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 'ppm()', 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 'ppm()', 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:

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

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

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

