Kmeasure              package:spatstat              R Documentation

_R_e_d_u_c_e_d _S_e_c_o_n_d _M_o_m_e_n_t _M_e_a_s_u_r_e

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

     Estimates the reduced second moment measure Kappa  from a point
     pattern in a window of arbitrary shape.

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

       Kmeasure(X, sigma, edge=TRUE)

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

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

   sigma: standard deviation sigma of the Gaussian smoothing kernel. 

    edge: logical value indicating whether an edge correction should be
          applied. 

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

     The reduced second moment measure Kappa of a stationary point
     process X is defined so that, for a `typical' point x of the
     process,  the expected number of other points y of the process
     such that the vector y - x lies in a region A, equals lambda
     Kappa(A). Here lambda is the intensity of the process, i.e. the
     expected number of points of X per unit area.

     The more familiar K-function K(t) is just the value of the reduced
     second moment measure for each disc centred at the origin; that
     is, K(t) = Kappa(b(0,t)).

     An estimate of Kappa derived from a spatial point pattern dataset
     can be useful in exploratory data analysis. Its advantage over the
     K-function is that it is also sensitive to anisotropy and
     directional effects. 

     This function computes an estimate of Kappa from a point pattern
     dataset 'X', which is assumed to be a realisation of a stationary
     point process, observed inside a known, bounded window. Marks are
     ignored.

     The algorithm approximates the point pattern and its window by
     binary pixel images, introduces an isotropic Gaussian smoothing
     kernel and uses the Fast Fourier Transform 'fft' to form a density
     estimate of Kappa. The calculation corresponds to the edge
     correction known as the ``translation correction''.

     The density estimate of Kappa is returned in the form of a
     real-valued pixel image. Pixel values are estimates of the
     integral of the second moment density over the pixel. (The uniform
     Poisson process would have values identically equal to a where a
     is the area of a pixel.) Sums of pixel values over a desired
     region A are estimates of the value of Kappa(A). The image 'x' and
     'y' coordinates are on the same scale as vector displacements in
     the original point pattern window. The point 'x=0, y=0'
     corresponds to the `typical point'. A peak in the image near
     '(0,0)' suggests clustering; a dip in the image near '(0,0)'
     suggests inhibition; peaks or dips at other positions suggest
     possible periodicity.

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

     A real-valued pixel image (an object of class '"im"', see
     'im.object') whose pixel values are estimates of the value of the
     reduced second moment measure for each pixel (i.e. estimates of
     the integral of the second moment density over each pixel).

_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:

     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:

     'Kest', 'spatstat.options', 'im.object'

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

      data(cells)
      image(Kmeasure(cells, 0.05))
      # shows pronounced dip around origin consistent with strong inhibition
      data(redwood)
      image(Kmeasure(redwood, 0.03), col=grey(seq(1,0,length=32)))
      # shows peaks at several places, reflecting clustering and ?periodicity

