Compute a kernel smoothed intensity function from a point pattern.

```
# S3 method for ppp
density(x, sigma=NULL, ...,
weights=NULL, edge=TRUE, varcov=NULL,
at="pixels", leaveoneout=TRUE,
adjust=1, diggle=FALSE, se=FALSE,
kernel="gaussian",
scalekernel=is.character(kernel),
positive=FALSE, verbose=TRUE)
```

By default, the result is
a pixel image (object of class `"im"`

).
Pixel values are estimated intensity values,
expressed in “points per unit area”.

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.

If `weights`

is a matrix with more than one column, then the
result is a list of images (if `at="pixels"`

) or a matrix of
numerical values (if `at="points"`

).

If `se=TRUE`

, the result is a list with two elements named

`estimate`

and `SE`

, each of the format described above.

- x
Point pattern (object of class

`"ppp"`

).- sigma
The smoothing bandwidth (the amount of smoothing). The standard deviation of the isotropic smoothing kernel. Either a numerical value, or a function that computes an appropriate value of

`sigma`

.- weights
Optional weights to be attached to the points. A numeric vector, numeric matrix, an

`expression`

, or a pixel image.- ...
Additional arguments passed to

`pixellate.ppp`

and`as.mask`

to determine the pixel resolution, or passed to`sigma`

if it is a function.- edge
Logical value indicating whether to apply edge correction.

- varcov
Variance-covariance matrix of anisotropic smoothing kernel. Incompatible with

`sigma`

.- at
String specifying whether to compute the intensity values at a grid of pixel locations (

`at="pixels"`

) or only at the points of`x`

(`at="points"`

).- leaveoneout
Logical value indicating whether to compute a leave-one-out estimator. Applicable only when

`at="points"`

.- adjust
Optional. Adjustment factor for the smoothing parameter.

- diggle
Logical. If

`TRUE`

, use the Jones-Diggle improved edge correction, which is more accurate but slower to compute than the default correction.- kernel
The smoothing kernel. A character string specifying the smoothing kernel (current options are

`"gaussian"`

,`"epanechnikov"`

,`"quartic"`

or`"disc"`

), or a pixel image (object of class`"im"`

) containing values of the kernel, or a`function(x,y)`

which yields values of the kernel.- scalekernel
Logical value. If

`scalekernel=TRUE`

, then the kernel will be rescaled to the bandwidth determined by`sigma`

and`varcov`

: this is the default behaviour when`kernel`

is a character string. If`scalekernel=FALSE`

, then`sigma`

and`varcov`

will be ignored: this is the default behaviour when`kernel`

is a function or a pixel image.- se
Logical value indicating whether to compute standard errors as well.

- positive
Logical value indicating whether to force all density values to be positive numbers. Default is

`FALSE`

.- verbose
Logical value indicating whether to issue warnings about numerical problems and conditions.

The amount of smoothing is determined by the arguments
`sigma`

, `varcov`

and `adjust`

.

if

`sigma`

is a single numerical value, this is taken as the standard deviation of the isotropic Gaussian kernel.alternatively

`sigma`

may be a function that computes an appropriate bandwidth from the data point pattern by calling`sigma(x)`

. To perform automatic bandwidth selection using cross-validation, it is recommended to use the functions`bw.diggle`

,`bw.CvL`

,`bw.scott`

or`bw.ppl`

.The smoothing kernel may be made anisotropic by giving the variance-covariance matrix

`varcov`

. The arguments`sigma`

and`varcov`

are incompatible.Alternatively

`sigma`

may be a vector of length 2 giving the standard deviations of the \(x\) and \(y\) coordinates, thus equivalent to`varcov = diag(rep(sigma^2, 2))`

.if neither

`sigma`

nor`varcov`

is specified, an isotropic Gaussian kernel will be used, with a default value of`sigma`

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 the bandwidth specified by any of the rules above. The value of`sigma`

will be multiplied by the factor`adjust`

. The matrix`varcov`

will be multiplied by`adjust^2`

. To double the smoothing bandwidth, set`adjust=2`

.An infinite bandwidth,

`sigma=Inf`

or`adjust=Inf`

, is permitted, and yields an intensity estimate which is constant over the spatial domain.

If `edge=TRUE`

, the intensity estimate is corrected for
edge effect bias in one of two ways:

If

`diggle=FALSE`

(the default) the intensity estimate is correted by dividing it by the convolution of the Gaussian kernel with the window of observation. This is the approach originally described in Diggle (1985). 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.If

`diggle=TRUE`

then the code uses the improved edge correction described by Jones (1993) and Diggle (2010, equation 18.9). This has been shown to have better performance (Jones, 1993) but is slightly slower to compute. 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.

In both cases, the edge correction term \(e(u)\) is the reciprocal of the kernel mass inside the window: $$ \frac{1}{e(u)} = \int_W k(v-u) \, {\rm d}v $$ where \(W\) is the observation window.

By default, smoothing is performed using a Gaussian kernel.

The choice of smoothing kernel is determined by the argument `kernel`

.
This should be a character string giving the name of a recognised
two-dimensional kernel
(current options are `"gaussian"`

, `"epanechnikov"`

,
`"quartic"`

or `"disc"`

),
or a pixel image (object of class `"im"`

)
containing values of the kernel, or a `function(x,y)`

which
yields values of the kernel. The default is a Gaussian kernel.

