77:, which makes contact with advanced concepts from measure theory. The key idea is to focus on the probabilities of the given random closed set hitting specified test sets. There arise questions of inference (for example, estimate the set which encloses a given point pattern) and theories of generalizations of means etc. to apply to random sets. Connections are now being made between this latter work and recent developments in geometric mathematical analysis concerning general metric spaces and their geometry. Good parametrizations of specific random sets can allow us to refer random object processes to the theory of marked point processes; object-point pairs are viewed as points in a larger product space formed as the product of the original space and the space of parametrization.
86:
however the theory may be mapped back into point process theory by representing each object by a point in a suitable representation space. For example, in the case of directed lines in the plane one may take the representation space to be a cylinder. A complication is that the
Euclidean motion symmetries will then be expressed on the representation space in a somewhat unusual way. Moreover, calculations need to take account of interesting spatial biases (for example, line segments are less likely to be hit by random lines to which they are nearly parallel) and this provides an interesting and significant connection to the hugely significant area of
65:, places a random compact object at each point of a Poisson point process. More complex versions allow interactions based in various ways on the geometry of objects. Different directions of application include: the production of models for random images either as set-union of objects, or as patterns of overlapping objects; also the generation of geometrically inspired models for the underlying point process (for example, the point pattern distribution may be biased by an exponential factor involving the area of the union of the objects; this is related to the Widom–Rowlinson model of statistical mechanics).
17:
85:
Suppose we are concerned no longer with compact objects, but with objects which are spatially extended: lines on the plane or flats in 3-space. This leads to consideration of line processes, and of processes of flats or hyper-flats. There can no longer be a preferred spatial location for each object;
97:
dividing space; hence for example one may speak of
Poisson line tessellations. A notable recent result proves that the cell at the origin of the Poisson line tessellation is approximately circular when conditioned to be large. Tessellations in stochastic geometry can of course be produced by other
142:
This brief description has focused on the theory of stochastic geometry, which allows a view of the structure of the subject. However, much of the life and interest of the subject, and indeed many of its original ideas, flow from a very wide range of applications, for example: astronomy,
90:, which in some respects can be viewed as yet another theme of stochastic geometry. It is often the case that calculations are best carried out in terms of bundles of lines hitting various test-sets, rather than by working in representation space.
510:
Piterbarg, V. I.; Wong, K. T. (2005). "Spatial-Correlation-Coefficient at the
Basestation, in Closed-Form Explicit Analytic Expression, Due to Heterogeneously Poisson Distributed Scatterers".
61:
The point pattern theory provides a major building block for generation of random object processes, allowing construction of elaborate random spatial patterns. The simplest version, the
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144:
21:
512:
119:
32:
is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of
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186:
171:). Most recently determinantal and permanental point processes (connected to random matrix theory) are beginning to play a role.
261:
415:
468:
Baccelli, F.; Klein, M.; Lebourges, M.; Zuyev, S. (1997). "Stochastic geometry and architecture of communication networks".
547:
Abdulla, M.; Shayan, Y. R. (2014). "Large-Scale Fading
Behavior for a Cellular Network with Uniform Spatial Distribution".
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Line and hyper-flat processes have their own direct applications, but also find application as one way of creating
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201:
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There are various models for point processes, typically based on but going beyond the classic homogeneous
584:
Stoyan, D.; Penttinen, A. (2000). "Recent
Applications of Point Process Methods in Forestry Statistics".
54:
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in 1963 as one of two suggestions for names of a theory of "random irregular structures" inspired by
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168:
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Georgii, H.-O.; Häggström, O.; Maes, C. (2001). "The random geometry of equilibrium phases".
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What is meant by a random object? A complete answer to this question requires the theory of
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298:"A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons"
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167:, and implementations of the theory in statistical computing (for example, spatstat in
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448:
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411:
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Frisch, H. L.; Hammersley, J. M. (1963). "Percolation processes and related topics".
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workshop, though antecedents for the theory stretch back much further under the name
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36:, hence notions of Palm conditioning, which extend to the more abstract setting of
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and connectivity constructed from randomly sized disks placed at random locations
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and variant constructions, and also by iterating various means of construction.
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257:"The analysis of the Widom-Rowlinson model by stochastic geometric methods"
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58:) to find expressive models which allow effective statistical methods.
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671:
Stochastic
Geometry Models in Image Analysis and Spatial Statistics
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151:, forestry, the statistical theory of shape, material science,
148:
754:
McCullagh, P.; Møller, J. (2006). "The permanental process".
721:"Spatstat: An R package for analyzing spatial point patterns"
126:. The term "stochastic geometry" was also used by Frisch and
467:
20:
A possible stochastic geometry model (Boolean model) for
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Journal of
Applied Mathematics and Stochastic Analysis
147:, wireless network modeling and analysis, modeling of
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361:
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753:
583:
217:Stochastic geometry models of wireless networks
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513:IEEE Antennas and Wireless Propagation Letters
333:Stoyan, D.; Kendall, W. S.; Mecke, J. (1987).
607:"A survey of the statistical theory of shape"
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549:Wireless Communications and Mobile Computing
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163:. There are links to statistical mechanics,
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498:Stochastic geometry for wireless networks
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701:Phase Transitions and Critical Phenomena
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335:Stochastic geometry and its applications
145:spatially distributed telecommunications
110:The name appears to have been coined by
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187:Spherical contact distribution function
782:
262:Communications in Mathematical Physics
441:Statistics of The Galaxy Distribution
327:
325:
105:
398:. Probability and Its Applications.
500:. Cambridge University Press, 2012.
365:SIAM Journal on Applied Mathematics
13:
439:Martinez, V. J.; Saar, E. (2001).
322:
255:; Chayes, L.; KoteckĂ˝, R. (1995).
14:
811:
719:Baddeley, A.; Turner, R. (2005).
396:Stochastic and Integral Geometry
232:Stochastic differential geometry
118:while preparing for a June 1969
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757:Advances in Applied Probability
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726:Journal of Statistical Software
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669:Van Lieshout, M. N. M. (1995).
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642:Random heterogeneous materials
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81:Line and hyper-flat processes
202:Continuum percolation theory
98:means, for example by using
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55:complete spatial randomness
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182:Nearest neighbour function
471:Telecommunication Systems
408:10.1007/978-3-540-78859-1
315:10.1155/S1048953399000283
296:Kovalenko, I. N. (1999).
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22:wireless network coverage
534:10.1109/LAWP.2005.857968
192:Factorial moment measure
165:Markov chain Monte Carlo
605:Kendall, D. G. (1989).
484:10.1023/A:1019172312328
222:Mathematical morphology
34:spatial point processes
770:10.1239/aap/1165414583
25:
740:10.18637/jss.v012.i06
640:Torquato, S. (2002).
626:10.1214/ss/1177012582
153:multivariate analysis
124:geometric probability
52:(the basic model for
50:Poisson point process
19:
790:Stochastic processes
227:Information geometry
612:Statistical Science
587:Statistical Science
526:2005IAWPL...4..385P
394:; Weil, W. (2008).
275:1995CMaPh.172..551C
30:stochastic geometry
673:. CWI Tract, 108.
445:Chapman & Hall
283:10.1007/BF02101808
212:Spatial statistics
132:percolation theory
106:Origin of the name
75:random closed sets
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800:Spatial processes
795:Integral geometry
708:. pp. 1–142.
417:978-3-540-78858-4
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128:Hammersley
88:stereology
562:1302.0891
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