Bootstrap Confidence Bands for Summary Function
Computes a bootstrap confidence band for a summary function of a point process.
lohboot(X, fun=c("pcf", "Kest", "Lest", "pcfinhom", "Kinhom", "Linhom"), ..., nsim=200, confidence=0.95, global=FALSE, type=7)
- A point pattern (object of class
- Name of the summary function for which confidence intervals are
desired: one of the strings
"Linhom". Alternatively, t
- Arguments passed to the corresponding local version of the summary function (see Details).
- Number of bootstrap simulations.
- Confidence level, as a fraction between 0 and 1.
- Logical. If
FALSE(the default), pointwise confidence intervals are constructed. If
TRUE, a global (simultaneous) confidence band is constructed.
- Integer. Argument passed to
quantilecontrolling the way the quantiles are calculated.
This algorithm computes
confidence bands for the true value of the summary function
fun using the bootstrap method of Loh (2008).
fun="pcf", for example, the algorithm computes a pointwise
(100 * confidence)% confidence interval for the true value of
the pair correlation function for the point process,
normally estimated by
It starts by computing the array of
local pair correlation functions,
localpcf, of the data pattern
This array consists of the contributions to the estimate of the
pair correlation function from each
data point. Then these contributions are resampled
with replacement; from each resampled dataset the total contribution
is computed, yielding
nsim random pair correlation functions.
1 - alpha/2 quantiles of
these functions are computed, where
alpha = 1 - confidence.
The average of the local functions is also computed as an estimate
of the pair correlation function.
To control the estimation algorithm, use the
..., which are passed to the local version
of the summary function, as shown below:
fun="Lest", the calculations are first performed
fun="Kest", and then the square-root transformation is
applied to obtain the $L$-function.
Note that the confidence bands computed by
lohboot(fun="pcf") may not contain the estimate of the
pair correlation function computed by
because of differences between the algorithm parameters
(such as the choice of edge correction)
If you are using
appropriate point estimate of the pair correlation itself is
the pointwise mean of the local estimates, which is provided
in the result of
lohboot and is shown in the default plot.
If the confidence bands seem unbelievably narrow,
this may occur because the point pattern has a hard core
(the true pair correlation function is zero for certain values of
distance) or because of an optical illusion when the
function is steeply sloping (remember the width of the confidence
bands should be measured vertically).
An alternative to
- A function value table
(object of class
"fv") containing columns giving the estimate of the summary function, the upper and lower limits of the bootstrap confidence interval, and the theoretical value of the summary function for a Poisson process.
Loh, J.M. (2008) A valid and fast spatial bootstrap for correlation functions. The Astrophysical Journal, 681, 726--734.
varblock for an alternative bootstrap technique.
p <- lohboot(simdat, stoyan=0.5) plot(p)