Kdot: Multitype K Function (i-to-any)
In spatstat.explore: Exploratory Data Analysis for the 'spatstat' Family
Kdot R Documentation
Multitype K Function (i-to-any)
Description
For a multitype point pattern,
estimate the multitype K function
which counts the expected number of other points of the process
within a given distance of a point of type i.
Usage
Kdot(X, i, r=NULL, breaks=NULL, correction, ..., ratio=FALSE, from)
Arguments
X
The observed point pattern,
from which an estimate of the multitype K function
K_{i\bullet}(r) will be computed.
It must be a multitype point pattern (a marked point pattern
whose marks are a factor). See under Details.
i
The type (mark value)
of the points in X from which distances are measured.
A character string (or something that will be converted to a
character string).
Defaults to the first level of marks(X).
r
numeric vector. The values of the argument r
at which the distribution function
K_{i\bullet}(r) should be evaluated.
There is a sensible default.
First-time users are strongly advised not to specify this argument.
See below for important conditions on r.
breaks
This argument is for internal use only.
correction
A character vector containing any selection of the
options "border", "bord.modif",
"isotropic", "Ripley", "translate",
"translation", "periodic",
"none" or "best".
It specifies the edge correction(s) to be applied.
Alternatively correction="all" selects all options.
...
Ignored.
ratio
Logical.
If TRUE, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns.
from
An alternative way to specify i.
Details
This function Kdot and its companions
Kcross and Kmulti
are generalisations of the function Kest
to multitype point patterns.
A multitype point pattern is a spatial pattern of
points classified into a finite number of possible
“colours” or “types”. In the spatstat package,
a multitype pattern is represented as a single
point pattern object in which the points carry marks,
and the mark value attached to each point
determines the type of that point.
The argument X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
It must be a marked point pattern, and the mark vector
X$marks must be a factor.
The argument i will be interpreted as a
level of the factor X$marks.
If i is missing, it defaults to the first
level of the marks factor, i = levels(X$marks)[1].
The “type i to any type” multitype K function
of a stationary multitype point process X is defined so that
\lambda K_{i\bullet}(r)
equals the expected number of
additional random points within a distance r of a
typical point of type i in the process X.
Here \lambda
is the intensity of the process,
i.e. the expected number of points of X per unit area.
The function K_{i\bullet} is determined by the
second order moment properties of X.
An estimate of K_{i\bullet}(r)
is a useful summary statistic in exploratory data analysis
of a multitype point pattern.
If the subprocess of type i points were independent
of the subprocess of points of all types not equal to i,
then K_{i\bullet}(r) would equal \pi r^2.
Deviations between the empirical K_{i\bullet} curve
and the theoretical curve \pi r^2
may suggest dependence between types.
This algorithm estimates the distribution function K_{i\bullet}(r)
from the point pattern X. It assumes that X can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in X as Window(X))
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in Kest,
using the chosen edge correction(s).
The argument r is the vector of values for the
distance r at which K_{i\bullet}(r) should be evaluated.
The values of r must be increasing nonnegative numbers
and the maximum r value must not exceed the radius of the
largest disc contained in the window.
The pair correlation function can also be applied to the
result of Kdot; see pcf.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r
the values of the argument r
at which the function K_{i\bullet}(r) has been estimated
theo
the theoretical value of K_{i\bullet}(r)
for a marked Poisson process, namely \pi r^2
together with a column or columns named
"border", "bord.modif",
"iso" and/or "trans",
according to the selected edge corrections. These columns contain
estimates of the function K_{i\bullet}(r)
obtained by the edge corrections named.
If ratio=TRUE then the return value also has two
attributes called "numerator" and "denominator"
which are "fv" objects
containing the numerators and denominators of each
estimate of K(r).
Warnings
The argument i is interpreted as
a level of the factor X$marks. It is converted to a character
string if it is not already a character string.
The value i=1 does not
refer to the first level of the factor.
The reduced sample estimator of K_{i\bullet} is pointwise approximately
unbiased, but need not be a valid distribution function; it may
not be a nondecreasing function of r.
Author(s)
\adrian
and \rolf.
References
Cressie, N.A.C. Statistics for spatial data.
John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns.
Academic Press, 1983.
Harkness, R.D and Isham, V. (1983)
A bivariate spatial point pattern of ants' nests.
Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982).
Methods for analysing spatial processes of several types of points.
J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes.
Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J.
