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

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

Usage

KdEnvelope(X, r = NULL, NumberOfSimulations = 100, Alpha = 0.05, ReferenceType, 
           NeighborType = ReferenceType, Weighted = FALSE, Original = TRUE, 
           Approximate = ifelse(X$n < 10000, 0, 1), Adjust = 1, MaxRange = "ThirdW", 
           StartFromMinR = FALSE,
           SimulationType = "RandomLocation", Global = FALSE, 
           verbose = interactive())

Arguments

X: A point pattern (wmppp.object) or a Dtable object.
r: A vector of distances. If NULL, a default value is set: 512 equally spaced values are used, and the first 256 are returned, corresponding to half the maximum distance between points (following Duranton and Overman, 2005).
NumberOfSimulations: The number of simulations to run, 100 by default.
Alpha: The risk level, 5% by default.
ReferenceType: One of the point types.
NeighborType: One of the point types. By default, the same as reference type.
Weighted: Logical; if TRUE, estimates the Kemp function.
Original: Logical; if TRUE (by default), the original bandwidth selection by Duranton and Overman (2005) following Silverman (2006: eq 3.31) is used. If FALSE, it is calculated following Sheather and Jones (1991), i.e. the state of the art. See bw.SJ for more details.
Approximate: if not 0 (1 is a good choice), exact distances between pairs of points are rounded to 1024 times Approximate single values equally spaced between 0 and the largest distance. This technique (Scholl and Brenner, 2015) allows saving a lot of memory when addressing large point sets (the default value is 1 over 10000 points). Increasing Approximate allows better precision at the cost of proportional memory use. Ignored if X is a Dtable object.
Adjust: Force the automatically selected bandwidth (following Silverman, 1986) to be multiplied by Adjust. Setting it to values lower than one (1/2 for example) will sharpen the estimation. If not 1, Original is ignored.
MaxRange: The maximum value of r to consider, ignored if r is not NULL. Default is "ThirdW", one third of the diameter of the window. Other choices are "HalfW", and "QuarterW" and "D02005". "HalfW", and "QuarterW" are for half or the quarter of the diameter of the window. "D02005" is for the median distance observed between points, following Duranton and Overman (2005). "ThirdW" should be close to "DO2005" but has the advantage to be independent of the point types chosen as ReferenceType and NeighborType, to simplify comparisons between different types. "D02005" is approximated by "ThirdW" if Approximate is not 0. if X is a Dtable object, the diameter of the window is taken as the max distance between points.
StartFromMinR: Logical; if TRUE, points are assumed to be further from each other than the minimum observed distance, So Kd will not be estimated below it: it is assumed to be 0. If FALSE, by default, distances are smoothed down to \(r=0\). Ignored if Approximate is not 0: then, estimation always starts from \(r=0\).
SimulationType: A string describing the null hypothesis to simulate. The null hypothesis may be "RandomLocation": points are redistributed on the actual locations (default); "RandomLabeling": randomizes point types, keeping locations and weights unchanged; "PopulationIndependence": keeps reference points unchanged, randomizes other point locations.
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.

Scholl, T. and Brenner, T. (2015) Optimizing distance-based methods for large data sets, Journal of Geographical Systems 17(4): 333-351.

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

Examples

data(paracou16)
autoplot(paracou16[paracou16$marks$PointType=="Q. Rosea"])


# Calculate confidence envelope
plot(KdEnvelope(paracou16, , ReferenceType="Q. Rosea", Global=TRUE))

# Center of the confidence interval
Kdhat(paracou16, ReferenceType="") -> kd
lines(kd$Kd ~ kd$r, lty=3, col="green")