Geyer                package:spatstat                R Documentation

_G_e_y_e_r'_s _S_a_t_u_r_a_t_i_o_n _P_o_i_n_t _P_r_o_c_e_s_s _M_o_d_e_l

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

     Creates an instance of Geyer's ``saturation point process'' model
     which can then be fitted to point pattern data.

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

       Geyer(r,sat)

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

       r: Interaction radius. A positive real number.

     sat: Saturation threshold. A positive real number.

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

     Geyer (1999) introduced the ``saturation process'', a modification
     of the Strauss process (see `Strauss') in which the total
     contribution to the potential  from each point (from its pairwise
     interaction with all  other points) is trimmed to a maximum value
     s.  This model is implemented in the function `Geyer()'.

     The saturation point process with interaction radius r, saturation
     threshold s, and  parameters beta and gamma, is the point process
     in which each point x[i] in the pattern X contributes a factor

                     beta gamma^min(s, t(x[i],X))

     to the probability density of the point pattern, where t(x[i],X)
     denotes the number of ``close neighbours'' of x[i] in the pattern
     X. A close neighbour of x[i] is a point x[j] with j != i such that
     the distance between  x[i] and x[j] is less than or equal to r.

     If the saturation threshold s is set to infinity, this model
     reduces to the Strauss process (see `Strauss') with interaction
     parameter gamma^2. If s = 0, the model reduces to the Poisson
     point process. If s is a finite positive number, then the
     interaction parameter gamma may take any positive value (unlike
     the case of the Strauss process), with values gamma < 1 describing
     an ``ordered'' or ``inhibitive'' pattern, and  values gamma > 1
     describing a ``clustered or ``attractive'' pattern.

     The nonstationary saturation process is similar except that  the
     value beta  is replaced by a function beta(x[i]) of location.

     The function `mpl()', which fits point process models to  point
     pattern data, requires an argument  of class `"interact"'
     describing the interpoint interaction structure of the model to be
     fitted.  The appropriate description of the saturation process
     interaction is yielded by `Geyer(r, sat)' where the arguments `r'
     and `sat' specify the Strauss interaction radius r and the
     saturation threshold s, respectively. See the examples below.

     Note the only arguments are the interaction radius `r' and the
     saturation threshold `sat'. When `r' and `sat' are fixed, the
     model becomes an exponential family. The canonical parameters
     log(beta) and log(gamma) are estimated by `mpl()', not fixed in
     `Geyer()'.

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

     An object of class `"interact"' describing the interpoint
     interaction structure of Geyer's ``saturation point process'' with
     interaction radius r and saturation threshold `sat'.

_A_u_t_h_o_r(_s):

     Adrian Baddeley adrian@maths.uwa.edu.au <URL:
     http://www.maths.uwa.edu.au/~adrian/> and Rolf Turner
     rolf@math.unb.ca <URL: http://www.math.unb.ca/~rolf>

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

     Geyer, C.J. (1999) Likelihood Inference for Spatial Point
     Processes. Chapter 3 in  O.E. Barndorff-Nielsen, W.S. Kendall and
     M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and
     Computation, Chapman and Hall / CRC,  Monographs on Statistics and
     Applied Probability, number 80. Pages 79-140.

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

     `mpl', `pairwise.family', `ppm.object', `Strauss'

_E_x_a_m_p_l_e_s:

        library(spatstat)
        data(cells) 
        mpl(cells, ~1, Geyer(r=0.07, sat=2), rbord=0.07)
        # fit the stationary saturation process to `cells'

