Kcross               package:spatstat               R Documentation

_M_u_l_t_i_t_y_p_e _K _F_u_n_c_t_i_o_n (_C_r_o_s_s-_t_y_p_e)

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

     For a multitype point pattern,  estimate the multitype K function
     which counts the expected number of points of type j within a
     given distance of a point of type i.

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

     Kcross(X, i=1, j=2)
     Kcross(X, i=1, j=2, correction=c("border", "isotropic", "Ripley", "translate"))
     Kcross(X, i=1, j=2, r, correction)
     Kcross(X, i=1, j=2, breaks)

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

       X: The observed point pattern,  from which an estimate of the
          cross type K function Kij(r) will be computed. It must be a
          multitype point pattern (a marked point pattern whose marks
          are a factor). See under Details. 

       i: Number or character string identifying the type (mark value)
          of the points in 'X' from which distances are measured. 

       j: Number or character string identifying the type (mark value)
          of the points in 'X' to which distances are measured. 

       r: numeric vector. The values of the argument r at which the
          distribution 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. 

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:

     This function 'Kcross' and its companions 'Kdot' and 'Kmulti' are
     generalisations of the function 'Kest' to multitype point
     patterns. 

     A multitype point pattern is a spatial pattern of points
     classified into a finite number of possible ``colours'' or
     ``types''. In the 'spatstat' package, a multitype pattern is
     represented as a single  point pattern object in which the points
     carry marks, and the mark value attached to each point determines
     the type of that point.

     The argument 'X' must be a point pattern (object of class '"ppp"')
     or any data that are acceptable to 'as.ppp'. It must be a marked
     point pattern, and the mark vector 'X$marks' must be a factor. The
     arguments 'i' and 'j' will be interpreted as levels of the factor
     'X$marks'. (Warning: this means that an integer value 'i=3' will
     be interpreted as the 3rd smallest level, not the number 3). 

     The ``cross-type'' (type i to type j) K function  of a stationary
     multitype point process X is defined so that lambda[j] Kij(r)
     equals the expected number of additional random points of type j
     within a distance r of a typical point of type i in the process X.
     Here lambda[j] is the intensity of the type j points, i.e. the
     expected number of points of type j per unit area. The function
     Kij is determined by the  second order moment properties of X.

     An estimate of Kij(r) is a useful summary statistic in exploratory
     data analysis of a multitype point pattern. If the process of type
     i points were independent of the process of type j points, then
     Kij(r) would equal pi * r^2. Deviations between the empirical Kij
     curve and the theoretical curve pi * r^2  may suggest dependence
     between the points of types i and j.

     This algorithm estimates the distribution function Kij(r)  from
     the point pattern 'X'. It 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', using the border correction.

     The argument 'r' is the vector of values for the distance r at
     which Kij(r) should be evaluated.  The values of r must be
     increasing nonnegative numbers and the maximum r value must exceed
     the radius of the largest disc contained in the window.

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

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

     An object of class '"fv"' (see 'fv.object').

     Essentially a data frame containing numeric columns  

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

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

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

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

     The arguments 'i' and 'j' are interpreted as levels of the factor
     'X$marks'. Beware of the usual trap with factors: numerical values
     are not interpreted in the same way as character values. See the
     first example.

     The reduced sample estimator of Kij is pointwise approximately 
     unbiased, but need not be a valid distribution function; it may 
     not be a nondecreasing function of r. Its range is always  within
     [0,1].

_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.

     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.

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

     'Kdot', 'Kest', 'Kmulti', 'pcf'

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

         data(betacells)
          # cat retina data
         K01 <- Kcross(betacells, "off", "on") 
         plot(K01)

         K10 <- Kcross(betacells, "on", "off")

         # synthetic example    
         pp <- runifpoispp(50)
         pp <- pp %mark% factor(sample(0:1, pp$n, replace=TRUE))
         K <- Kcross(pp, "0", "1") # note: "0" not 0

