Stochastic geometry
Encyclopedia
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.
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
of statistical mechanics).
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.
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 tessellation
s 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 means,
for example by using Voronoi
and variant constructions, and also by iterating
various means of construction.
David Kendall
and Klaus Krickeberg (see remarks of David Kendall in the Foreword to ) 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
as one of two suggestions for names of a theory of "random irregular structures" inspired by percolation 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, spatially distributed
telecommunications,
modeling of channel fading, forestry, the statistical theory of shape, material science, multivariate analysis, problems in image
analysis and stereology.
There are links to statistical mechanics,
Markov chain Monte Carlo, and implementations
of the theory in statistical computing
(for example, spatstat in R
).
Most recently determinantal and permanental point processes
(connected to random matrix theory) are beginning to play a role.
spatial point processes
Point process
In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces...
, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.
Models
There are various models for point processes,typically based on but going beyond the classic homogeneous
Poisson point process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
(the basic model for complete spatial randomness
Complete spatial randomness
Complete spatial randomness describes a point process whereby point events occur within a given study area in a completely random fashion. Such a process is often modeled using only one parameter, i.e. the density of points, \rho within the defined area...
)
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
of statistical mechanics).
Random object
What is meant by a random object? A complete answer to this question requires the theory ofrandom closed sets
Random compact set
In mathematics, a random compact set is essentially a compact set-valued random variable. Random compact sets are useful in the study of attractors for random dynamical systems.-Definition:...
, 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
Suppose we are concerned no longer with compact objects, but with objects which arespatially 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
Stereology
Stereology was originally defined as "the spatial interpretation of sections". It is an interdisciplinary field that is largely concerned with the three-dimensional interpretation of planar sections of materials or tissues...
, 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 tessellation
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
s 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 means,
for example by using Voronoi
Voronoi diagram
In mathematics, a Voronoi diagram is a special kind of decomposition of a given space, e.g., a metric space, determined by distances to a specified family of objects in the space...
and variant constructions, and also by iterating
various means of construction.
Origin of the name
The name appears to have been coined byDavid Kendall
David George Kendall
David George Kendall FRS was an English statistician, who spent much of his academic life in the University of Oxford and the University of Cambridge. He worked with M. S...
and Klaus Krickeberg (see remarks of David Kendall in the Foreword to ) while preparing for a June 1969
Oberwolfach
Mathematical Research Institute of Oberwolfach
The Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany, was founded by mathematician Wilhelm Süss in 1944...
workshop, though antecedents
for the theory stretch back much further under the name geometric probability
Geometric probability
Problems of the following type, and their solution techniques, were firststudied in the 19th century, and the general topic became known as geometric probability....
.
The term "stochastic geometry" was also used by Frisch and Hammersley
John Hammersley
John Michael Hammersley was a British mathematician best known for his foundational work in the theory of self-avoiding walks and percolation theory. He was born in Helensburgh in Dunbartonshire, and educated at Sedbergh School. He started reading mathematics at Emmanuel College, Cambridge but was...
in 1963
as one of two suggestions for names of a theory of "random irregular structures" inspired by percolation theory
Percolation theory
In mathematics, percolation theory describes the behavior of connected clusters in a random graph. The applications of percolation theory to materials science and other domains are discussed in the article percolation.-Introduction:...
.
Applications
This brief description has focused on the theoryof 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, spatially distributed
telecommunications,
modeling of channel fading, forestry, the statistical theory of shape, material science, multivariate analysis, problems in image
analysis and stereology.
There are links to statistical mechanics,
Markov chain Monte Carlo, and implementations
of the theory in statistical computing
(for example, spatstat in R
R (programming language)
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
).
Most recently determinantal and permanental point processes
(connected to random matrix theory) are beginning to play a role.