Statistical Geometry| Statistical Geometry |
In General
* Idea: Includes statistical techniques for
studying a geometry, usually Euclidean (random sampling/sprinkling), and the study of properties
of stochastically distributed subsets of a geometry ("stochastic geometry").
Point Process
> s.a. Poisson Distribution; random
process [including departures from randomness]; Sprinkling.
* Random sprinkling: It can be
defined if we have a volume element on a set X, as a random process;
It corresponds to a random measure on the set.
* Stationary: The statistical
properties of the point process do not depend on location (what we would call
"homogeneous").
* Types: Homogeneous (the number
variance in a given region grows like the number mean), super-homogeneous (the
number variance grows slower than the number mean), hyper-uniform (variance
growth saturates).
* Moments: Each moment
corresponds to a measure on the set, like the process itself.
* Palm distribution: Gives the
conditional probability of point process events, given that a point is observed
at a specific location.
* Operations on point processes:
Thinning (a type of "coarse-graining"), clustering (replacing points
by clusters), superposition (union).
* Other constructions: From a point
distribution, one can get a (Delaunay) triangulation, and a (Voronoi) cell complex.
@ General references: Macchi AAP(75);
Ambartzumian 90;
van Hameren & Kleiss NPB(98)mp,
et al NPB(99)
[quantum field theory methods]; Barndorff-Nielsen et al 98;
Ramiche AAP(00) [of phase-type];
Daley & Vere-Jones 07;
Gabrielli et al PRE(08)-a0711 [superhomogeneous];
Møller & Schoenberg AAP(10) [random thinning];
Kendall & Molchanov ed-10;
Nehring JMP(13),
et al JMP(13) [method of cluster expansion].
@ Poisson point process: Cowan et al AAP(03) [gamma-distributed domains];
Bhattacharyya & Chakrabarti EJP(08) [distance to nth neighbor];
Balister et al AAP(09) [k-nearest-neighbour model, critical constant];
Chatterjee et al AM(10) [with allocation of measure to points];
Davydov et al AAP(10) [peeling procedure];
Serinaldi & Kilsby PhyA(13) [the Allan factor as an estimator of homogeneity];
Sevilla a1310 [Poisson processes with pile-up];
Cristina CQG(16)-a1603 [in Minkowski space, and Noldus limit];
Last & Penrose 18;
> s.a. Wikipedia page.
@ Other point processes: van Lieshout 00 [Markov point processes];
Hahn et al AAP(03) [inhomogeneous];
in Vuletić IMRN-mp/07 [Pfaffian];
Kuna et al AAP(11)-a0910 [realizability];
Gupta & Iyer AAP(10) [with exponentially decaying density];
Caron et al AAP(11) [conditional distributions];
Jansen a1807 [Gibbs point processes, cluster expansions].
@ Correlations:
Kerscher A&A(99)ap/98 [correlation estimators];
de Coninck et al PhyA(07) [correlation structure];
Lenz & Moody CMP(09)-a0902 [correlations].
@ Related topics: Barbour & Månsson AAP(00) [clustering of points];
Soshnikov RMS-m.PR/00,
AAP-m.PR/00 [determinantal random point fields];
Valdarnini ASP-ap/01 [analysis of point distributions];
Lytvynov RVMP(02)mp/01 [fermion and boson];
Chiu & Molchanov AAP(03) [nearest neighbors, degree];
Koyama & Shinomoto JPA(05) [Bayesian interpretation];
Cowan AAP(06) [complementary theorem for n-tuples];
Sangaletti et al JPA(07) [high-d, Cox probabilities];
Kuna et al JSP(07) [realizability of functions as correlation functions];
Bárány BAMS(08) [convex polytopes];
Majumdar et al JSP(10) [properties of convex hulls];
Møller & Berthelsen AAP(12) [superposition of spatial point processes];
Rohrmann & Zurbriggen PRE(12) [conditional pair distributions];
> s.a. Betti Numbers.
Other Processes and Applications > s.a. cover [coverage
process]; e; random tiling.
* Buffon's needle: An
experimental method for determining the value of π, used by Georges Leclerc,
Count de Buffon, in 1777; Rule a series of equidistant lines on a sheet of paper,
a distance d apart; Drop a needle of length l < d on
the paper so it falls in a random position; The probability it will cross a line
is 2l / πd.
@ Buffon's needle: in Gardner 81,
127-128 & ref [use (fudged)]; {> s.a. #Lazzarini}.
@ Related topics: Donetti & Destri JPA(04)cm/03 [scale-free random trees];
Vickers & Brown PRS(01) [projected area and perimeter of solids];
Roberts & Garboczi JMPS(02) [elastic properties of solids];
Arsuaga et al JPA(07) [uniform random polygons, linking].
> Online resources:
Garboczi page on elastic properties of solids.
Results and Special Cases
* 2D flat: For any non-concave
2D figure, the average width over all orientations is exactly perimeter/π;
For any non-concave solid, the average projected area
on a plane over all orientations is (surface area)/4.
* 2D curved:
* 3D flat:
* 3D curved:
@ On spheres: Tu & Fischbach mp/00/JMP,
JPA(02)mp [n dimensions, distances between random points];
comp.graphics.algorithms page(06).
@ On other manifolds: Parry & Fischbach JMP(00) [distances on an ellipsoid].
References > s.a. ergodic theory.
@ General: Meijering PRR(53);
Smith & Guttman JoM(53);
Gilbert AMS(62);
Miles MB(70)-mr;
Harding & Kendall ed-74 [see intro];
Matheron 75; Santaló 76;
Solomon 78;
Stoyan et al 95;
Beneš & Rataj 04.
@ Related topics: Schindler CG(94) [and equivariant mappings];
Grimmett a1110 [three theorems].
Computational Geometry > s.a. geometry.
* Simulating binomial point processes:
* Simulating Poisson point processes:
@ General references: Preparata & Shamos 85;
Ripley 87,
in Stoyan et al 87 [statistical simulations];
O'Rourke 98 [in C];
Boucetta & Morvan ed-05;
de Berg et al 08;
Devadoss & O'Rourke 11;
Joswig & Theobald 13 [polyhedral and algebraic methods];
Goodman et al 17.
@ Graphs:
Di Battista et al CG(94) [drawing algorithms].
@ Voronoi diagrams: Bespamyatnikh & Snoeyink CG(00) [queries with segments].
@ Delaunay triangulations: Su & Drysdale CG(97) [algorithm comparison];
Mücke et al CG(99) [point location, 2D and 3D];
Lemaire & Moreau CG(00);
Hjelle & Dæhlen 06.
@ Greedy triangulations:
Dickerson et al CG(97) [algorithms];
Levcopoulos & Krznaric CG(99).
@ Related topics:
Mehlhorn et al CG(98) [higher-dimensional].
And Physics > s.a. causal sets;
lattice field theory; semiclassical
quantum gravity.
@ References: David et al ed-96 [fluctuating geometry and statistical mechanics];
Requardt & Roy CQG(01) [fuzzy lumps].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 31 may 2019