# density.ppp

0th

Percentile

##### Kernel Smoothed Intensity of Point Pattern

Compute a kernel smoothed intensity function from a point pattern.

Keywords
methods, smooth, spatial
##### Usage
## S3 method for class 'ppp':
density(x, sigma, \dots,
weights, edge=TRUE, varcov=NULL,
at="pixels", leaveoneout=TRUE)
##### Arguments
x
Point pattern (object of class "ppp").
sigma
Standard deviation of isotropic Gaussian smoothing kernel.
weights
Optional vector of weights to be attached to the points. May include negative values.
...
Arguments passed to as.mask to determine the pixel resolution.
edge
Logical flag: if TRUE, apply edge correction.
varcov
Variance-covariance matrix of anisotropic Gaussian 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".
##### Details

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. 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 by dividing it by the convolution of the Gaussian kernel with the window of observation.

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(sigma^2).

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 $x_i$ are the weights.

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 $$\hat\lambda(x_i) = e(x_i) \sum_j k(x_j - x_i) w_i$$ 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.

##### Value

• 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.

##### Note

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 the workshop notes (Baddeley, 2008) or Diggle (2003).

##### References

Baddeley, A. (2008) Analysing spatial point patterns in R. Workshop notes. CSIRO online technical publication. URL: www.csiro.au/resources/pf16h.html

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.

smooth.ppp, adaptive.density, ppp.object, im.object

• density.ppp
##### Examples
data(cells)
Z <- density(cells, 0.05)
plot(Z)
density(cells, 0.05, at="points")
Documentation reproduced from package spatstat, version 1.16-3, License: GPL (>= 2)

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