Jump to content

Stochastic geometry: Difference between revisions

From Wikipedia, the free encyclopedia
Browse history interactively
← Previous edit
Content deleted Content added
VisualWikitext
m →Applications: Rm WP link of publisher; irrelevant.
 
(61 intermediate revisions by 35 users not shown)
Line 1: Line 1:
[[Image:BooleanCellCoverage.svg|thumb|right|A possible stochastic geometry model (Boolean model) for [[Stochastic geometry models of wireless networks|wireless network coverage]] and connectivity constructed from randomly sized disks placed at random locations]]
{{Expert-subject|mathematics|date=May 2009}}


In mathematics, '''stochastic geometry''' 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
In mathematics, '''stochastic geometry''' 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 [[Point process|spatial point processes]], hence notions of Palm conditioning, which extend to the more abstract setting of [[random measure]]s.
[[Point process|spatial point processes]], hence notions of Palm conditioning, which extend to the more abstract setting of [[random measure]]s.


==Models==
==Models==


There are various models for point processes, typically based on but going beyond the classic homogeneous [[Poisson process|Poisson point process]] (the basic model for ''[[complete spatial randomness]]'') to find expressive models which allow effective statistical methods.
There are various models for point processes,
typically based on but going beyond the classic homogeneous
[[Poisson process|Poisson point process]] (the basic model for ''complete spatial randomness'')
to find expressive models which allow effective statistical methods.


The point pattern theory provides a major building block for generation of random object processes, allowing construction of elaborate random spatial patterns.
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 [[Boolean model (probability theory)|Boolean model]], 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<ref>
The simplest version, the [[Boolean model (probability theory)|Boolean model]], places a random compact object at each point of a
{{cite journal
Poisson point process. More complex versions allow interactions based in various ways on the
|last1=Chayes |first1=J. T. | author1-link = Jennifer Tour Chayes
geometry of objects. Different directions of application include:
|last2=Chayes |first2=L.
the production of models for random images either as set-union of objects,
|last3=Kotecký |first3=R.
or as patterns of overlapping objects; also the generation of geometrically
|year=1995
inspired models for the underlying point process
|title=The analysis of the Widom-Rowlinson model by stochastic geometric methods
(for example, the point pattern distribution may be biased by an exponential
|journal=[[Communications in Mathematical Physics]]
factor involving the area of the union of the objects;
|volume=172 |issue=3 |pages=551&ndash;569
this is related to the Widom-Rowlinson model
|bibcode=1995CMaPh.172..551C
<ref> {{cite journal |author = Chayes, J.T. and Chayes, L. and Kotecký, R.|title=The analysis of the Widom-Rowlinson model by stochastic geometric methods.|journal= Comm. Math. Phys. |volume= 172 |year = 1995|pages=551&ndash;569 |doi = 10.1007/BF02101808}}</ref>
|doi=10.1007/BF02101808
of statistical mechanics).
|url=http://projecteuclid.org/euclid.cmp/1104274315 }}</ref> of statistical mechanics).


==Random object==
==Random object==


What is meant by a random object? A complete answer to this question requires the theory of [[Random compact set|random closed sets]], 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.
What is meant by a random object? A complete answer to this question requires the theory of
[[Random compact set|random closed sets]], 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.


==Line and hyper-flat processes==
==Line and hyper-flat processes==


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; 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 [[stereology]], 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.
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; 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 [[stereology]], 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.


Line and hyper-flat processes have their own direct applications, but also find
Line and hyper-flat processes have their own direct applications, but also find application as one way of creating [[tessellation]]s dividing space; hence for example one may speak of Poisson line tessellations. A notable recent result<ref>
{{cite journal
application as one way of creating [[tessellation]]s dividing space; hence for example one may speak of Poisson
|last1=Kovalenko |first=I. N.
line tessellations.
|year=1999
A notable recent result<ref> {{cite journal |author = Kovalenko, I. N.|title=A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons.|journal= J. Appl. Math. Stochastic Anal. |volume= 12 |year = 1999|pages=301&ndash;310 |doi = 10.1155/S1048953399000283}}</ref>
|title=A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons
proves that the cell at the origin of the
|journal=Journal of Applied Mathematics and Stochastic Analysis
Poisson line tessellation is approximately circular
|volume=12 |issue=4 |pages=301&ndash;310
when conditioned to be large.
|doi=10.1155/S1048953399000283
Tessellations in stochastic geometry can of course be produced by other means,
|doi-access=free
for example by using [[Voronoi diagram|Voronoi]] and variant constructions, and also by iterating
}}</ref> 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 means, for example by using [[Voronoi diagram|Voronoi]] and variant constructions, and also by iterating various means of construction.
various means of construction.


