dbmss (version 2.1.2)

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

Description

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), 
           SimulationType = "RandomLocation", Global = FALSE)

Arguments

X

A point pattern (wmppp.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 o

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, 2013) allows saving a lot of mem

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 weigh

Global

Logical; if TRUE, a global envelope sensu Duranton and Overman (2005) is calculated.

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 enveloppe.

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 enveloppe 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.

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 (2012). A typology of distance-based measures of spatial concentration. HAL SHS. 00679993. Scholl, T. and Brenner, T. (2013) Optimizing Distance-Based Methods for Big Data Analysis, Working Papers on Innovation and Space No 2013-09, Philipps University Marburg Silverman, B. W. (1986). Density estimation for statistics and data analysis. Chapman and Hall, London.

See Also

Kdhat

Examples

data(paracou16)
plot(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")