lambdahat
Kernel Density Estimation of Intensity Function
Kernel density estimation of the intensity function of a two-dimensional point process.
- Keywords
- multivariate, regression, smooth, spatial, nonparametric
Usage
lambdahat(pts, h, gpts = NULL, poly = NULL, edge = TRUE)Arguments
- pts
matrix containing the
x,y-coordinates of the data point locations.- h
numeric value of the bandwidth used in the kernel smoothing.
- gpts
matrix containing the
x,y-coordinates of point locations at which to calculate the intensity function, usually a fine grid points withinpoly, defaultNULLto estimate intensity function at data locations.- poly
matrix containing the
x,y-coordinates of the vertices of the polygon boundary in an anticlockwise order.- edge
logical, with default
TRUEto do edge-correction.
Details
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.
Value
A list with components
- lambda
numeric vector of the estimated intensity function.
- ...
copy of the arguments
pts, gpts, h, poly, edge.
Note
In principle, the double adaptive double integration algorithm of Zheng, P. et\ al (2004) can be applied to other kernel functions. Furthermore, the area at the present is enclosed by a simple polygon which could be generalized into a complex area with polygonal holes inside. For instance, a large lake lays within the land area of study.
Other source codes used in the implementation of the double integration algorithm include
Laurie, D.P. (1982) adaptive cubature code in Fortran;
Shewchuk, J.R. triangulation code in C;
Alan Murta's polygon intersection code in C (Project: Generic Polygon Clipper).
References
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