lambdahat           package:spatialkernel           R Documentation

_K_e_r_n_e_l _D_e_n_s_i_t_y _E_s_t_i_m_a_t_i_o_n _o_f _I_n_t_e_n_s_i_t_y _F_u_n_c_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

       lambdahat(pts, h, gpts = NULL, poly = NULL, edge = TRUE)

_A_r_g_u_m_e_n_t_s:

     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.

_D_e_t_a_i_l_s:

     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'.

_V_a_l_u_e:

     A list with components 

  lambda: numeric vector of the estimated intensity function.

     ...: copy of the arguments 'pts, gpts, h, poly, edge'.

_N_o_t_e:

     In principle, the _double adaptive_ double integration algorithm 
     of Zheng, P. _et al_ (2004) can be applied to other kernel
     functions. 

     Other source codes used in the implementation of the double
     integration  algorithm include \itemize {

     *  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_). }

_R_e_f_e_r_e_n_c_e_s:

        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{\o}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 <URL:
           http://www-2.cs.cmu.edu/~quake/triangle.html>.

        6.  Alan Murta, _General Polygon Clipper_ at <URL:
           http://www.cs.man.ac.uk/~toby/alan/software/#gpc>.

        7.  NAG's Numerical Library. _Chapter 11: Quadrature_,  NAG's
           Fortran 90 Library. (<URL:
           http://www.nag.co.uk/numeric/fn/manual/html/c11_fn03.html>)

_S_e_e _A_l_s_o:

     'setkernel', 'kinhat', 'density'

