Jmulti               package:spatstat               R Documentation

_M_a_r_k_e_d _J _F_u_n_c_t_i_o_n

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

     For a marked point pattern,  estimate the multitype J function
     summarising dependence between the points in subset 'I' and those
     in subset J.

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

     Jmulti(X, I, J)
     Jmulti(X, I, J, eps, r)
     Jmulti(X, I, J, eps, breaks)

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

       X: The observed point pattern,  from which an estimate of the
          multitype distance distribution function JIJ(r) will be
          computed. It must be a marked point pattern. See under
          Details. 

       I: Subset of points of 'X' from which distances are measured.  

       J: Subset of points in 'X' to which distances are measured. 

     eps: A positive number. The pixel resolution of the discrete
          approximation to Euclidean distance (see 'Jest'). There is a
          sensible default. 

       r: numeric vector. The values of the argument r at which the
          distribution function JIJ(r) should be evaluated. There is a
          sensible default. First-time users are strongly advised not
          to specify this argument. See below for important conditions
          on r. 

  breaks: An alternative to the argument 'r'. Not normally invoked by
          the user. See the *Details* section. 

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

     The function 'Jmulti' generalises 'Jest' (for unmarked point
     patterns) and 'Jdot' and 'Jcross' (for multitype point patterns)
     to arbitrary marked point patterns.

     Suppose X[I], X[J] are subsets, possibly overlapping, of a marked
     point process. Define

                  JIJ(r) = (1 - GIJ(r))/(1 - FJ(r))

     where FJ(r) is the cumulative distribution function of the
     distance from a fixed location to the nearest point of X[J], and
     GJ(r) is the distribution function of the distance from a typical
     point of  X[I] to the nearest distinct point of X[J]. 

     The argument 'X' must be a point pattern (object of class '"ppp"')
     or any data that are acceptable to 'as.ppp'.

     The arguments 'I' and 'J' specify two subsets of the point
     pattern. They may be logical vectors of length equal to 'X$n', or
     integer vectors with entries in the range 1 to 'X$n', etc.

     It is assumed that 'X' can be treated as a realisation of a
     stationary (spatially homogeneous)  random spatial point process
     in the plane, observed through a bounded window. The window (which
     is specified in 'X' as 'X$window') may have arbitrary shape.
     Biases due to edge effects are treated in the same manner as in
     'Jest'.

     The argument 'r' is the vector of values for the distance r at
     which JIJ(r) should be evaluated.  It is also used to determine
     the breakpoints (in the sense of 'hist') for the computation of
     histograms of distances. The reduced-sample and Kaplan-Meier
     estimators are computed from histogram counts.  In the case of the
     Kaplan-Meier estimator this introduces a discretisation error
     which is controlled by the fineness of the breakpoints.

     First-time users would be strongly advised not to specify 'r'.
     However, if it is specified, 'r' must satisfy 'r[1] = 0',  and
     'max(r)' must be larger than the radius of the largest disc 
     contained in the window. Furthermore, the successive entries of
     'r' must be finely spaced.

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

     An object of class '"fv"' (see 'fv.object').

     Essentially a data frame containing six numeric columns  

       r: the values of the argument r  at which the function JIJ(r)
          has been  estimated 

      rs: the ``reduced sample'' or ``border correction'' estimator of
          JIJ(r) 

      km: the spatial Kaplan-Meier estimator of JIJ(r) 

      un: the uncorrected estimate of JIJ(r), formed by taking the
          ratio of uncorrected empirical estimators of 1 - GIJ(r) and 1
          - FJ(r), see 'Gdot' and 'Fest'. 

    theo: the theoretical value of JIJ(r) for a marked Poisson process
          with the same estimated intensity, namely 1. 

_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:

     Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of
     dependence between types in multivariate point patterns.
     _Scandinavian Journal of Statistics_ *26*, 511-532.

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

     'Jcross', 'Jdot', 'Jest'

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

         data(longleaf)
          # Longleaf Pine data: marks represent diameter
         
         Jm <- Jmulti(longleaf, longleaf$marks <= 15, longleaf$marks >= 25)
         plot(Jm)

