AreaInter: The Area Interaction Point Process Model
In spatstat.model: Parametric Statistical Modelling and Inference for the 'spatstat' Family
AreaInter R Documentation
The Area Interaction Point Process Model
Description
Creates an instance of the Area Interaction point process model
(Widom-Rowlinson penetrable spheres model)
which can then be fitted to point pattern data.
Usage
AreaInter(r)
Arguments
r
The radius of the discs in the area interaction process
Details
This function defines the interpoint interaction structure of a point
process called the Widom-Rowlinson penetrable sphere model
or area-interaction process. It can be used to fit this model to
point pattern data.
The function ppm(), which fits point process models to
point pattern data, requires an argument
of class "interact" describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the area interaction structure is
yielded by the function AreaInter(). See the examples below.
In standard form, the area-interaction process
(Widom and Rowlinson, 1970;
Baddeley and Van Lieshout, 1995) with disc radius r,
intensity parameter \kappa and interaction parameter
\gamma is a point process with probability density
f(x_1,\ldots,x_n) =
\alpha \kappa^{n(x)} \gamma^{-A(x)}
for a point pattern x, where
x_1,\ldots,x_n represent the
points of the pattern, n(x) is the number of points in the
pattern, and A(x) is the area of the region formed by
the union of discs of radius r centred at the points
x_1,\ldots,x_n.
Here \alpha is a normalising constant.
The interaction parameter \gamma can be any positive number.
If \gamma = 1 then the model reduces to a Poisson
process with intensity \kappa.
If \gamma < 1 then the process is regular,
while if \gamma > 1 the process is clustered.
Thus, an area interaction process can be used to model either
clustered or regular point patterns. Two points interact if the
distance between them is less than 2r.
The standard form of the model, shown above, is a little
complicated to interpret in practical applications.
For example, each isolated point of the pattern x contributes a factor
\kappa \gamma^{-\pi r^2}
to the probability density.
In spatstat, the model is parametrised in a different form,
which is easier to interpret.
In canonical scale-free form, the probability density is rewritten as
f(x_1,\ldots,x_n) =
\alpha \beta^{n(x)} \eta^{-C(x)}
where \beta is the new intensity parameter,
\eta is the new interaction parameter, and
C(x) = B(x) - n(x) is the interaction potential. Here
B(x) = \frac{A(x)}{\pi r^2}
is the normalised area (so that the discs have unit area).
In this formulation, each isolated point of the pattern contributes a
factor \beta to the probability density (so the
first order trend is \beta). The quantity
C(x) is a true interaction potential, in the sense that
C(x) = 0 if the point pattern x does not contain any
points that lie close together (closer than 2r units
apart).
When a new point u is added to an existing point pattern
x, the rescaled potential -C(x) increases by
a value between 0 and 1.
The increase is zero if u is not close to any point of x.
The increase is 1 if the disc of radius r centred at u
is completely contained in the union of discs of radius r
centred at the data points x_i. Thus, the increase in
potential is a measure of how close the new point u is to the
existing pattern x. Addition of the point
u contributes a factor \beta \eta^\delta
to the probability density, where \delta is the
increase in potential.
The old parameters \kappa,\gamma of the
standard form are related to
the new parameters \beta,\eta of the canonical
scale-free form, by
\beta = \kappa \gamma^{-\pi r^2} = \kappa /\eta
and
\eta = \gamma^{\pi r^2}
provided \gamma and \kappa are positive and finite.
In the canonical scale-free form, the parameter \eta
can take any nonnegative value. The value \eta = 1
again corresponds to a Poisson process, with intensity \beta.
If \eta < 1 then the process is regular,
while if \eta > 1 the process is clustered.
The value \eta = 0 corresponds to a hard core process
with hard core radius r (interaction distance 2r).
The nonstationary area interaction process is similar except that
the contribution of each individual point x_i
is a function \beta(x_i)
of location, rather than a constant beta.
Note the only argument of AreaInter() is the disc radius r.
When r is fixed, the model becomes an exponential family.
The canonical parameters \log(\beta)
and \log(\eta)
are estimated by ppm(), not fixed in
AreaInter().
Value
An object of class "interact"
describing the interpoint interaction
structure of the area-interaction process with disc radius r.
Warnings
The interaction distance of this process is equal to
2 * r. Two discs of radius r overlap if their centres
are closer than 2 * r units apart.
The estimate of the interaction parameter \eta
is unreliable if the interaction radius r is too small
or too large. In these situations the model is approximately Poisson
so that \eta is unidentifiable.
As a rule of thumb, one can inspect the empty space function
of the data, computed by Fest. The value F(r)
of the empty space function at the interaction radius r should
be between 0.2 and 0.8.
Author(s)
\adrian
and
\rolf
References
Baddeley, A.J. and Van Lieshout, M.N.M. (1995).
Area-interaction point processes.
Annals of the Institute of Statistical Mathematics
47 (1995) 601–619.
Widom, B. and Rowlinson, J.S. (1970).
New model for the study of liquid-vapor phase transitions.
The Journal of Chemical Physics
52 (1970) 1670–1684.
See Also
ppm,
pairwise.family,
ppm.object
ragsAreaInter and rmh for simulation
of area-interaction models.
Examples
# prints a sensible description of itself
AreaInter(r=0.1)
# Note the reach is twice the radius
reach(AreaInter(r=1))
# Fit the stationary area interaction process to Swedish Pines data
ppm(swedishpines ~1, AreaInter(r=7))
# Fit the stationary area interaction process to `cells'
ppm(cells ~1, AreaInter(r=0.06))
# eta=0 indicates hard core process.
# Fit a nonstationary area interaction with log-cubic polynomial trend
ppm(swedishpines ~polynom(x/10,y/10,3), AreaInter(r=7))
spatstat.model documentation built on Sept. 30, 2024, 9:26 a.m.
