Nonparametric Estimate of Spatially-Varying Relative Risk

Given a multitype point pattern, this function estimates the spatially-varying probability of each type of point, using kernel smoothing. The default smoothing bandwidth is selected by cross-validation.

methods, smooth, spatial
relrisk(X, sigma = NULL, ..., varcov = NULL, at = "pixels", casecontrol=TRUE)
A multitype point pattern (object of class "ppp" which has factor valued marks).
Optional. Standard deviation of isotropic Gaussian smoothing kernel.
Arguments passed to bw.relrisk to select the bandwidth, or passed to density.ppp to control the pixel resolution.
Optional. Variance-covariance matrix of anisotopic Gaussian smoothing kernel. Incompatible with sigma.
String specifying whether to compute the probability values at a grid of pixel locations (at="pixels") or only at the points of X (at="points").
Logical. Whether to treat a bivariate point pattern as consisting of cases and controls. See Details.

If X is a bivariate point pattern (a multitype point pattern consisting of two types of points) then by default, the points of the first type (the first level of marks(X)) are treated as controls or non-events, and points of the second type are treated as cases or events. Then this command computes the spatially-varying risk of an event, i.e. the probability $p(u)$ that a point at spatial location $u$ will be a case.

If X is a multitype point pattern with $m > 2$ types, or if X is a bivariate point pattern and casecontrol=FALSE, then this command computes, for each type $j$, a nonparametric estimate of the spatially-varying risk of an event of type $j$. This is the probability $p_j(u)$ that a point at spatial location $u$ will belong to type $j$.

If at = "pixels" the calculation is performed for every spatial location $u$ on a fine pixel grid, and the result is a pixel image representing the function $p(u)$ or a list of pixel images representing the functions $p_j(u)$ for $j = 1,\ldots,m$.

If at = "points" the calculation is performed only at the data points $x_i$. The result is a vector of values $p(x_i)$ giving the estimated probability of a case at each data point, or a matrix of values $p_j(x_i)$ giving the estimated probability of each possible type $j$ at each data point.

Estimation is performed by a simple Nadaraja-Watson type kernel smoother (Diggle, 2003). If sigma and varcov are both missing or null, then the smoothing bandwidth sigma is selected by cross-validation using bw.relrisk.


  • If X consists of only two types of points, the result is a pixel image (if at="pixels") or a vector of probabilities (if at="points").

    If X consists of more than two types of points, the result is:

    • (ifat="pixels") a list of pixel images, with one image for each possible type of point. The result also belongs to the class"listof"so that it can be printed and plotted.
    • (ifat="points") a matrix of probabilities, with rows corresponding to data points$x_i$, and columns corresponding to types$j$.


Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.

See Also

bw.relrisk, density.ppp, smooth.ppp,

  • relrisk
   p <- relrisk(urkiola, 20)
   if(interactive()) {
      plot(p, main="proportion of oak")
      plot( > 0.3), main="More than 30 percent oak")
      z <- relrisk(lansing)
      plot(z, main="Lansing Woods")
      plot(, main="Most common species")
Documentation reproduced from package spatstat, version 1.31-2, License: GPL (>= 2)

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