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, r, correction=c("border", "isotropic", "Ripley", "translate"))
       Kest(X, breaks=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. 

correction: A character vector containing any selection of the options
          '"border"', '"bord.modif"', '"isotropic"', '"Ripley"' or
          '"translate"'. It specifies the edge correction(s) to be
          applied. 

_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 corrections implemented here are

     _b_o_r_d_e_r the border method or ``reduced sample'' estimator (see
          Ripley, 1988). This is the least efficient (statistically)
          and the fastest to compute. It can be computed for a window
          of arbitrary shape.

     _i_s_o_t_r_o_p_i_c/_R_i_p_l_e_y Ripley's isotropic correction (see Ripley, 1988;
          Ohser, 1983). This is currently implemented only for
          rectangular windows.

     _t_r_a_n_s_l_a_t_e Translation correction (Ohser, 1983). Implemented for
          all window geometries, but slow for complex windows. 

     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:

     An object of class '"fv"', see 'fv.object', which can be plotted
     directly using 'plot.fv'.

     Essentially a data frame containing columns 

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

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

     together with columns named  '"border"', '"bord.modif"', '"iso"'
     and/or '"trans"', according to the selected edge corrections.
     These columns contain estimates of the function K(r) obtained by
     the edge corrections named.

_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>

_R_e_f_e_r_e_n_c_e_s:

     Baddeley, A.J. Spatial sampling and censoring. In O.E.
     Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds) 
     _Stochastic Geometry: Likelihood and Computation_. Chapman and
     Hall, 1998. Chapter 2, pages 37-78.

     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.

     Ohser, J. (1983) On estimators for the reduced second moment
     measure of point processes. _Mathematische Operationsforschung und
     Statistik, series Statistics_, *14*, 63 - 71.

     Ripley, B.D. _Statistical inference for spatial processes_.
     Cambridge University Press, 1988.

     Stoyan, D, Kendall, W.S. and Mecke, J. (1995) _Stochastic geometry
     and its applications_. 2nd edition. Springer Verlag.

     Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point
     fields: methods of geometrical statistics. John Wiley and Sons.

_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:

      pp <- runifpoint(50)
      K <- Kest(pp)
      data(cells)
      K <- Kest(cells, correction="isotropic")
      plot(K)
      plot(K, main="K function for cells")
      # plot the L function
      plot(K, sqrt(iso/pi) ~ r)
      plot(K, sqrt(./pi) ~ r, ylab="L(r)", main="L function for cells")

