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.
spatcov(X, Y=X, ..., correlation=FALSE, isotropic = TRUE,
clip = TRUE, pooling=TRUE)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.
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 function \(C_2\) 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 function \(C_1\)
is computed, assuming the random field is stationary and isotropic.
When pooling=FALSE, the estimate of \(C_1\)
is the rotational average of the naive estimate of \(C_2\).
When pooling=TRUE (the default), the estimate of \(C_1\)
is the ratio of the rotational averages of the numerator and
denominator which form the naive estimate of \(C_2\).
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.
imcov, setcov
if(offline <- !interactive()) op <- spatstat.options(npixel=32)
D <- density(cells)
plot(spatcov(D))
if(offline) spatstat.options(op)
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