Read More
Date: 14-4-2022
![]()
Date: 13-5-2022
![]()
Date: 8-5-2022
![]() |
In most modern literature, a Boolean model is a probabilistic model of continuum percolation theory characterized by the existence of a stationary point process and a random variable
which independently determine the centers and the random radii of a collection of closed balls in
for some
.
In this case, the model is said to be driven by .
Worth noting is that the most intuitive ideas about constructing a feasible model using and
often lead to unexpected and undesirable results (Meester and Roy 1996). For that reason, some more sophisticated machinery and quite a bit of care is needed to translate from the language of
and
into a reasonable model of continuum percolation. The formal construction is as follows.
Let be a stationary point process as discussed above and suppose that
is defined on a probability space
. Next, define the space
to be the product space
(1) |
and define associated to the usual product sigma-algebra and product measure
where here, all the marginal probabilities are given by some probability measure
on
. Finally, define
, equip
with the product measure
and usual product
-algebra. Under this construction, a Boolean model is a measurable mapping from
into
defined by
(2) |
where here, denotes the set of all counting measures on the
-algebra
of Borel sets in
which assign finite measure to all bounded Borel sets and which assign values of at most 1 to points
.
One then transitions to percolation by first defining the collection of so-called binary cubes of order
(3) |
for all ,
, and by noting that each point
is contained in a unique binary cube
of order
. Moreover, for each
, there is a unique smallest number
such that
contains no other points of
-almost surely. This fact allows one to define the radius
of the ball centered at
to be
(4) |
where is the notation used to denote an element
. Using this construction, one gets a collection of overlapping
-dimensional closed spheres whose radii are independent of the point process
and for which different points have balls with independent and identically-distributed radii.
It is not uncommon for a general Boolean model constructed in this way to be denoted or
, interchangeably. In the particular instance that
is a Poisson process with density
, the measure
is sometimes written
while the probability of an event
is then written
or
interchangeably.
In Boolean models, the space is partitioned into two regions, namely the occupied region-the subset of
covered by at least one ball, denoted
-and its complement, the vacant region. These two regions are similar in that both consist of connected components (the occupied components and the vacant components, respectively) and the notation
is used to denote the union of all occupied components having non-empty intersection with a subset
. For
, the notation
is used, and in the event of vacancy, the same notation is used throughout with
instead of
. Two points
which are in the same occupied component are said to be connected in the occupied region, sometimes denoted
(5) |
Connectedness in the vacant region is defined analogously and denoted
(6) |
If for some
and
are in the same occupied, respectively vacant, component of
, respectively of
, the notation
in
, respectively
in
is used.
The above figure illustrates a realization of a Boolean model, illustrating some of the terminology related to thereto. In this figure, the shaded region is while the darker shaded region is
. Note that
is empty due to the fact that
is non-empty. Moreover, the path joining
and
lies entirely in
; this indicates that
are in the same vacant component of
, whereby it follows that
in
.
Historically, the term Boolean model was also used to refer to what's now known as the Boolean-Poisson model (Hanisch 1981).
Hanisch, K. H. "On Classes of Random Sets and Point Process Models." Serdica Bulgariacae Mathematicae Publicationes 7, 160-166, 1981.
Meester, R. and Roy, R. Continuum Percolation. New York: Cambridge University Press, 2008.
|
|
للعاملين في الليل.. حيلة صحية تجنبكم خطر هذا النوع من العمل
|
|
|
|
|
"ناسا" تحتفي برائد الفضاء السوفياتي يوري غاغارين
|
|
|
|
|
المجمع العلمي يقيم ورشة تطويرية ودورة قرآنية في النجف والديوانية
|
|
|