## S3 method for class 'ppp':
density(x, sigma, \dots,
weights, edge=TRUE, varcov=NULL,
at="pixels", leaveoneout=TRUE,
adjust=1, diggle=FALSE)"ppp").as.mask to determine
the pixel resolution.TRUE, apply edge correction.sigma.at="pixels") or
only at the points of x (at="points").at="points".TRUE, use Diggle's edge correction,
which is more accurate but slower to compute than the
correction described under Details."im").
Pixel values are estimated intensity values,
expressed in If at="points", the result is a numeric vector
of length equal to the number of points in x.
Values are estimated intensity values at the points of x.
In either case, the return value has attributes
"sigma" and "varcov" which report the smoothing
bandwidth that was used.
density. It computes a fixed-bandwidth kernel estimate
(Diggle, 1985) of the intensity function of the point process
that generated the point pattern x.
By default it computes the convolution of the
isotropic Gaussian kernel of standard deviation sigma
with point masses at each of the data points in x.
Each point has unit weight, unless the argument weights is
given (it should be a numeric vector; weights can be negative or zero).
If edge=TRUE, the intensity estimate is corrected for
edge effect bias in one of two ways:
diggle=FALSE(the default) the intensity estimate is
correted by dividing it by the convolution of the
Gaussian kernel with the window of observation.
Thus the intensity value at a point$u$is$$\hat\lambda(u) = e(u) \sum_i k(x_i - u) w_i$$where$k$is the Gaussian smoothing kernel,$e(u)$is an edge correction factor,
and$w_i$are the weights.diggle=TRUEthen the method of Diggle (1985)
is followed exactly.
The intensity value at a point$u$is$$\hat\lambda(u) = \sum_i k(x_i - u) w_i e(x_i)$$where again$k$is the Gaussian smoothing kernel,$e(x_i)$is an edge correction factor,
and$w_i$are the weights.
This computation is slightly slower but more accurate. Instead of the isotropic Gaussian kernel with standard deviation
sigma, the smoothing kernel may be chosen to be any Gaussian
kernel, by giving the variance-covariance matrix varcov.
The arguments sigma and varcov are incompatible.
Also sigma may be a vector of length 2 giving the
standard deviations of two independent Gaussian coordinates,
thus equivalent to varcov = diag(rep(sigma^2, 2)).
The smoothing parameter sigma has a default value
calculated by a simple rule of thumb
that depends only on the size of the window.
The argument adjust makes it easy for the user to change this
default. The value of sigma will be multiplied by
the factor adjust. To double the smoothing parameter, set
adjust=2.
By default the intensity values are
computed at every location $u$ in a fine grid,
and are returned as a pixel image.
Computation is performed using the Fast Fourier Transform.
Accuracy depends on the pixel resolution, controlled by the arguments
... passed to as.mask.
If at="points", the intensity values are computed
to high accuracy at the points of x only. Computation is
performed by directly evaluating and summing the Gaussian kernel
contributions without discretising the data. The result is a numeric
vector giving the density values.
The intensity value at a point $x_i$ is (if diggle=FALSE)
$$\hat\lambda(x_i) = e(x_i) \sum_j k(x_j - x_i) w_j$$
or (if diggle=TRUE)
$$\hat\lambda(x_i) = \sum_j k(x_j - x_i) w_j e(x_j)$$
If leaveoneout=TRUE (the default), then the sum in the equation
is taken over all $j$ not equal to $i$,
so that the intensity value at a
data point is the sum of kernel contributions from
all other data points.
If leaveoneout=FALSE then the sum is taken over all $j$,
so that the intensity value at a data point includes a contribution
from the same point.
To perform spatial interpolation of values that were observed
at the points of a point pattern, use smooth.ppp.
For adaptive nonparametric estimation, see
adaptive.density.
For data sharpening, see sharpen.ppp.
To compute a relative risk surface or probability map for
two (or more) types of points, use relrisk.
www.csiro.au/resources/pf16h.htmlDiggle, P.J. (1985) A kernel method for smoothing point process data. Applied Statistics (Journal of the Royal Statistical Society, Series C) 34 (1985) 138--147.
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
smooth.ppp,
sharpen.ppp,
adaptive.density,
relrisk,
ppp.object,
im.objectdata(cells)
if(interactive()) {
opa <- par(mfrow=c(1,2))
plot(density(cells, 0.05))
plot(density(cells, 0.05, diggle=TRUE))
par(opa)
v <- diag(c(0.05, 0.07)^2)
plot(density(cells, varcov=v))
}
<testonly>Z <- density(cells, 0.05)
Z <- density(cells, 0.05, diggle=TRUE)
Z <- density(cells, varcov=diag(c(0.05^2, 0.07^2)))</testonly>
density(cells, 0.05, at="points")Run the code above in your browser using DataLab