Kest                package:spatstat                R Documentation

_K-_f_u_n_c_t_i_o_n

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

     Estimates the reduced second moment function K(r)  from a point
     pattern in a window of arbitrary shape.

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

       Kest(X)
       Kest(X, r)
       Kest(X, breaks)

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

       X: The observed point pattern,  from which an estimate of K(r)
          will be computed. An object of class `"ppp"', or data in any
          format acceptable to `as.ppp()'. 

       r: vector of values for the argument r at which K(r)  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:

     The K function (variously called ``Ripley's K-function'' and the
     ``reduced second moment function'') of a stationary point process
     X is defined so that lambda K(r) equals the expected number of
     additional random points within a distance r of a typical random
     point of X. Here lambda is the intensity of the process, i.e. the
     expected number of points of X per unit area. The K function is
     determined by the  second order moment properties of X.

     An estimate of K derived from a spatial point pattern dataset can
     be used in exploratory data analysis and formal inference about
     the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1988). In
     exploratory analyses, the estimate of K is a useful statistic 
     summarising aspects of inter-point ``dependence'' and
     ``clustering''. For inferential purposes, the estimate of K is
     usually compared to the  true value of K for a completely random
     (Poisson) point process, which is K(r) = pi * r^2. Deviations
     between the empirical and theoretical K curves may suggest spatial
     clustering or spatial regularity.

     This routine `Kest' estimates the K function of a stationary point
     process, given observation of the process inside a known, bounded
     window.  The argument `X' is interpreted as a point pattern object
      (of class `"ppp"', see `ppp.object') and can be supplied in any
     of the formats recognised by `as.ppp()'.

     The estimation of K is hampered by edge effects arising from  the
     unobservability of points of the random pattern outside the
     window.  An edge correction is needed to reduce bias (Baddeley,
     1998; Ripley, 1988).  The correction implemented here is the
     border method or ``reduced sample'' estimator (Ripley, 1988).
     Although this is less efficient than some alternative corrections,
     it has the advantage that it can be computed for a window of
     arbitrary shape.

     Note that the estimator assumes the process is stationary
     (spatially homogeneous). For inhomogeneous point patterns, see
     `Kinhom'.

     The estimator `Kest' ignores marks. Its counterparts for multitype
     point patterns are `Kcross', `Kdot', and for general marked point
     patterns see `Kmulti'. 

     Some writers, particularly Stoyan (1994, 1995) advocate the use of
     the ``pair correlation function''

                     g(r) = K'(r)/ ( 2 * pi * r)

     where K'(r) is the derivative of K(r). See `pcf' on how to
     estimate this function.

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

     A data frame containing 

       r: the vector of values of the argument r  at which the function
          K has been  estimated 

  border: the ``reduced sample'' or ``border corrected'' estimator of
          K(r) 

    theo: the theoretical value K(r) = pi * r^2 for a stationary
          Poisson process 

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

     The estimator of K(r) is approximately unbiased for each fixed r.
     Bias increases with r and depends on the window geometry. For a
     rectangular window it is prudent to restrict the r values to a
     maximum of 1/4 of the smaller side length of the rectangle. Bias
     may become appreciable for point patterns consisting of  fewer
     than 15 points.

     While K(r) is always a non-decreasing function, the estimator  of
     K is not guaranteed to be non-decreasing. This is rarely  a
     problem in practice.

_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>

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

     `Fest', `Gest', `Jest', `pcf', `reduced.sample', `Kcross', `Kdot',
     `Kinhom', `Kmulti'

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

      library(spatstat)
      pp <- runifpoint(50)
      Kpp <- Kest(pp)

      plot(Kpp$r, Kpp$border, type="l", xlab="r", ylab="K(r)", ylim=c(0,1),
            main = "K-function")
      r <- Kpp$r
      lines(r, Kpp$theo, lty=2)
      legend(0.5, 2, c("empirical (border correction)", "Poisson process"), lty=c(1,2))

      data(cells)
      K <- Kest(cells)

      plot(K$r, K$border, type="l")
      plot(border ~ r, type="l", data=K)
      # restrict the plot to values of r less than 0.1
      plot(border ~ r, type="l", data=K[K$r <= 0.1, ])
      plot(border ~ r, type="l", data=K, subset=(r <= 0.1))


