Estimate Variance of Summary Statistic by Subdivision
This command estimates the variance of any summary statistic (such as the $K$-function) by spatial subdivision of a single point pattern dataset.
varblock(X, fun = Kest, blocks = quadrats(X, nx = nx, ny = ny), ..., nx = 3, ny = nx, confidence=0.95)
- Point pattern dataset (object of class
- Function that computes the summary statistic.
- Optional. A tessellation that specifies the division of the space into blocks.
- Arguments passed to
- Optional. Number of rectangular blocks
in the $x$ and $y$ directions.
- Confidence level, as a fraction between 0 and 1.
This command computes an estimate of the variance of
the summary statistic
fun(X) from a single point pattern
X using a subdivision method.
It can be used to plot confidence intervals
for the true value of a summary function such as the $K$-function.
The window containing
X is divided into pieces by
nx * ny array of rectangles
(or is divided into pieces of more general shape,
according to the argument
blocks if it is present).
The summary statistic
fun is applied to each of the
corresponding sub-patterns of
X as described below.
Then the pointwise
sample mean, sample variance and sample standard deviation
of these summary statistics are computed. Then
pointwise confidence intervals are computed, for the specified level
of confidence, defaulting to 95 percent.
The variance is estimated by equation (4.21) of Diggle (2003, page 52).
This assumes that the point pattern
X is stationary.
For further details see Diggle (2003, pp 52--53).
The estimate of the summary statistic
from each block is computed as follows.
For most functions
the estimate from block
is computed by finding the subset of
X consisting of
points that fall inside
fun to these points, by calling
fun is the $K$-function
or any function which has an argument called
the estimate for each block
B is computed
fun(X, domain=B). In the case of the
$K$-function this means that the estimate from block
is computed by counting pairs of
points in which the first point lies in
while the second point may lie anywhere.
- A function value table (object of class
"fv") that contains the result of
fun(X)as well as the sample mean, sample variance and sample standard deviation of the block estimates, together with the upper and lower two-standard-deviation confidence limits.
If the blocks are too small, there may be insufficient data
in some blocks, and the function
fun may report an error.
If this happens, you need to take larger blocks.
An error message about incompatibility may occur.
The different function estimates may be incompatible in some cases,
for example, because they use different default edge corrections
(typically because the tiles of the tessellation are not the same kind
of geometric object as the window of
X, or because the default
edge correction depends on the number of points). To prevent
this, specify the choice of edge correction,
correction argument to
fun, if it has one.
An alternative to
varblock is Loh's mark bootstrap
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
v <- varblock(amacrine, Kest, nx=4, ny=2) v <- varblock(amacrine, Kcross, nx=4, ny=2) if(interactive()) plot(v, iso ~ r, shade=c("hiiso", "loiso"))