Kernel density estimation of the intensity function of a two-dimensional point process.
lambdahat(pts, h, gpts = NULL, poly = NULL, edge = TRUE)matrix containing the x,y-coordinates of the
data point locations.
numeric value of the bandwidth used in the kernel smoothing.
matrix containing the x,y-coordinates of point
locations at which to calculate the intensity function, usually
a fine grid points within poly, default NULL to
estimate intensity function at data locations.
matrix containing the x,y-coordinates of the
vertices of the polygon boundary in an anticlockwise order.
logical, with default TRUE to do edge-correction.
A list with components
numeric vector of the estimated intensity function.
copy of the arguments pts, gpts, h, poly, edge.
Kernel smoothing methods are widely used to estimate the intensity of a spatial point process. One problem which arises is the need to handle edge effects. Several methods of edge-correction have been proposed. The adjustment factor proposed in Berman and Diggle (1989) is a double integration \(\int_AK[(x-x_0)/h]/h^2\), where \(A\) is a polygonal area, \(K\) is the smoothing kernel and \(h\) is the bandwidth used for the smoothing. Zheng, P. et\ al (2004) proposed an algorithm for fast calculate of Berman and Diggle's adjustment factor.
When gpts is NULL, lambdahat uses a leave-one-out
estimator for the intensity at each of the data points, as been suggested
in Baddeley et al (2000). This leave-one-out estimate at each of the
data points then can be used in the inhomogeneous K function estimation
kinhat when the true intensity function is unknown.
The default kernel is the Gaussian. The kernel function is selected
by calling setkernel.
M. Berman and P. Diggle (1989) Estimating weighted integrals of the second-order intensity of a spatial point process, J. R. Stat. Soc. B, 51, 81--92.
P. Zheng, P.A. Durr and P.J. Diggle (2004) Edge--correction for Spatial Kernel Smoothing --- When Is It Necessary? Proceedings of the GisVet Conference 2004, University of Guelph, Ontario, Canada, June 2004.
Baddeley, A. J. and Møller, J. and Waagepetersen R. (2000) Non and semi-parametric estimation of interaction in inhomogeneous point patterns, Statistica Neerlandica, 54, 3, 329--350.
Laurie, D.P. (1982). Algorithm 584 CUBTRI: Adaptive Cubature over a Triangle. ACM--Trans. Math. Software, 8, 210--218.
Jonathan R. Shewchuk, Triangle, a Two-Dimensional Quality Mesh Generator and Delaunay Triangulator at http://www-2.cs.cmu.edu/~quake/triangle.html.
Alan Murta, General Polygon Clipper at http://www.cs.man.ac.uk/~toby/alan/software/#gpc.
NAG's Numerical Library. Chapter 11: Quadrature, NAG's Fortran 90 Library. http://www.nag.co.uk/numeric/fn/manual/html/c11_fn03.html