==Origin of the name==
==Origin of the name==
The name appears to have been coined in 1969 by
The name appears to have been coined by [[David George Kendall|David Kendall]] and [[Klaus Krickeberg]]<ref name="SKM">See foreword in {{cite book
|last1=Stoyan |first1=D.
[[David George Kendall|David Kendall]] and Klaus Krickeberg (see remarks of David Kendall in the Foreword to <ref name="SKM"> {{cite book
| author = Stoyan, D., Kendall, W.S. and Mecke, J.
|last2=Kendall |first2=W. S.
|last3=Mecke |first3=J.
| title = Stochastic geometry and its applications
| year = 1987
|year=1987
|title=Stochastic geometry and its applications
| publisher = Wiley
|publisher=[[John Wiley & Sons|Wiley]]
| isbn = 0-471-90519-4}}
|isbn=0-471-90519-4
</ref>) while preparing for a June 1969
[[Mathematical Research Institute of Oberwolfach|Oberwolfach]] workshop, though antecedents
}}</ref> while preparing for a June 1969 [[Mathematical Research Institute of Oberwolfach|Oberwolfach]] workshop, though antecedents for the theory stretch back much further under the name [[geometric probability]]. The term "stochastic geometry" was also used by Frisch and [[John Hammersley|Hammersley]] in 1963<ref>
{{Cite journal
for the theory stretch back much further under the name [[geometric probability]].
|last1=Frisch |first1=H. L.
The term "stochastic geometry" was also used by Frisch and [[John_Hammersley|Hammersley]] in 1963
|last2=Hammersley |first2=J. M.
{{cite
|year=1963
| author = Frisch, H.L. and Hammersley, J.M.
| title = Percolation processes and related topics
|title=Percolation processes and related topics
|journal=[[SIAM Journal on Applied Mathematics]]
| year = 1963
|volume=11 |issue=4 |pages=894–918
| journal = J. Soc. Indust. Appl. Math.
|doi=10.1137/0111066
| volume = 11
}}</ref> as one of two suggestions for names of a theory of "random irregular structures" inspired by [[percolation theory]].
| pages = 894--918}}
as one of two suggestions for names of a theory of "random irregular structures" inspired by [[percolation theory]].


