Estimation of the confidence envelope of the K function under its null hypothesis

Simulates point patterns according to the null hypothesis and returns the envelope of K according to the confidence level.

Usage

KEnvelope(X, r = NULL, NumberOfSimulations = 100, Alpha = 0.05, 
          ReferenceType = "", NeighborType = ReferenceType, 
          SimulationType = "RandomPosition", Precision = 0, Global = FALSE,
          verbose = interactive())

Arguments

X: A point pattern (wmppp.object).
r: A vector of distances. If NULL, a sensible default value is chosen (512 intervals, from 0 to half the diameter of the window) following spatstat.
NumberOfSimulations: The number of simulations to run, 100 by default.
Alpha: The risk level, 5% by default.
ReferenceType: One of the point types. Default is all point types.
NeighborType: One of the point types. By default, the same as reference type.
SimulationType: A string describing the null hypothesis to simulate. The null hypothesis may be "RandomPosition": points are drawn in a Poisson process (default); "RandomLabeling": randomizes point types, keeping locations unchanged; "PopulationIndependence": keeps reference points unchanged, shifts other point locations.
Precision: Accuracy of point coordinates, measured as a part of distance unit. See rRandomPositionK. Default is 0 for no approximation.
Global: Logical; if TRUE, a global envelope sensu Duranton and Overman (2005) is calculated.
verbose: Logical; if TRUE, print progress reports during the simulations.

Details

This envelope is local by default, that is to say it is computed separately at each distance. See Loosmore and Ford (2006) for a discussion.

The global envelope is calculated by iteration: the simulations reaching one of the upper or lower values at any distance are eliminated at each step. The process is repeated until Alpha / Number of simulations simulations are dropped. The remaining upper and lower bounds at all distances constitute the global envelope. Interpolation is used if the exact ratio cannot be reached.

Value

An envelope object (envelope). There are methods for print and plot for this class.

The fv contains the observed value of the function, its average simulated value and the confidence envelope.

References

Duranton, G. and Overman, H. G. (2005). Testing for Localisation Using Micro-Geographic Data. Review of Economic Studies 72(4): 1077-1106.

Kenkel, N. C. (1988). Pattern of Self-Thinning in Jack Pine: Testing the Random Mortality Hypothesis. Ecology 69(4): 1017-1024.

Loosmore, N. B. and Ford, E. D. (2006). Statistical inference using the G or K point pattern spatial statistics. Ecology 87(8): 1925-1931.

Marcon, E. and F. Puech (2017). A typology of distance-based measures of spatial concentration. Regional Science and Urban Economics. 62:56-67.

Silverman, B. W. (1986). Density estimation for statistics and data analysis. Chapman and Hall, London.

Examples

data(paracou16)
# Keep only 20% of points to run this example
X <- as.wmppp(rthin(paracou16, 0.2))
autoplot(X, 
  labelSize = expression("Basal area (" ~cm^2~ ")"), 
  labelColor = "Species")


# Calculate confidence envelope (should be 1000 simulations, reduced to 20 to save time)
r <- 0:30
NumberOfSimulations <- 20
# Plot the envelope
autoplot(KEnvelope(X, r, NumberOfSimulations), ./(pi*r^2) ~ r)