Kmulti               package:spatstat               R Documentation

_M_a_r_k_e_d _K-_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 K function
     which counts the expected number of points of subset J within a
     given distance from a typical point in subset `I'.

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

     Kmulti(X, I, J)
     Kmulti(X, I, J, r)
     Kmulti(X, I, J, breaks)

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

       X: The observed point pattern,  from which an estimate of the
          multitype K function KIJ(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. 

       r: numeric vector. The values of the argument r at which the
          multitype K function KIJ(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 `Kmulti' generalises `Kest' (for unmarked point
     patterns) and `Kdot' and `Kcross' (for multitype point patterns)
     to arbitrary marked point patterns.

     Suppose X[I], X[J] are subsets, possibly overlapping, of a marked
     point process. The multitype K function  is defined so that
     lambda[J] KIJ(r) equals the expected number of additional random
     points of X[J]  within a distance r of a typical point of  X[I].
     Here lambda[J] is the intensity of X[J]  i.e. the expected number
     of points of X[J] per unit area. The function KIJ is determined by
     the  second order moment properties of X.

     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.

     The argument `r' is the vector of values for the distance r at
     which KIJ(r) should be evaluated.  It is also used to determine
     the breakpoints (in the sense of `hist') for the computation of
     histograms of distances.

     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. 

     This algorithm 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
     `Kest'.

     There are several alternative estimators of K based on different
     ``edge corrections''.  The current implementation computes only
     the ``border correction'' (or ``reduced sample'') estimator
     (Ripley, 1988; Baddeley, 2000).

     The pair correlation function can also be applied to the result of
     `Kmulti'; see `pcf'.

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

     A data frame containing three numeric columns  

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

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

    theo: the theoretical value of  KIJ(r) for a marked Poisson
          process, namely pi * r^2 

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

     The function KIJ is not necessarily differentiable.

     The border correction (reduced sample) estimator of KIJ used here
     is pointwise approximately  unbiased, but need not be a
     nondecreasing function of r, while the true  KIJ must be
     nondecreasing.

_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:

     Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons,
     1991.

     Diggle, P.J. Statistical analysis of spatial point patterns.
     Academic Press, 1983.

     Diggle, P. J. (1986). Displaced amacrine cells in the retina of a
     rabbit : analysis of a bivariate spatial point pattern.  J.
     Neurosci. Meth. 18, 115-125.

     Harkness, R.D and Isham, V. (1983) A bivariate spatial point
     pattern of ants' nests. Applied Statistics 32, 293-303

     Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing
     spatial processes of several types of points. J. Royal Statist.
     Soc. Ser. B 44, 406-413.

     Ripley, B.D. Statistical inference for spatial processes.
     Cambridge University Press, 1988.

     Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its
     applications. 2nd edition. Springer Verlag, 1995.

     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:

     `Kcross', `Kdot', `Kest', `pcf'

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

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

         K <- Kmulti(longleaf, longleaf$marks <= 15, longleaf$marks >= 25)


         plot(K$r, K$border,
                       xlab="r", ylab="Kmulti(r)",
                       type="l", ylim=c(0,1))

         # Poisson theoretical curve
         lines(K$r, K$theo, lty=2)

