spatstat.explore (version 3.4-2)

bw.stoyan: Stoyan's Rule of Thumb for Bandwidth for Estimating Pair Correlation

Description

Computes a rough estimate of the appropriate bandwidth for kernel smoothing estimators of the pair correlation function.

Usage

bw.stoyan(X, co=0.15, extrapolate=FALSE, ...)

Value

A finite positive numerical value giving the selected bandwidth (the standard deviation of the smoothing kernel).

Arguments

X

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

co

Coefficient appearing in the rule of thumb. See Details.

extrapolate

Logical value specifying whether to use the extrapolated version of the rule. See Details.

...

Ignored.

Author

Adrian Baddeley , Rolf Turner , Tilman Davies , Martin Hazelton and Ya-Mei Chang .

Details

Estimation of the pair correlation function (and similar quantities) by smoothing methods requires a choice of the smoothing bandwidth. Stoyan and Stoyan (1995, equation (15.16), page 285) proposed a rule of thumb for choosing the smoothing bandwidth.

For the Epanechnikov kernel, the rule of thumb is to set the kernel's half-width \(h\) to \(0.15/\sqrt{\lambda}\) where \(\lambda\) is the estimated intensity of the point pattern, typically computed as the number of points of X divided by the area of the window containing X.

For a general kernel, the corresponding rule is to set the standard deviation of the kernel to \(\sigma = 0.15/\sqrt{5\lambda}\).

The coefficient \(0.15\) can be tweaked using the argument co.

To ensure the bandwidth is finite, an empty point pattern is treated as if it contained 1 point.

The original version of Stoyan's rule, stated above, was developed by experience with patterns of 30 to 100 points. For patterns with larger numbers of points, the bandwidth should be smaller: the theoretically optimal bandwidth decreases in proportion to \(n^{-1/5}\) where \(n\) is the number of points in the pattern. In the ‘extrapolated’ version of Stoyan's rule proposed by Baddeley, Davies and Hazelton (2025), the value \(\sigma\) calculated above is multiplied by \((100/n)^{1/5}\). The extrapolated rule is applied if extrapolate=TRUE.

References

Baddeley, A., Davies, T.M. and Hazelton, M.L. (2025) An improved estimator of the pair correlation function of a spatial point process. Biometrika, to appear.

Stoyan, D. and Stoyan, H. (1995) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

See Also

pcf, bw.relrisk

Examples

  bw.stoyan(shapley)
  bw.stoyan(shapley, extrapolate=TRUE)