Jcross               package:spatstat               R Documentation

_M_u_l_t_i_t_y_p_e _J _F_u_n_c_t_i_o_n (_i-_t_o-_j)

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

     For a multitype point pattern,  estimate the multitype J function 
     summarising the interpoint dependence between points of type i and
     of type j.

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

     Jcross(X, i=1, j=2)
     Jcross(X, i=1, j=2, eps, r)
     Jcross(X, i=1, j=2, eps, breaks)

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

       X: The observed point pattern,  from which an estimate of the
          multitype J function Jij(r) will be computed. It must be a
          multitype point pattern (a marked point pattern whose marks
          are a factor). See under Details. 

       i: Number or character string identifying the type (mark value)
          of the points in 'X' *from* which distances are measured. 

       j: Number or character string identifying the type (mark value)
          of the points in 'X' *to* which distances are measured. 

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

       r: numeric vector. The values of the argument r at which the
          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:

     This function 'Jcross' and its companions 'Jdot' and 'Jmulti' are
     generalisations of the function 'Jest' to multitype point
     patterns. 

     A multitype point pattern is a spatial pattern of points
     classified into a finite number of possible ``colours'' or
     ``types''. In the 'spatstat' package, a multitype pattern is
     represented as a single  point pattern object in which the points
     carry marks, and the mark value attached to each point determines
     the type of that point.

     The argument 'X' must be a point pattern (object of class '"ppp"')
     or any data that are acceptable to 'as.ppp'. It must be a marked
     point pattern, and the mark vector 'X$marks' must be a factor. The
     argument 'i' will be interpreted as a level of the factor
     'X$marks'. (Warning: this means that an integer value 'i=3' will
     be interpreted as the 3rd smallest level, not the number 3). 

     The ``type i to type j'' multitype J function  of a stationary
     multitype point process X was introduced by Van lieshout and
     Baddeley (1999). It is defined by

                   Jij(r) = (1 - Gij(r))/(1-Fj(r))

     where Gij(r) is the distribution function of the distance from a
     type i point to the nearest point of type j, and Fj(r) is the
     distribution function of the distance from a fixed point in space
     to the nearest point of type j in the pattern.

     An estimate of Jij(r) is a useful summary statistic in exploratory
     data analysis of a multitype point pattern.  If the subprocess of
     type i points is independent of the subprocess of points of type
     j, then Jij(r) = 1. Hence deviations of the empirical estimate of
     Jij from the value 1 may suggest dependence between types.

     This algorithm estimates Jij(r)  from the point pattern 'X'. It
     assumes 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',
     using the Kaplan-Meier and border corrections. The main work is
     done by 'Gmulti' and 'Fest'.

     The argument 'r' is the vector of values for the distance r at
     which Jij(r) should be evaluated.  The values of r must be
     increasing nonnegative numbers and the maximum r value must exceed
     the radius of the largest disc contained in the window.

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

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

     Essentially a data frame containing six numeric columns  

       J: the recommended estimator of Jij(r), currently the
          Kaplan-Meier estimator. 

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

      km: the Kaplan-Meier  estimator of Jij(r) 

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

      un: the ``uncorrected''  estimator 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, namely 1. 

     The result also has two attributes '"G"' and '"F"' which are
     respectively the outputs of 'Gcross' and 'Fest' for the point
     pattern.

_W_a_r_n_i_n_g_s:

     The argument 'i' is interpreted as a level of the factor
     'X$marks'. Beware of the usual trap with factors: numerical values
     are not interpreted in the same way as character values. See the
     first example.

_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. (1996) A nonparametric
     measure of spatial interaction in point patterns. _Statistica
     Neerlandica_ *50*, 344-361.

     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:

     'Jdot', 'Jest', 'Jmulti'

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

          # Lansing woods data: 6 types of trees
         data(lansing)

         
         Jhm <- Jcross(lansing, "hickory", "maple")
         # diagnostic plot for independence between hickories and maples
         plot(Jhm)

         # synthetic example with two marks "a" and "b"
         pp <- runifpoispp(50)
         pp <- pp %mark% sample(c("a","b"), pp$n, replace=TRUE)
         J <- Jcross(pp, "a", "b")

