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 discontinuity points for the potential

_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 `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 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
     `mpl')  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 reduces to the Strauss process,
     see `Strauss'.

     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:

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

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

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

