PairPiece              package:spatstat              R Documentation

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

     Creates an instance of a pairwise interaction point process model
     with piecewise constant potential function. The model can then be
     fitted to point pattern data.

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

       PairPiece(r)

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

       r: vector of jump points for the potential function

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

     A pairwise interaction point process in a bounded region is a
     stochastic point process with probability density of the form

 f(x_1,...,x_n) = alpha . product { b(x[i]) } product { h(x_i, x_j) }

     where x[1],...,x[n] represent the  points of the pattern. The
     first product on the right hand side is over all points of the
     pattern; the second product is over all unordered pairs of points
     of the pattern.

     Thus each point x[i] of the pattern contributes a factor  b(x[i])
     to the probability density, and each pair of points x[i], x[j]
     contributes a factor h(x[i], x[j]) to the density.

     The pairwise interaction term h(u, v) is called piecewise constant
     if it depends only on the distance between u and v, say h(u,v) =
     H(||u-v||), and H is a piecewise constant function (a function
     which is constant except for jumps at a finite number of places). 

     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 piecewise constant
     pairwise interaction is yielded by the function 'PairPiece()'. See
     the examples below.

     The entries of 'r' must be strictly increasing, positive numbers.
     They are interpreted as the points of discontinuity of H. It is
     assumed that H(s) =1 for all s > rmax where rmax is the maximum
     value in 'r'. Thus the model has as many regular parameters (see
     'ppm')  as there are entries in 'r'. The i-th regular parameter
     theta[i] is the logarithm of the value of the interaction function
     H on the interval [r[i-1],r[i]). 

     If 'r' is a single number, this model is similar to the  Strauss
     process, see 'Strauss'. The difference is that in 'PairPiece' the
     interaction function is continuous on the right, while in
     'Strauss' it is continuous on the left.

     The analogue of this model for multitype point processes has not
     yet been implemented.

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

     An object of class '"interact"' describing the interpoint
     interaction structure of a point process. The process is a
     pairwise interaction process, whose interaction potential is
     piecewise constant, with jumps at the distances given in the
     vector r.

_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>

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

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

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

        PairPiece(c(0.1,0.2))
        # prints a sensible description of itself
        data(cells) 
        ppm(cells, ~1, PairPiece(r = c(0.05, 0.1, 0.2)), rbord=0.2)
        # fit a stationary piecewise constant pairwise interaction process
        ppm(cells, ~polynom(x,y,3), PairPiece(c(0.05, 0.1)), rbord=0.1)
        # nonstationary process with log-cubic polynomial trend

