Diffusion Estimate of Point Pattern Intensity
Computes the diffusion estimate of the intensity of a point pattern.
## S3 method for class 'ppp'
densityHeat(x, sigma, ..., weights=NULL,
connect=8, symmetric=FALSE,
sigmaX=NULL, k=1, show=FALSE, se=FALSE,
at=c("pixels", "points"),
leaveoneout = TRUE,
extrapolate = FALSE, coarsen = TRUE,
verbose=TRUE, internal=NULL)x |
Point pattern (object of class |
sigma |
Smoothing bandwidth. A single number giving the equivalent
standard deviation of the smoother.
Alternatively, a pixel image (class |
... |
Arguments passed to |
weights |
Optional numeric vector of weights associated with each point of
|
connect |
Grid connectivity: either 4 or 8. |
symmetric |
Logical value indicating whether to force the algorithm to use a symmetric random walk. |
sigmaX |
Numeric vector of bandwidths, one associated with each data point in
|
k |
Integer. Calculations will be performed by repeatedly multiplying
the current state by the |
show |
Logical value indicating whether to plot successive iterations. |
se |
Logical value indicating whether to compute standard errors. |
at |
Character string specifying whether to compute values
at a grid of pixels ( |
leaveoneout |
Logical value specifying whether to compute a leave-one-out
estimate at each data point, when |
extrapolate |
Logical value specifying whether to use Richardson extrapolation to improve the accuracy of the computation. |
coarsen |
Logical value, controlling the calculation performed when
|
verbose |
Logical value specifying whether to print progress reports. |
internal |
Developer use only. |
This command computes a diffusion kernel estimate
of point process intensity from the observed point pattern x.
The function densityHeat is generic,
with methods for point patterns in two dimensions
(class "ppp") and point patterns on a linear network
(class "lpp"). The function densityHeat.ppp described
here is the method for class "ppp". Given a two-dimensional
point pattern x, it computes a diffusion kernel estimate
of the intensity of the point process which generated x.
Diffusion kernel estimates were developed by Botev et al (2010), Barry and McIntyre (2011) and Baddeley et al (2021).
Barry and McIntyre (2011) proposed an estimator for point process intensity based on a random walk on the pixel grid inside the observation window. Baddeley et al (2021) showed that the Barry-McIntyre method is a special case of the diffusion estimator proposed by Botev et al (2010).
The original Barry-McIntyre algorithm assumes a symmetric random walk
(i.e. each possible transition has the same probability p)
and requires a square pixel grid (i.e. equal
spacing in the x and y directions). Their original
algorithm is used if symmetric=TRUE. Use the ...
arguments to ensure a square grid: for example, the argument
eps specifies a square grid with spacing eps units.
The more general algorithm used here (Baddeley et al, 2021)
does not require a square grid of pixels.
If the pixel grid is not square, and if symmetric=FALSE
(the default), then the random walk is not symmetric,
in the sense that the probabilities of different jumps will be
different, in order to ensure that the smoothing is isotropic.
This implementation also includes two generalizations to the case of adaptive smoothing (Baddeley et al, 2021).
In the first version of adaptive smoothing, the bandwidth is
spatially-varying.
The argument sigma should be a pixel image (class "im")
or a function(x,y) specifying the bandwidth at each spatial
location. The smoothing is performed by solving the
heat equation with spatially-varying parameters.
In the second version of adaptive smoothing, each data point in
x is smoothed using a separate bandwidth.
The argument sigmaX should be a numeric vector
specifying the bandwidth for each point of x.
The smoothing is performed using the lagged arrival algorithm.
The argument sigma can be omitted.
If extrapolate=FALSE (the default), calculations are performed
using the Euler scheme for the heat equation.
If extrapolate=TRUE, the accuracy of the result will be
improved by applying Richardson extrapolation (Baddeley et al, 2021, Section
4). After computing the intensity estimate using the Euler scheme
on the desired pixel grid, another estimate is computed using the same
method on another pixel grid, and the two estimates are combined by
Richardson extrapolation to obtain a more accurate result.
The second grid is coarser than the original grid if
coarsen=TRUE (the default), and finer than the original grid
if coarsen=FALSE. Setting extrapolate=TRUE increases
computation time by 35% if coarsen=TRUE and by 400% if
coarsen=FALSE.
Pixel image (object of class "im") giving the estimated
intensity of the point process.
If se=TRUE, the result has an attribute "se"
which is another pixel image giving the estimated standard error.
If at="points" then the result is a numeric vector
with one entry for each point of x.
Adrian Baddeley and Tilman Davies.
Baddeley, A., Davies, T., Rakshit, S., Nair, G. and McSwiggan, G. (2021) Diffusion smoothing for spatial point patterns. Statistical Science, in press.
Barry, R.P. and McIntyre, J. (2011) Estimating animal densities and home range in regions with irregular boundaries and holes: a lattice-based alternative to the kernel density estimator. Ecological Modelling 222, 1666–1672.
Botev, Z.I., Grotowski, J.F. and Kroese, D.P. (2010) Kernel density estimation via diffusion. Annals of Statistics 38, 2916–2957.
density.ppp for the usual kernel estimator,
and adaptive.density for the
tessellation-based estimator.
online <- interactive()
if(!online) op <- spatstat.options(npixel=32)
X <- runifpoint(25, letterR)
Z <- densityHeat(X, 0.2)
if(online) {
plot(Z, main="Diffusion estimator")
plot(X, add=TRUE, pch=16)
integral(Z) # should equal 25
}
Z <- densityHeat(X, 0.2, se=TRUE)
Zse <- attr(Z, "se")
if(online) plot(solist(estimate=Z, SE=Zse), main="")
Zex <- densityHeat(X, 0.2, extrapolate=TRUE)
ZS <- densityHeat(X, 0.2, symmetric=TRUE, eps=0.125)
if(online) {
plot(ZS, main="fixed bandwidth")
plot(X, add=TRUE, pch=16)
}
sig <- function(x,y) { (x-1.5)/10 }
ZZ <- densityHeat(X, sig)
if(online) {
plot(ZZ, main="adaptive (I)")
plot(X, add=TRUE, pch=16)
}
sigX <- sig(X$x, X$y)
AA <- densityHeat(X, sigmaX=sigX)
if(online) {
plot(AA, main="adaptive (II)")
plot(X, add=TRUE, pch=16)
}
if(!online) spatstat.options(op)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.