Iest                package:spatstat                R Documentation

_E_s_t_i_m_a_t_e _t_h_e _I-_f_u_n_c_t_i_o_n

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

     Estimates the summary function I(r) for a multitype point pattern.

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

       Iest(X)
       Iest(X, eps)
       Iest(X, eps, r)
       Iest(X, eps, breaks)

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

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

     eps: the resolution of the discrete approximation to Euclidean
          distance (see below). There is a sensible default. 

       r: vector of values for the argument r at which I(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 Details section. 

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

     The I function  summarises the dependence between types in a
     multitype point process (Van Lieshout and Baddeley, 1999) It is
     based on the concept of the J function for an unmarked point
     process (Van Lieshout and Baddeley, 1996). See 'Jest' for
     information about the J function.

     The I function is defined as 

                   I(r) = (sum p[i] Jii(r)) - J(r)

     where J is the J function for the entire point process ignoring
     the marks, while  Jii is the J function for the process consisting
     of points of type i only, and p[i] is the proportion of points
     which are of type i.

     The I function is designed to measure dependence between points of
     different types, even if the points are not Poisson. Let X be a
     stationary multitype point process, and write X[i] for the process
     of points of type i. If the processes X[i] are independent of each
     other, then the I-function is identically equal to 0.  Deviations
     I(r) < 1 or I(r) > 1 typically indicate negative and positive
     association, respectively, between types. See Van Lieshout and
     Baddeley (1999) for further information.

     An estimate of I derived from a multitype spatial point pattern
     dataset can be used in exploratory data analysis and formal
     inference about the pattern. The estimate of I(r) is compared
     against the  constant function 0. Deviations I(r) < 1 or I(r) > 1
     may suggest negative and positive association, respectively.

     This algorithm estimates the I-function from the multitype point
     pattern 'X'. It assumes that 'X' can be treated as a realisation
     of a stationary (spatially homogeneous)  random spatial marked
     point process in the plane, observed through a 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()'. It must be a multitype point
     pattern (it must have a 'marks' vector which is a 'factor').

     The function 'Jest' is called to  compute estimates of the J
     functions in the formula above. In fact three different estimates
     are computed using different edge corrections. See 'Jest' for
     information.

_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 

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

      rs: the ``reduced sample'' or ``border correction'' estimator of
          I(r) computed from the border-corrected estimates of J
          functions

      km: the spatial Kaplan-Meier estimator of I(r) computed from the
          Kaplan-Meier estimates of J functions

      un: the uncorrected estimate of I(r) computed from the
          uncorrected estimates of J 

    theo: the theoretical value of I(r) for a stationary Poisson
          process: identically equal to 0 

_N_o_t_e:

     Sizeable amounts of memory may be needed during the calculation.

_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:

     Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric
     measure of spatial interaction in point patterns. _Statistica
     Neerlandica_ *50*, 344-361.

     Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of
     dependence between types in multivariate point patterns.
     _Scandinavian Journal of Statistics_ *26*, 511-532.

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

     'Jest'

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

        data(amacrine)
        Ic <- Iest(amacrine)
        plot(Ic, main="Amacrine Cells data")
        # values are below I= 0, suggesting negative association
        # between 'on' and 'off' cells.

