable. In the region of the secondary neighbors the same behavior is repeated but is less in magnitude. This local fluctuation in the distribution of neighbors disappears beyond the secondary neighbors. The velocity autocorrelation function and its spectrum f(ω) are found to have characteristics related to these local fluctuations. f(ω) is found to have a solidlike part having the same Debye ω as that in the solid before melting. The diffusive part of the velocity autocorrelation function arises out of the particle's taking advantage of the local fluctuation in the configuration of its neighbors to move in the ``easy'' direction afforded to it by the fluctuation.

The convex hull of a random set of points

SUMMARY Various expectations concerning the convex hull of N independently and identically distributed random points in the plane or in space are evaluated. Integral expressions are given for the expected area, expected perimeter, expected probability content and expected number of sides. These integrals are shown to be particularly simple when the underlying distribution is normal or uniform over a disk or sphere. In two recent papers Renyi & Sulanki (1963, 1964) have given expressions for the expected area, perimeter, and number of vertices of the convex hull of N independently and identically selected random points in the plane. In these papers, limit theorems for asymptotically large N receive the greatest attention. Here the emphasis will be on the development of convenient formulae for fixed values of N. Some new results for random convex hulls in the plane are derived, such as the expected probability content, and also various expectations concerned with random convex hulls in three and higher dimensions. Special attention is given to the case of normally distributed points and also to the case of points drawn uniformly from an ellipse or an ellipsoid. Historically, calculating the expected probability content of three random points in two dimensions is known as 'Sylvester's problem', and has been solved explicitly for many different distributions by Deltheil (1926, p. 42). The corresponding problem of four points in three dimensions is connected with the name of Hostinsky (1925). A discussion of Sylvester's problem is given in Kendall & Moran's monograph, Geometrical Probability (1963). 2. THE EXPECTED NUMBER OF VERTICES, FACES, AND EDGES