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",
"Kcross", "Lcross", "Kdot", "Ldot",
"Kcross.inhom", "Lcross.inhom"),
...,
block=FALSE, global=FALSE, basicboot=FALSE, Vcorrection=FALSE,
confidence=0.95, nx = 4, ny = nx, nsim=200, type=7)X |
A point pattern (object of class |
fun |
Name of the summary function for which confidence intervals are
desired: one of the strings |
... |
Arguments passed to the corresponding local version of the summary function (see Details). |
block |
Logical value indicating whether to use Loh's block bootstrap
as originally proposed. Default is |
global |
Logical. If |
basicboot |
Logical value indicating whether to use the so-called basic bootstrap confidence interval. See Details. |
Vcorrection |
Logical value indicating whether to use a variance correction
when |
confidence |
Confidence level, as a fraction between 0 and 1. |
nx,ny |
Integers.
If |
nsim |
Number of bootstrap simulations. |
type |
Integer. Type of quantiles.
Argument passed to |
This algorithm computes
confidence bands for the true value of the summary function
fun using the bootstrap method of Loh (2008)
and a modification described in Baddeley, Rubak, Turner (2015).
If 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 pcf.
It starts by computing the array of
local pair correlation functions,
localpcf, of the data pattern X.
This array consists of the contributions to the estimate of the
pair correlation function from each
data point.
If block=FALSE, these contributions are resampled nsim times
with replacement as described in Baddeley, Rubak, Turner (2015);
from each resampled dataset the total contribution
is computed, yielding nsim random pair correlation functions.
If block=TRUE, the calculation is performed as originally
proposed by Loh (2008, 2010). The (bounding box of the) window is divided
into nx * ny rectangles (blocks).
The average contribution of a block
is obtained by averaging the contribution of each point included in the block.
Then, the average contributions on each block are resampled nsim times
with replacement as described in Loh (2008) and Loh (2010);
from each resampled dataset the total contribution
is computed, yielding nsim random pair correlation functions.
Notice that for non-rectangular windows any blocks not fully contained in the
window are discarded before doing the resampling, so the effective number of
blocks may be substantially smaller than nx * ny in this case.
The pointwise alpha/2 and 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.
There are several ways to define a bootstrap confidence interval.
If basicbootstrap=TRUE,
the so-called basic confidence bootstrap interval
is used as described in Loh (2008).
It has been noticed in Loh (2010) that
when the intensity of the point process is unknown,
the bootstrap error estimate is larger than it should be.
When the K function is used,
an adjustment procedure has been proposed in Loh (2010)
that is used if Vcorrection=TRUE.
In this case, the basic confidence bootstrap interval is implicitly used.
To control the estimation algorithm, use the
arguments ..., which are passed to the local version
of the summary function, as shown below:
For fun="Lest", the calculations are first performed
as if fun="Kest", and then the square-root transformation is
applied to obtain the L-function. Similarly for
fun="Linhom", "Lcross", "Ldot", "Lcross.inhom".
Note that the confidence bands computed by
lohboot(fun="pcf") may not contain the estimate of the
pair correlation function computed by pcf,
because of differences between the algorithm parameters
(such as the choice of edge correction)
in localpcf and pcf.
If you are using lohboot, the
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 lohboot is varblock.
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.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk and Christophe Biscio.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Loh, J.M. (2008) A valid and fast spatial bootstrap for correlation functions. The Astrophysical Journal, 681, 726–734.
Loh, J.M. (2010) Bootstrapping an inhomogeneous point process. Journal of Statistical Planning and Inference, 140, 734–749.
Summary functions
Kest,
pcf,
Kinhom,
pcfinhom,
localK,
localpcf,
localKinhom,
localpcfinhom,
localKcross,
localKdot,
localLcross,
localLdot.
localKcross.inhom,
localLcross.inhom.
See varblock for an alternative bootstrap technique.
p <- lohboot(simdat, stoyan=0.5) g <- lohboot(simdat, stoyan=0.5, block=TRUE) g plot(g)
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