==Applications==
==Applications==
This brief description has focused on the theory
This brief description has focused on the theory<ref name="SKM"/><ref>
{{cite book
<ref name="SKM"/>
|last1=Schneider |first1=R.|author-link1=Rolf Schneider
<ref> {{cite book
|last2=Weil |first2=W.
| author = Schneider, R. and Weil, W.
|year=2008
| title = Stochastic and Integral Geometry
|title=Stochastic and Integral Geometry
| year = 2008
|series=Probability and Its Applications
| publisher = Springer
|publisher=[[Springer (publisher)|Springer]]
| isbn = 978-3-540-78858-4}}
|doi=10.1007/978-3-540-78859-1
</ref>
|isbn=978-3-540-78858-4
of stochastic geometry,
|mr=2455326
which is the best perspective from which to view the structure of the subject.
However. much of the life and interest of the subject, and indeed many of its original ideas,
}}</ref> 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,<ref>
{{cite book
flow from a very wide range of applications, for example: astronomy<ref> {{cite book
| author = Martinez, V. J. and Saar, E.
|last1=Martinez |first1=V. J.
|last2=Saar |first2=E.
| title = Statistics Of The Galaxy Distribution
| year = 2001
|year=2001
|title=Statistics of The Galaxy Distribution
| publisher = Chapman & Hall
|publisher=Chapman & Hall
| isbn = 1-58488-084-8
|isbn=1-58488-084-8
}}
</ref>, spatially distributed
}}</ref> [[Stochastic geometry models of wireless networks|spatially distributed telecommunications]],<ref>
{{cite journal
telecommunications<ref> {{cite journal |author = Baccelli. F., Klein, M., Lebourges, M. and Zuyev, S.|title=Stochastic geometry and architecture of communication networks.|journal= Telecommunication Systems |volume= 7 |year = 1997|pages=209&ndash;227 |doi = 10.1023/A:1019172312328}}</ref>,
|last1=Baccelli |first1=F.
modeling of channel fading<ref> {{cite journal |author = V. I. Piterbarg & K. T. Wong
|last2=Klein |first2=M.
|title=Spatial-Correlation-Coefficient at the Basestation, in Closed-Form Explicit Analytic Expression, Due to Heterogeneously Poisson Distributed Scatterers
|last3=Lebourges |first3=M.
|journal= IEEE Antennas & Wireless Propagation Letters
|last4=Zuyev |first4=S.
|volume= 4
|year=1997
|year = 2005
|title=Stochastic geometry and architecture of communication networks
|pages=385&ndash;388
|journal=[[Telecommunication Systems]]
|doi = 10.1109/LAWP.2005.857968}}
|volume=7 |pages=209&ndash;227
</ref>
|doi=10.1023/A:1019172312328
, forestry<ref> {{cite journal |author = Stoyan, D. and Penttinen, A. |title=Recent Applications of Point Process Methods in Forestry Statistics|journal= Statistical Science, |volume= 15 |year = 2000|pages=61&ndash;78}}</ref>, the statistical theory of shape<ref> {{cite journal |author = Kendall, D.G. |title=A survey of the statistical theory of shape|journal= Statistical Science, |volume= 4 |year = 1989|pages=87&ndash;99 |doi = 10.1214/ss/1177012582}}</ref>, material science<ref> {{cite book
}}</ref> wireless network modeling and analysis,<ref name="haenggi2012stochastic">M. Haenggi. ''Stochastic geometry for wireless networks''. Cambridge University Press, 2012.</ref> modeling of [[Fading|channel fading]],<ref>
| author = Torquato, S.
{{cite journal
| title = Random heterogeneous materials
|last1=Piterbarg |first1=V. I.
| year = 2002
|last2=Wong |first2=K. T.
| publisher = Springer-Verlag
|year=2005
| isbn = 0-387-95167-9}}
|title=Spatial-Correlation-Coefficient at the Basestation, in Closed-Form Explicit Analytic Expression, Due to Heterogeneously Poisson Distributed Scatterers
</ref>, multivariate analysis, problems in image
|journal=[[IEEE Antennas and Wireless Propagation Letters]]
analysis<ref> {{cite book
|volume=4 |issue= 1|pages=385&ndash;388
| author = Van Lieshout, M. N. M.
|bibcode=2005IAWPL...4..385P
| year = 1995
|doi=10.1109/LAWP.2005.857968
| title = Stochastic Geometry Models in Image Analysis and Spatial Statistics
}}</ref><ref>
| series= CWI Tract, 108
{{cite journal
| publisher = CWI
|last1=Abdulla |first1=M.
| isbn = 90-6196-453-9}}
|last2=Shayan |first2=Y. R.
</ref> and stereology.
|year=2014
There are links to statistical mechanics<ref> {{cite conference
|title=Large-Scale Fading Behavior for a Cellular Network with Uniform Spatial Distribution
| author = Georgii, H.-O., Häggström, O., Maes, C.
|journal= Wireless Communications and Mobile Computing
| year = 2001
|volume=4 |issue= 7|pages=1&ndash;17
| title = The random geometry of equilibrium phases
|arxiv=1302.0891
| booktitle = Phase transitions and critical phenomena
|doi=10.1002/WCM.2565
| volume = 18
}}</ref> forestry,<ref>
| pages = 1&ndash;142
{{cite journal
| publisher = Academic Press
|last1=Stoyan |first1=D.
}}
|last2=Penttinen |first2=A.
</ref>,
|year=2000
Markov chain Monte Carlo, and implementations
|title=Recent Applications of Point Process Methods in Forestry Statistics
of the theory in statistical computing
|journal=[[Statistical Science]]
(for example, spatstat<ref> {{cite journal
|volume=15 |pages=61&ndash;78
| author = Baddeley, A. and Turner, R.
}}</ref> the statistical theory of shape,<ref>
| title = Spatstat: an R package for analyzing spatial point patterns
{{cite journal
| journal = Journal of Statistical Software
|last1=Kendall |first1=D. G.
| volume = 12
| year = 2005
|year=1989
|title=A survey of the statistical theory of shape
| pages = 1&ndash;42}}
|journal=[[Statistical Science]]
</ref> in [[R (programming language)|R]]).
|volume=4 |issue=2 |pages=87&ndash;99
Most recently determinantal and permanental point processes
|doi=10.1214/ss/1177012582
(connected to random matrix theory) are beginning to play a role<ref> {{cite journal
|doi-access=free
| author = McCullagh, P. and Møller, J.
}}</ref> material science,<ref>
| title = The permanental process
{{cite book
| journal = Advances in Applied Probability
|last1=Torquato |first1=S.
| volume = 38
| year = 2006
|year=2002
|title=Random heterogeneous materials
| pages = 873&ndash;888
|publisher=[[Springer-Verlag]]
| doi = 10.1239/aap/1165414583}}
|isbn=0-387-95167-9
</ref>.
}}</ref> [[multivariate analysis]], problems in [[image analysis]]<ref>
{{cite book
|last1=Van Lieshout |first1=M. N. M.
|year=1995
|title=Stochastic Geometry Models in Image Analysis and Spatial Statistics
|series=CWI Tract, 108
|publisher=[[Centrum Wiskunde & Informatica|CWI]]
|isbn=90-6196-453-9
}}</ref> and [[stereology]]. There are links to statistical mechanics,<ref>
{{cite conference
|last1=Georgii |first1=H.-O.
|last2=Häggström |first2=O.
|last3=Maes |first3=C.
|year=2001
|title=The random geometry of equilibrium phases
|book-title=[[Phase Transitions and Critical Phenomena]]
|volume=18 |pages=1&ndash;142
|publisher=[[Academic Press]]
}}</ref> [[Markov chain Monte Carlo]], and implementations of the theory in statistical computing (for example, spatstat<ref>
{{cite journal
|last1=Baddeley |first1=A.
|last2=Turner |first2=R.
|year=2005
|title=Spatstat: An R package for analyzing spatial point patterns
|journal=[[Journal of Statistical Software]]
|volume=12 |issue= 6|pages=1&ndash;42
|doi=10.18637/jss.v012.i06
|doi-access=free
}}</ref> in [[R (programming language)|R]]). Most recently determinantal and permanental point processes (connected to random matrix theory) are beginning to play a role.<ref>
{{cite journal
|last1=McCullagh |first1=P.
|last2=Møller |first2=J.
|year=2006
|title=The permanental process
|journal=[[Advances in Applied Probability]]
|volume=38 |issue=4 |pages=873&ndash;888
|doi=10.1239/aap/1165414583
}}</ref>

