StraussHard             package:spatstat             R Documentation

_T_h_e _S_t_r_a_u_s_s / _H_a_r_d _C_o_r_e _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 the ``Strauss/ hard core'' point process
     model which can then be fitted to point pattern data.

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

       StraussHard(r, hc)

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

       r: The interaction radius of the Strauss interaction

      hc: The hard core distance

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

     A Strauss/hard core process with interaction radius r, hard core
     distance h < r, and  parameters beta and gamma, is a pairwise
     interaction point process in which

        *  distinct points are not allowed to come closer than a
           distance h apart

        *  each pair of points closer than r units apart contributes a
           factor gamma to the probability density.

     This is a hybrid of the Strauss process and the hard core process.

     The probability density is zero if any pair of points is closer
     than h units apart, and otherwise equals

            f(x_1,...,x_n) = alpha . beta^n(x) gamma^s(x)

     where x[1],...,x[n] represent the  points of the pattern, n(x) is
     the number of points in the pattern, s(x) is the number of
     distinct unordered pairs of points that are closer than r units
     apart, and alpha is the normalising constant.

     The interaction parameter gamma may take any positive value
     (unlike the case for the Strauss process). If gamma = 1, the
     process reduces to a classical hard core process. If gamma < 1,
     the model describes an ``ordered'' or ``inhibitive'' pattern. If
     gamma > 1, the model is ``ordered'' or ``inhibitive'' up to the
     distance h, but has an ``attraction'' between points lying at
     distances in the range between h and r.

     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 Strauss/hard core
     process pairwise interaction is yielded by the function
     'StraussHard()'. See the examples below.

     The canonical parameter log(gamma) is estimated by 'ppm()', not
     fixed in 'StraussHard()'.

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

     An object of class '"interact"' describing the interpoint
     interaction structure of the ``Strauss/hard core'' process with
     Strauss interaction radius r and hard core distance 'hc'.

_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:

     Baddeley, A. and Turner, R. (2000) Practical maximum
     pseudolikelihood for spatial point patterns. _Australian and New
     Zealand Journal of Statistics_ *42*, 283-322.

     Ripley, B.D. (1981)  _Spatial statistics_. John Wiley and Sons.

     Strauss, D.J. (1975) A model for clustering. _Biometrika_ *63*,
     467-475.

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

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

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

        StraussHard(r=1,hc=0.02)
        # prints a sensible description of itself

        data(cells) 
        ppm(cells, ~1, StraussHard(r=0.1, hc=0.05), rbord=0.1)
        # fit the stationary Strauss/hard core  process to `cells'

        ppm(cells, ~ polynom(x,y,3), StraussHard(r=0.1, hc=0.05), rbord=0.1)
        # fit a nonstationary Strauss/hard core process
        # with log-cubic polynomial trend
        

