Kernel Smoothed Intensity of Point Pattern

Description

Compute a kernel smoothed intensity function from a point pattern.

Usage

  ## S3 method for class 'ppp'
density(x, sigma, ...,
        weights=NULL, edge=TRUE, varcov=NULL,
        at="pixels", leaveoneout=TRUE,
        adjust=1, diggle=FALSE)

Arguments

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. Anisotropic Gaussian kernels are also supported. Each point has unit weight, unless the argument weights is given.

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. Thus the intensity value at a point u is

  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 method of Diggle (1985) is followed exactly. The intensity value at a point u is

  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.

In both cases, the edge correction term e(u) is the reciprocal of the kernel mass inside the window:

1/e(u) = integral[v in W] k(v-u) dv

where W is the observation window.

The smoothing kernel 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 for the isotropic Gaussian kernel 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 or bw.ppl.

• 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.

• Alternatively 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)).

• 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.

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)

lambda(x[i]) = e(x[i]) sum[j] k(x[j] - x[i]) w[j]

or (if diggle=TRUE)

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.

To select the bandwidth sigma automatically by cross-validation, use bw.diggle 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.

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.

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").

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).

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.

Author(s)

Adrian Baddeley Adrian.Baddeley@uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner r.turner@auckland.ac.nz

References

Baddeley, A. (2010) Analysing spatial point patterns in R. Workshop notes. CSIRO online technical publication. URL: www.uwa.edu.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.

See Also

bw.diggle, bw.ppl, Smooth.ppp, sharpen.ppp, adaptive.density, relrisk, ppp.object, im.object

Examples

  data(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))
  }
  
  # 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")