Fest                package:spatstat                R Documentation

_E_s_t_i_m_a_t_e _t_h_e _e_m_p_t_y _s_p_a_c_e _f_u_n_c_t_i_o_n _F

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

     Estimates the empty space function F(r) from a point pattern in a 
     window of arbitrary shape.

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

     Fest(X, eps)
     Fest(X, eps, r)
     Fest(X, eps, breaks)

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

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

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

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

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

     The empty space function  (also called the ``spherical contact
     distribution'' or the ``point-to-nearest-event'' distribution) of
     a stationary point process X is the cumulative distribution
     function F of the distance from a fixed point in space to the
     nearest point of X.

     An estimate of F 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 F is a useful statistic 
     summarising the sizes of gaps in the pattern. For inferential
     purposes, the estimate of F is usually compared to the  true value
     of F for a completely random (Poisson) point process, which is

                 F(r) = 1 - exp( - lambda * pi * r^2)

     where lambda is the intensity (expected number of points per unit
     area). Deviations between the empirical and theoretical F curves
     may suggest spatial clustering or spatial regularity.

     This algorithm estimates the empty space function F  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') may have arbitrary
     shape. 

     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 algorithm uses two discrete approximations which are
     controlled by the parameter `eps' and by the spacing of values of
     `r' respectively. (See below for details.) First-time users are
     strongly advised not to specify these arguments.

     The estimation of F 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 two edge corrections implemented here
     are the border method or "reduced sample" estimator, and the
     spatial Kaplan-Meier estimator (Baddeley and Gill, 1997). 

     Our implementation makes essential use of the distance transform
     algorithm of image processing (Borgefors, 1986). A fine grid of
     pixels is  created in the observation window. The Euclidean
     distance between two pixels is approximated by the length of the
     shortest path joining them in the grid, where a path is a sequence
     of steps between adjacent pixels, and  horizontal, vertical and
     diagonal steps have length 1, 1 and sqrt(2) respectively in pixel
     units. If the pixel grid is sufficiently fine then this is an
     accurate approximation. 

     The parameter `eps' is the pixel width of the rectangular raster
     used to compute the distance transform (see below). It must not be
     too large: the absolute error in distance values due to
     discretisation is bounded by `eps'.

     If `eps' is not specified, the function checks whether the window
     `X$window' contains pixel raster information. If so, then `eps' is
     set equal to the  pixel width of the raster; otherwise, `eps'
     defaults to 1/100 of the width of the observation window.

     The argument `r' is the vector of values for the distance r at
     which F(r) should be evaluated.  It is also used to determine the
     breakpoints (in the sense of `hist' for the computation of
     histograms of distances. The reduced-sample and Kaplan-Meier
     estimators are computed from histogram counts.  In the case of the
     Kaplan-Meier estimator this introduces a discretisation error
     which is controlled by the fineness of the breakpoints.

     First-time users would be strongly advised not to specify `r'.
     However, if it is specified, `r' must satisfy `r[1] = 0',  and
     `max(r)' must be larger than the radius of the largest disc 
     contained in the window. Furthermore, the spacing of successive
     `r' values must be very fine (ideally not greater than `eps/4').

     The algorithm also returns an estimate of the hazard rate
     function,  lambda(r), of F(r). The hazard rate is defined by

                  lambda(r) = - (d/dr) log(1 - F(r))

     The hazard rate of F has been proposed as a useful exploratory
     statistic (Baddeley and Gill, 1994). The estimate of lambda(r)
     given here is a discrete approximation to the hazard rate of the 
     Kaplan-Meier estimator of F. Note that F is  absolutely continuous
     (for any stationary point process X),  so the hazard function
     always exists (Baddeley and Gill, 1997). 

     The naive empirical distribution of distances from each location
     in the window to the nearest point of the data pattern, is a
     biased estimate of F. However this is also returned by the
     algorithm, as it is sometimes useful in other contexts. Care
     should be taken not to use the uncorrected empirical F as if it
     were an unbiased estimator of F.

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

     A data frame containing six columns: 

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

      rs: the ``reduced sample'' or ``border correction'' estimator of
          F(r) 

      km: the spatial Kaplan-Meier estimator of F(r) 

  hazard: the hazard rate lambda(r) of F(r) by the spatial Kaplan-Meier
          method 

     raw: the uncorrected estimate of F(r), i.e. the empirical
          distribution of the distance from  a random point in the
          window to the nearest point of the data pattern `X' 

    theo: the theoretical value of F(r) for a stationary Poisson
          process of the same estimated intensity. 

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

     The reduced sample (border method) estimator of F 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].

     The spatial Kaplan-Meier estimator of F is always nondecreasing
     but its maximum value may be less than 1.

     The estimate of lambda(r) returned by the algorithm is an
     approximately unbiased estimate for the integral of lambda() over
     the corresponding histogram cell. It may exhibit oscillations due
     to discretisation effects. We recommend modest smoothing, such as
     kernel smoothing with  kernel width equal to the width of a
     histogram cell.

_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:

     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.

     Baddeley, A.J. and Gill, R.D.  The empty space hazard of a spatial
     pattern. Research Report 1994/3, Department of Mathematics,
     University of Western Australia, May 1994.

     Baddeley, A.J. and Gill, R.D. Kaplan-Meier estimators of
     interpoint distance distributions for spatial point processes.
     Annals of Statistics 25 (1997) 263-292.

     Borgefors, G. Distance transformations in digital images. Computer
     Vision, Graphics and Image Processing 34 (1986) 344-371.

     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.

     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:

     `Gest', `Jest', `Kest', `kmrs', `reduced.sample', `kaplan.meier'

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

        library(spatstat)
        data(cells)
        Fc <- Fest(cells, 0.01)

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

        # simulated data
        pp <- runifpoint(50)
        Fpp <- Fest(pp, 0.01)

             plot(Fpp$r, Fpp$km, type="l", xlab="r", ylab="F(r)", ylim=c(0,1),
             main = "empty space function")
             r <- Fpp$r
             lines(r, Fpp$theo, lty=2)
             legend(0.5, 2, c("Kaplan-Meier estimator", "Poisson process"), 
                        lty=c(1,2))