==See also==
{{div col|colwidth=20em}}
* [[Nearest neighbour function]]
* [[Spherical contact distribution function]]
* [[Factorial moment measure]]
* [[Moment measure]]
* [[Continuum percolation theory]]
* [[Random graphs]]
* [[Spatial statistics]]
* [[Stochastic geometry models of wireless networks]]
* [[Mathematical morphology]]
* [[Information geometry]]
* [[Stochastic differential geometry]]
{{div col end}}


==References==
==References==


{{Reflist}}
<references/>


{{Authority control}}
[[Category:Probability theory]]

[[Category:Stochastic processes]]
[[Category:Integral geometry]]
[[Category:Integral geometry]]
[[Category:Spatial processes]]
[[Category:Spatial processes]]

[[de:Stochastische Geometrie]]

Latest revision as of 00:45, 27 November 2024

A possible stochastic geometry model (Boolean model) for wireless network coverage and connectivity constructed from randomly sized disks placed at random locations

In mathematics, stochastic geometry 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 spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.

Models

[edit]

There are various models for point processes, typically based on but going beyond the classic homogeneous Poisson point process (the basic model for complete spatial randomness) to find expressive models which allow effective statistical methods.

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 Boolean model, 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[1] of statistical mechanics).

Random object

[edit]

What is meant by a random object? A complete answer to this question requires the theory of random closed sets, 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.

Line and hyper-flat processes

[edit]

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; 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 stereology, 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.

Line and hyper-flat processes have their own direct applications, but also find application as one way of creating tessellations dividing space; hence for example one may speak of Poisson line tessellations. A notable recent result[2] 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 means, for example by using Voronoi and variant constructions, and also by iterating various means of construction.

Origin of the name

[edit]

The name appears to have been coined by David Kendall and Klaus Krickeberg[3] while preparing for a June 1969 Oberwolfach workshop, though antecedents for the theory stretch back much further under the name geometric probability. The term "stochastic geometry" was also used by Frisch and Hammersley in 1963[4] as one of two suggestions for names of a theory of "random irregular structures" inspired by percolation theory.

