sharpen: Data Sharpening of Point Pattern

Description

Performs Choi-Hall data sharpening of a spatial point pattern.

Usage

sharpen(X, …)
# S3 method for ppp
sharpen(X, sigma=NULL, …,
                      varcov=NULL, edgecorrect=FALSE)

Arguments

A marked point pattern (object of class "ppp").

sigma

Standard deviation of isotropic Gaussian smoothing kernel.

varcov

Variance-covariance matrix of anisotropic Gaussian kernel. Incompatible with sigma.

edgecorrect

Logical value indicating whether to apply edge effect bias correction.

…

Arguments passed to density.ppp to control the pixel resolution of the result.

Value

A point pattern (object of class "ppp") in the same window as the original pattern X, and with the same marks as X.

Details

Choi and Hall (2001) proposed a procedure for data sharpening of spatial point patterns. This procedure is appropriate for earthquake epicentres and other point patterns which are believed to exhibit strong concentrations of points along a curve. Data sharpening causes such points to concentrate more tightly along the curve.

If the original data points are $X_{1}, \dots, X_{n}$ then the sharpened points are ${\hat{X}}_{i} = \frac{\sum_{j} X_{j} k (X_{j} - X_{i})}{\sum_{j} k (X_{j} - X_{i})}$ where $k$ is a smoothing kernel in two dimensions. Thus, the new point ${\hat{X}}_{i}$ is a vector average of the nearby points $X [j]$ .

The function sharpen is generic. It currently has only one method, for two-dimensional point patterns (objects of class "ppp").

If sigma is given, the smoothing kernel is the isotropic two-dimensional Gaussian density with standard deviation sigma in each axis. If varcov is given, the smoothing kernel is the Gaussian density with variance-covariance matrix varcov.

The data sharpening procedure tends to cause the point pattern to contract away from the boundary of the window. That is, points X_iX[i] that lie `quite close to the edge of the window of the point pattern tend to be displaced inward. If edgecorrect=TRUE then the algorithm is modified to correct this vector bias.

References

Choi, E. and Hall, P. (2001) Nonparametric analysis of earthquake point-process data. In M. de Gunst, C. Klaassen and A. van der Vaart (eds.) State of the art in probability and statistics: Festschrift for Willem R. van Zwet, Institute of Mathematical Statistics, Beachwood, Ohio. Pages 324--344.

Examples

Run this code

# NOT RUN {
   data(shapley)
   X <- unmark(shapley)
   
# }
# NOT RUN {
   Y <- sharpen(X, sigma=0.5)
   Z <- sharpen(X, sigma=0.5, edgecorrect=TRUE)
   opa <- par(mar=rep(0.2, 4))
   plot(solist(X, Y, Z), main= " ",
        main.panel=c("data", "sharpen", "sharpen, correct"),
        pch=".", equal.scales=TRUE, mar.panel=0.2)
   par(opa)
# }

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