If `scalekernel=TRUE`

then the kernel values will be rescaled
according to the arguments `sigma`

, `varcov`

and
`adjust`

as explained above, effectively treating
`kernel`

as the template kernel with standard deviation equal to 1.
This is the default behaviour when `kernel`

is a character string.
If `scalekernel=FALSE`

, the kernel values will not be altered,
and the arguments `sigma`

, `varcov`

and `adjust`

are ignored. This is the default behaviour when `kernel`

is a
pixel image or a function.

If `at="pixels"`

(the default), intensity values are
computed at every location \(u\) in a fine grid,
and are returned as a pixel image. The point pattern is first discretised
using `pixellate.ppp`

, then the intensity is
computed using the Fast Fourier Transform.
Accuracy depends on the pixel resolution and the discretisation rule.
The pixel resolution is controlled by the arguments
`...`

passed to `as.mask`

(specify the number of
pixels by `dimyx`

or the pixel size by `eps`

).
The discretisation rule is controlled by the arguments
`...`

passed to `pixellate.ppp`

(the default rule is that each point is allocated to the nearest
pixel centre; this can be modified using the arguments
`fractional`

and `preserve`

).

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 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.

If `weights`

is a matrix with more than one column, then the
calculation is effectively repeated for each column of weights. The
result is a list of images (if `at="pixels"`

) or a matrix of
numerical values (if `at="points"`

).

The argument `weights`

can also be an `expression`

.
It will be evaluated in the data frame `as.data.frame(x)`

to obtain a vector or matrix of weights. The expression may involve
the symbols `x`

and `y`

representing the Cartesian
coordinates, the symbol `marks`

representing the mark values
if there is only one column of marks, and the names of the columns of
marks if there are several columns.

The argument `weights`

can also be a pixel image
(object of class `"im"`

). numerical weights for the data points
will be extracted from this image (by looking up the pixel values
at the locations of the data points in `x`

).

This function is often misunderstood.

The result of `density.ppp`

is not a spatial smoothing
of the marks or weights attached to the point pattern.
To perform spatial interpolation of values that were observed
at the points of a point pattern, use `Smooth.ppp`

.

The result of `density.ppp`

is not a probability density.
It is an estimate of the *intensity function* of the
point process that generated the point pattern data.
Intensity is the expected number of random points
per unit area.
The units of intensity are “points per unit area”.
Intensity is usually a function of spatial location,
and it is this function which is estimated by `density.ppp`

.
The integral of the intensity function over a spatial region gives the
expected number of points falling in this region.

Inspecting an estimate of the intensity function is usually the first step in exploring a spatial point pattern dataset. For more explanation, see Baddeley, Rubak and Turner (2015) or Diggle (2003, 2010).

If you have two (or more) types of points, and you want a
probability map or relative risk surface (the spatially-varying
probability of a given type), use `relrisk`

.

Negative and zero values of the density estimate are possible
when `at="pixels"`

because of numerical errors in finite-precision
arithmetic.

By default, `density.ppp`

does not try to repair such errors.
This would take more computation time and is not always needed.
(Also it would not be appropriate if `weights`

include negative values.)

To ensure that the resulting density values are always positive,
set `positive=TRUE`

.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk

This is a method for the generic function `density`

.

It computes a fixed-bandwidth kernel estimate
(Diggle, 1985) of the intensity function of the point process
that generated the point pattern `x`

.

The amount of smoothing is controlled by `sigma`

if it is specified.

By default, smoothing is performed using a Gaussian kernel.
The resulting density estimate is the convolution of the
isotropic Gaussian kernel, of standard deviation `sigma`

,
with point masses at each of the data points in `x`

.

Anisotropic kernels, and non-Gaussian kernels, are also supported.
Each point has unit weight, unless the argument `weights`

is
given.

If `edge=TRUE`

(the default), the intensity estimate is corrected
for edge effect bias.

If `at="pixels"`

(the default), the result is a pixel image
giving the estimated intensity at each pixel in a grid.
If `at="points"`

, the result is a numeric vector giving the
estimated intensity at each of the original data points in `x`

.

Baddeley, A., Rubak, E. and Turner, R. (2015) *Spatial Point Patterns: Methodology and Applications with R*. Chapman and Hall/CRC Press.

Diggle, 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.

Diggle, P.J. (2010)
Nonparametric methods.
Chapter 18, pp. 299--316 in
A.E. Gelfand, P.J. Diggle, M. Fuentes and P. Guttorp (eds.)
*Handbook of Spatial Statistics*,
CRC Press, Boca Raton, FL.

Jones, M.C. (1993)
Simple boundary corrections for kernel density estimation.
*Statistics and Computing* **3**, 135--146.

To select the bandwidth `sigma`

automatically by
cross-validation, use
`bw.diggle`

,
`bw.CvL`

,
`bw.scott`

or
`bw.ppl`

.

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`

.

For information about the data structures, see
`ppp.object`

,
`im.object`

.

```
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))
}
# automatic bandwidth selection
plot(density(cells, sigma=bw.diggle(cells)))
# equivalent:
plot(density(cells, bw.diggle))
# evaluate intensity at points
density(cells, 0.05, at="points")
# non-Gaussian kernel
plot(density(cells, sigma=0.4, kernel="epanechnikov"))
if(interactive()) {
# see effect of changing pixel resolution
opa <- par(mfrow=c(1,2))
plot(density(cells, sigma=0.4))
plot(density(cells, sigma=0.4, eps=0.05))
par(opa)
}
# relative risk calculation by hand (see relrisk.ppp)
lung <- split(chorley)$lung
larynx <- split(chorley)$larynx
D <- density(lung, sigma=2)
plot(density(larynx, sigma=2, weights=1/D))
```

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