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| AreaInter | R Documentation |
The Area Interaction Point Process Model
Description
Creates an instance of the Area Interaction point process model (Widom-Rowlinson penetrable spheres model) which can then be fitted to point pattern data.
Usage
AreaInter(r)
Arguments
r |
The radius of the discs in the area interaction process |
Details
This function defines the interpoint interaction structure of a point process called the Widom-Rowlinson penetrable sphere model or area-interaction process. It can be used to fit this model to point pattern data.
The function ppm(), which fits point process models to
point pattern data, requires an argument
of class "interact" describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the area interaction structure is
yielded by the function AreaInter(). See the examples below.
In standard form, the area-interaction process
(Widom and Rowlinson, 1970;
Baddeley and Van Lieshout, 1995) with disc radius r,
intensity parameter \kappa and interaction parameter
\gamma is a point process with probability density
f(x_1,\ldots,x_n) =
\alpha \kappa^{n(x)} \gamma^{-A(x)}
for a point pattern x, where
x_1,\ldots,x_n represent the
points of the pattern, n(x) is the number of points in the
pattern, and A(x) is the area of the region formed by
the union of discs of radius r centred at the points
x_1,\ldots,x_n.
Here \alpha is a normalising constant.
The interaction parameter \gamma can be any positive number.
If \gamma = 1 then the model reduces to a Poisson
process with intensity \kappa.
If \gamma < 1 then the process is regular,
while if \gamma > 1 the process is clustered.
Thus, an area interaction process can be used to model either
clustered or regular point patterns. Two points interact if the
distance between them is less than 2r.
The standard form of the model, shown above, is a little
complicated to interpret in practical applications.
For example, each isolated point of the pattern x contributes a factor
\kappa \gamma^{-\pi r^2}
to the probability density.
In spatstat, the model is parametrised in a different form, which is easier to interpret. In canonical scale-free form, the probability density is rewritten as
f(x_1,\ldots,x_n) =
\alpha \beta^{n(x)} \eta^{-C(x)}
where \beta is the new intensity parameter,
\eta is the new interaction parameter, and
C(x) = B(x) - n(x) is the interaction potential. Here
B(x) = \frac{A(x)}{\pi r^2}
is the normalised area (so that the discs have unit area).
In this formulation, each isolated point of the pattern contributes a
factor \beta to the probability density (so the
first order trend is \beta). The quantity
C(x) is a true interaction potential, in the sense that
C(x) = 0 if the point pattern x does not contain any
points that lie close together (closer than 2r units
apart).
When a new point u is added to an existing point pattern
x, the rescaled potential -C(x) increases by
a value between 0 and 1.
The increase is zero if u is not close to any point of x.
The increase is 1 if the disc of radius r centred at u
is completely contained in the union of discs of radius r
centred at the data points x_i. Thus, the increase in
potential is a measure of how close the new point u is to the
existing pattern x. Addition of the point
u contributes a factor \beta \eta^\delta
to the probability density, where \delta is the
increase in potential.
The old parameters \kappa,\gamma of the
standard form are related to
the new parameters \beta,\eta of the canonical
scale-free form, by
\beta = \kappa \gamma^{-\pi r^2} = \kappa /\eta
and
\eta = \gamma^{\pi r^2}
provided \gamma and \kappa are positive and finite.
In the canonical scale-free form, the parameter \eta
can take any nonnegative value. The value \eta = 1
again corresponds to a Poisson process, with intensity \beta.
If \eta < 1 then the process is regular,
while if \eta > 1 the process is clustered.
The value \eta = 0 corresponds to a hard core process
with hard core radius r (interaction distance 2r).
The nonstationary area interaction process is similar except that
the contribution of each individual point x_i
is a function \beta(x_i)
of location, rather than a constant beta.
Note the only argument of AreaInter() is the disc radius r.
When r is fixed, the model becomes an exponential family.
The canonical parameters \log(\beta)
and \log(\eta)
are estimated by ppm(), not fixed in
AreaInter().
Value
An object of class "interact"
describing the interpoint interaction
structure of the area-interaction process with disc radius r.
Warnings
The interaction distance of this process is equal to
2 * r. Two discs of radius r overlap if their centres
are closer than 2 * r units apart.
The estimate of the interaction parameter \eta
is unreliable if the interaction radius r is too small
or too large. In these situations the model is approximately Poisson
so that \eta is unidentifiable.
As a rule of thumb, one can inspect the empty space function
of the data, computed by Fest. The value F(r)
of the empty space function at the interaction radius r should
be between 0.2 and 0.8.
Author(s)
\adrianand \rolf
References
Baddeley, A.J. and Van Lieshout, M.N.M. (1995). Area-interaction point processes. Annals of the Institute of Statistical Mathematics 47 (1995) 601–619.
Widom, B. and Rowlinson, J.S. (1970). New model for the study of liquid-vapor phase transitions. The Journal of Chemical Physics 52 (1970) 1670–1684.
See Also
ppm,
pairwise.family,
ppm.object
ragsAreaInter and rmh for simulation
of area-interaction models.
Examples
# prints a sensible description of itself
AreaInter(r=0.1)
# Note the reach is twice the radius
reach(AreaInter(r=1))
# Fit the stationary area interaction process to Swedish Pines data
ppm(swedishpines ~1, AreaInter(r=7))
# Fit the stationary area interaction process to `cells'
ppm(cells ~1, AreaInter(r=0.06))
# eta=0 indicates hard core process.
# Fit a nonstationary area interaction with log-cubic polynomial trend
ppm(swedishpines ~polynom(x/10,y/10,3), AreaInter(r=7))
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