Applications

[edit]

This brief description has focused on the theory[3][5] 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,[6] spatially distributed telecommunications,[7] wireless network modeling and analysis,[8] modeling of channel fading,[9][10] forestry,[11] the statistical theory of shape,[12] material science,[13] multivariate analysis, problems in image analysis[14] and stereology. There are links to statistical mechanics,[15] Markov chain Monte Carlo, and implementations of the theory in statistical computing (for example, spatstat[16] in R). Most recently determinantal and permanental point processes (connected to random matrix theory) are beginning to play a role.[17]

See also

[edit]

References

[edit]
  1. ^ Chayes, J. T.; Chayes, L.; Kotecký, R. (1995). "The analysis of the Widom-Rowlinson model by stochastic geometric methods". Communications in Mathematical Physics. 172 (3): 551–569. Bibcode:1995CMaPh.172..551C. doi:10.1007/BF02101808.
  2. ^ Kovalenko, I. N. (1999). "A simplified proof of a conjecture of D. G. Kendall concerning shapes of random polygons". Journal of Applied Mathematics and Stochastic Analysis. 12 (4): 301–310. doi:10.1155/S1048953399000283.
  3. ^ a b See foreword in Stoyan, D.; Kendall, W. S.; Mecke, J. (1987). Stochastic geometry and its applications. Wiley. ISBN 0-471-90519-4.
  4. ^ Frisch, H. L.; Hammersley, J. M. (1963). "Percolation processes and related topics". SIAM Journal on Applied Mathematics. 11 (4): 894–918. doi:10.1137/0111066.
  5. ^ Schneider, R.; Weil, W. (2008). Stochastic and Integral Geometry. Probability and Its Applications. Springer. doi:10.1007/978-3-540-78859-1. ISBN 978-3-540-78858-4. MR 2455326.
  6. ^ Martinez, V. J.; Saar, E. (2001). Statistics of The Galaxy Distribution. Chapman & Hall. ISBN 1-58488-084-8.
  7. ^ Baccelli, F.; Klein, M.; Lebourges, M.; Zuyev, S. (1997). "Stochastic geometry and architecture of communication networks". Telecommunication Systems. 7: 209–227. doi:10.1023/A:1019172312328.
  8. ^ M. Haenggi. Stochastic geometry for wireless networks. Cambridge University Press, 2012.
  9. ^ 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". IEEE Antennas and Wireless Propagation Letters. 4 (1): 385–388. Bibcode:2005IAWPL...4..385P. doi:10.1109/LAWP.2005.857968.
  10. ^ Abdulla, M.; Shayan, Y. R. (2014). "Large-Scale Fading Behavior for a Cellular Network with Uniform Spatial Distribution". Wireless Communications and Mobile Computing. 4 (7): 1–17. arXiv:1302.0891. doi:10.1002/WCM.2565.
  11. ^ Stoyan, D.; Penttinen, A. (2000). "Recent Applications of Point Process Methods in Forestry Statistics". Statistical Science. 15: 61–78.
  12. ^ Kendall, D. G. (1989). "A survey of the statistical theory of shape". Statistical Science. 4 (2): 87–99. doi:10.1214/ss/1177012582.
  13. ^ Torquato, S. (2002). Random heterogeneous materials. Springer-Verlag. ISBN 0-387-95167-9.
  14. ^ Van Lieshout, M. N. M. (1995). Stochastic Geometry Models in Image Analysis and Spatial Statistics. CWI Tract, 108. CWI. ISBN 90-6196-453-9.
  15. ^ Georgii, H.-O.; Häggström, O.; Maes, C. (2001). "The random geometry of equilibrium phases". Phase Transitions and Critical Phenomena. Vol. 18. Academic Press. pp. 1–142.
  16. ^ Baddeley, A.; Turner, R. (2005). "Spatstat: An R package for analyzing spatial point patterns". Journal of Statistical Software. 12 (6): 1–42. doi:10.18637/jss.v012.i06.
  17. ^ McCullagh, P.; Møller, J. (2006). "The permanental process". Advances in Applied Probability. 38 (4): 873–888. doi:10.1239/aap/1165414583.
Retrieved from "https://en.wikipedia.org/enwiki/w/index.php?title=Stochastic_geometry&oldid=1259781515"