Kinhom               package:spatstat               R Documentation

_I_n_h_o_m_o_g_e_n_e_o_u_s _K-_f_u_n_c_t_i_o_n

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

     Estimates the inhomogeneous K function of a non-stationary point
     pattern.

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

       Kinhom(X, lambda)
       Kinhom(X, lambda, r)
       Kinhom(X, lambda, breaks)

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

       X: The observed data point pattern, from which an estimate of
          the inhomogeneous K function will be computed. An object of
          class `"ppp"' or in a format recognised by `as.ppp()' 

  lambda: Vector of values of the estimated intensity function,
          evaluated at the points of the pattern `X' 

       r: vector of values for the argument r at which the
          inhomogeneous K function should be evaluated. There is a
          sensible default. 

  breaks: 

     An alternative to the argument `r'. Not normally invoked by the
     user. See Details.

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

     This computes a generalisation of the K function for inhomogeneous
     point patterns, proposed by Baddeley, Moller and Waagepetersen
     (2000).

     The ``ordinary'' K function (variously known as the reduced second
     order moment function and Ripley's K function), is described under
     `Kest'. It is defined only for stationary point processes.

     The inhomogeneous K function Kinhom(r) is a direct generalisation
     to nonstationary point processes. Suppose x is a point process
     with non-constant intensity lambda(u) at each location u. Define
     Kinhom(r) to be the expected value, given that u is a point of x,
     of the sum of all terms 1/lambda(u)lambda(x[j]) over all points
     x[j] in the process separated from u by a distance less than r.
     This reduces to the ordinary K function if lambda() is constant.
     If x is an inhomogeneous Poisson process with intensity function
     lambda(u), then Kinhom(r) = pi * r^2.

     This allows us to inspect a point pattern for evidence of 
     interpoint interactions after allowing for spatial inhomogeneity
     of the pattern. Values  Kinhom(r) > pi * r^2 are suggestive of
     clustering.

     The argument `lambda' must be a vector of length equal to the
     number of points in the pattern `X'. It will be interpreted as
     giving the (estimated) values of lambda(x[i]) for each point x[i]
     of the pattern x.

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

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

     A data frame containing 

       r: the vector of values of the argument r  at which the pair
          correlation function g(r) has been  estimated 

       K: vector of values of Kinhom(r) 

    theo: vector of values of pi * r^2, the theoretical value of
          Kinhom(r) for an inhomogeneous Poisson process 

_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:

     Baddeley, A., Moller, J. and Waagepetersen, R. (2000) Non- and
     semiparametric estimation of interaction in inhomogeneous point
     patterns. Statistica Neerlandica 54, 329-350.

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

     `Kest', `pcf'

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


       library(spatstat)


       data(lansing)
       # inhomogeneous pattern of maples
       X <- unmark(lansing[lansing$marks == "maple",])

       # fit spatial trend
       fit <- mpl(X, ~ polynom(x,y,2), Poisson())
       # predict intensity values at points themselves
       lambda <- predict(fit, newdata=data.frame(x=X$x, y=X$y), type="trend")
       # inhomogeneous K function
       Ki <- Kinhom(X, lambda)

         conspire(Ki, cbind(K, theo) ~ r, subset="r <= 0.4")


       # SIMULATED DATA
       # known intensity function
       lamfun <- function(x,y) { 100 * x }
       # inhomogeneous Poisson process
       Y <- rpoispp(lamfun, 100, owin())
       # evaluate intensity at points of pattern
       lambda <- lamfun(Y$x, Y$y)
       # inhomogeneous K function
       Ki <- Kinhom(Y, lambda)

         conspire(Ki, cbind(K, theo) ~ r, subset="r <= 0.2")