Stochastic geometry and its applications.
2nd edition. Springer Verlag, 1995.
See Also
Kdot,
Kest,
Kmulti,
pcf
Examples
# Lansing woods data: 6 types of trees
woods <- lansing
Kh. <- Kdot(woods, "hickory")
# diagnostic plot for independence between hickories and other trees
plot(Kh.)
# synthetic example with two marks "a" and "b"
pp <- runifpoispp(50)
pp <- pp %mark% factor(sample(c("a","b"), npoints(pp), replace=TRUE))
K <- Kdot(pp, "a")
spatstat.explore documentation built on Oct. 22, 2024, 9:07 a.m.
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| Kdot | R Documentation |
Multitype K Function (i-to-any)
Description
For a multitype point pattern,
estimate the multitype K function
which counts the expected number of other points of the process
within a given distance of a point of type i.
Usage
Kdot(X, i, r=NULL, breaks=NULL, correction, ..., ratio=FALSE, from)
Arguments
X |
The observed point pattern,
from which an estimate of the multitype |
i |
The type (mark value)
of the points in |
r |
numeric vector. The values of the argument |
breaks |
This argument is for internal use only. |
correction |
A character vector containing any selection of the
options |
... |
Ignored. |
ratio |
Logical.
If |
from |
An alternative way to specify |
Details
This function Kdot and its companions
Kcross and Kmulti
are generalisations of the function Kest
to multitype point patterns.
A multitype point pattern is a spatial pattern of points classified into a finite number of possible “colours” or “types”. In the spatstat package, a multitype pattern is represented as a single point pattern object in which the points carry marks, and the mark value attached to each point determines the type of that point.
The argument X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
It must be a marked point pattern, and the mark vector
X$marks must be a factor.
The argument i will be interpreted as a
level of the factor X$marks.
If i is missing, it defaults to the first
level of the marks factor, i = levels(X$marks)[1].
The “type i to any type” multitype K function
of a stationary multitype point process X is defined so that
\lambda K_{i\bullet}(r)
equals the expected number of
additional random points within a distance r of a
typical point of type i in the process X.
Here \lambda
is the intensity of the process,
i.e. the expected number of points of X per unit area.
The function K_{i\bullet} is determined by the
second order moment properties of X.
An estimate of K_{i\bullet}(r)
is a useful summary statistic in exploratory data analysis
of a multitype point pattern.
If the subprocess of type i points were independent
of the subprocess of points of all types not equal to i,
then K_{i\bullet}(r) would equal \pi r^2.
Deviations between the empirical K_{i\bullet} curve
and the theoretical curve \pi r^2
may suggest dependence between types.
This algorithm estimates the distribution function K_{i\bullet}(r)
from the point pattern X. It assumes that X can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in X as Window(X))
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in Kest,
using the chosen edge correction(s).
The argument r is the vector of values for the
distance r at which K_{i\bullet}(r) should be evaluated.
The values of r must be increasing nonnegative numbers
and the maximum r value must not exceed the radius of the
largest disc contained in the window.
The pair correlation function can also be applied to the
result of Kdot; see pcf.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r |
the values of the argument |
theo |
the theoretical value of |
together with a column or columns named
"border", "bord.modif",
"iso" and/or "trans",
according to the selected edge corrections. These columns contain
estimates of the function K_{i\bullet}(r)
obtained by the edge corrections named.
If ratio=TRUE then the return value also has two
attributes called "numerator" and "denominator"
which are "fv" objects
containing the numerators and denominators of each
estimate of K(r).
Warnings
The argument i is interpreted as
a level of the factor X$marks. It is converted to a character
string if it is not already a character string.
The value i=1 does not
refer to the first level of the factor.
The reduced sample estimator of K_{i\bullet} is pointwise approximately
unbiased, but need not be a valid distribution function; it may
not be a nondecreasing function of r.
Author(s)
\adrianand \rolf.
References
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants' nests. Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.
See Also
Kdot,
Kest,
Kmulti,
pcf
Examples
# Lansing woods data: 6 types of trees
woods <- lansing
Kh. <- Kdot(woods, "hickory")
# diagnostic plot for independence between hickories and other trees
plot(Kh.)
# synthetic example with two marks "a" and "b"
pp <- runifpoispp(50)
pp <- pp %mark% factor(sample(c("a","b"), npoints(pp), replace=TRUE))
K <- Kdot(pp, "a")
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