# 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 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

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

##### See Also

*Documentation reproduced from package spatialkernel, version 0.4-23, License: CC BY-NC-SA 4.0*