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:

     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:

         library(spatstat)
         data(longleaf)
          # Longleaf Pine data: marks represent diameter

         Jm <- Jmulti(longleaf, longleaf$marks <= 15, longleaf$marks >= 25)



              plot(Jm$r, Jm$km,
                       xlab="r", ylab="Jmulti(r)",
                       type="l")
              # Poisson theoretical curve
              abline(h=1, lty=2)


