dbmss (version 2.2.3)

Kdhat: Estimation of the Kd function

Description

Estimates the Kd function

Usage

Kdhat(X, r = NULL, ReferenceType, NeighborType = ReferenceType, Weighted = FALSE,
       Original = TRUE, Approximate = ifelse(X$n < 10000, 0, 1), Adjust = 1,
       MaxRange = "ThirdW", CheckArguments = TRUE)

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, from the smallest distance between points to half the diameter of the window.

ReferenceType

One of the point types. If "", all points are considered (this is not the default value; NeighborType is ignored then) to estimate the average value of simulated Kd values under the null hypothesis of RandomLocation (Marcon

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 (1986: 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

Adjust

Force the automatically selected bandwidth (following Original) to be multiplied by Adjust. Setting it to values lower than one (1/2 for example) will sharpen the estimation.

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

CheckArguments

Logical; if TRUE, the function arguments are verified. Should be set to FALSE to save time in simulations for example, when the arguments have been checked elsewhere.

Value

• An object of class fv, see fv.object, which can be plotted directly using plot.fv.

Details

Kd is a density, absolute measure of a point pattern structure. Kd is computed efficiently by building a matrix of distances between point pairs and calculating the density of their distribution (the default values of r are those of the density function). The kernel estimator is Gaussian. The weighted Kd function has been named Kemp (emp is for employees) by Duranton and Overman (2005). The maximum value of r is obtained from the geometry of the window rather than caculating the median distance between points as suggested by Duranton and Overman (2005) to save (a lot of) calculation time.

References

Duranton, G. and Overman, H. G. (2005). Testing for Localisation Using Micro-Geographic Data. Review of Economic Studies 72(4): 1077-1106. 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 Sheather, S. J. and Jones, M. C. (1991) A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society series B, 53, 683-690. Silverman, B. W. (1986). Density estimation for statistics and data analysis. Chapman and Hall, London.

See Also

KdEnvelope, Mhat

Examples

data(paracou16)
plot(paracou16)

# Calculate Kd
(Paracou <- Kdhat(paracou16, , "Q. Rosea", "V. Americana"))
# Plot
plot(Paracou)