# lambdahat: Kernel Density Estimation of Intensity Function In spatialkernel: Non-Parametric Estimation of Spatial Segregation in a Multivariate Point Process

## Description

Kernel density estimation of the intensity function of a two-dimensional point process.

## Usage

 `1` ```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 within `poly`, default `NULL` to 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 `TRUE` to 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

1. 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.

2. 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.

3. Baddeley, A. J. and M&#248;ller, J. and Waagepetersen R. (2000) Non and semi-parametric estimation of interaction in inhomogeneous point patterns, Statistica Neerlandica, 54, 3, 329–350.

4. Laurie, D.P. (1982). Algorithm 584 CUBTRI: Adaptive Cubature over a Triangle. ACM–Trans. Math. Software, 8, 210–218.

5. Jonathan R. Shewchuk, Triangle, a Two-Dimensional Quality Mesh Generator and Delaunay Triangulator at http://www-2.cs.cmu.edu/~quake/triangle.html.

6. Alan Murta, General Polygon Clipper at http://www.cs.man.ac.uk/~toby/alan/software/#gpc.

7. NAG's Numerical Library. Chapter 11: Quadrature, NAG's Fortran 90 Library. http://www.nag.co.uk/numeric/fn/manual/html/c11_fn03.html

`setkernel`, `kinhat`, `density`