spatcov {spatstat.explore} | R Documentation |
Estimate the Spatial Covariance Function of a Random Field
Description
Given a pixel image, calculate an estimate of the spatial covariance function. Given two pixel images, calculate an estimate of their spatial cross-covariance function.
Usage
spatcov(X, Y=X, ..., correlation=FALSE, isotropic = TRUE,
clip = TRUE, pooling=TRUE)
Arguments
X |
A pixel image (object of class |
Y |
Optional. Another pixel image. |
correlation |
Logical value specifying whether to standardise so that the spatial correlation function is returned. |
isotropic |
Logical value specifying whether to assume the covariance is isotropic, so that the result is a function of the lag distance. |
clip |
Logical value specifying whether to restrict the results to the range of spatial lags where the estimate is reliable. |
pooling |
Logical value specifying the estimation method when |
... |
Ignored. |
Details
In normal usage, only the first argument X
is given.
Then the pixel image X
is treated as a realisation of a stationary
random field, and its spatial covariance function is estimated.
Alternatively if Y
is given,
then X
and Y
are assumed to be
jointly stationary random fields, and their spatial cross-covariance
function is estimated.
For any random field X
, the spatial covariance
is defined for any two spatial locations u
and v
by
C(u,v) = \mbox{cov}(X(u), X(v))
where X(u)
and X(v)
are the values of the random field
at those locations. Here\mbox{cov}
denotes the
statistical covariance, defined for any random variables
A
and B
by
\mbox{cov}(A,B) = E(AB) - E(A) E(B)
where E(A)
denotes the expected value of A
.
If the random field is assumed to be stationary (at least second-order
stationary) then the spatial covariance C(u,v)
depends only on the lag vector v-u
:
C(u,v) = C_2(v-u)
C(u,v) = C2(v-u)
where C_2
is a function of a single vector argument.
If the random field is stationary and isotropic, then the spatial
covariance depends only on the lag distance
\| v - u \|
:
C_2(v-u) = C_1(\|v-u\|)
where C_1
is a function of distance.
The function spatcov
computes estimates of the
covariance function C_1
or C_2
as follows:
-
If
isotropic=FALSE
, an estimate of the covariance functionC_2
is computed, assuming the random field is stationary, using the naive moment estimator,C2 = imcov(X-mean(X))/setcov(Window(X))
. The result is a pixel image. -
If
isotropic=TRUE
(the default) an estimate of the covariance functionC_1
is computed, assuming the random field is stationary and isotropic.-
When
pooling=FALSE
, the estimate ofC_1
is the rotational average of the naive estimate ofC_2
. -
When
pooling=TRUE
(the default), the estimate ofC_1
is the ratio of the rotational averages of the numerator and denominator which form the naive estimate ofC_2
.
The result is a function object (class
"fv"
). -
If the argument Y
is given, it should be a pixel image
compatible with X
. An estimate of the spatial cross-covariance function
between X
and Y
will be computed.
Value
If isotropic=TRUE
(the default), the result is a function value
table (object of class "fv"
) giving the estimated values of the
covariance function or spatial correlation function
for a sequence of values of the spatial lag
distance r
.
If isotropic=FALSE
, the result is a pixel image
(object of class "im"
) giving the estimated values of the
spatial covariance function or spatial correlation function
for a grid of values of the spatial lag vector.
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
See Also
Examples
if(offline <- !interactive()) op <- spatstat.options(npixel=32)
D <- density(cells)
plot(spatcov(D))
if(offline) spatstat.options(op)