mpl                 package:spatstat                 R Documentation

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

     Fits a point process model to an observed point pattern by the
     method of maximum pseudolikelihood.

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

      mpl(Q, trend=~1, interaction=NULL, data, correction="border", rbord=0, use.gam=F)

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

       Q: A data point pattern (of class `"ppp"') to which the model
          will be fitted, or preferably a quadrature scheme (of class
          `"quad"') containing this pattern. 

   trend: An R formula object specifying the spatial trend to be
          fitted.  The default formula, `~1', indicates the model is
          stationary and no trend is to be fitted.  

interaction: An object of class `"interact"' describing the point
          process interaction structure, or `NULL' indicating that a
          Poisson process (stationary or nonstationary) should be
          fitted. 

    data: An optional data frame of spatial covariates (evaluated at
          the locations given in the quadrature scheme `Q'). 

correction: The name of the edge correction to be used. The default  is
          `"border"' indicating the border correction. 

   rbord: If `correction = "border"' this argument specifies the
          distance by which the window should be eroded for the border
          correction. 

 use.gam: Logical flag; if `TRUE' then computations are performed using
          `gam' instead of `glm'. 

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

     This function fits a point process model to an observed point
     pattern by the method of maximum pseudolikelihood (Besag, 1975).
     The model may include spatial trend, interpoint interaction, and
     dependence on covariates.

     The algorithm is an implementation of the method of Baddeley and
     Turner (2000), which approximates the pseudolikelihood by a
     special type of quadrature sum generalising the Berman-Turner
     (1992)  approximation. 

     The argument `Q' should be either a point pattern or a quadrature
     scheme. If it is a point pattern, it is converted into a
     quadrature scheme.

     A quadrature scheme is an object of class `"quad"' (see
     `quad.object') which specifies both the data point pattern and the
     dummy points for the quadrature scheme, as well as the quadrature
     weights associated with these points. If `Q' is simply a point
     pattern (of class `"ppp"', see `ppp.object') then it is
     interpreted as specifying the data points only; a set of dummy
     points specified by `default.dummy()' is added, and the default
     weighting rule is invoked to compute the quadrature weights.

     The usage of `mpl()' is closely analogous to the Splus/R functions
     `glm' and `gam'. The analogy is:

       glm        mpl
       `formula'  `trend'
       `family'   `interaction'

     The point process model to be fitted is specified by the 
     arguments `trend' and `interaction' which are respectively
     analogous to the `formula' and `family' arguments of glm(). 

     Systematic effects (spatial trend and/or dependence on  spatial
     covariates) are specified by the argument `trend'. This is an
     Splus/R formula object, which may be expressed in terms of the
     Cartesian coordinates `x', `y', the marks `marks', or the
     variables in the data frame `data' (if supplied), or both. It
     specifies the logarithm of the first order potential of the
     process. The formula should not use the names `Y', `V', `W', or
     `SUBSET', which are reserved for internal use. If `trend' is
     absent or equal to the default, `~1', then the model to be fitted
     is stationary (or at least, its first order  potential is
     constant). 

     Stochastic interactions between random points of the point process
     are defined by the argument `interaction'. This is an object of
     class `"interact"' which is initialised in a very similar way to
     the usage of family objects in `glm' and `gam'. See the examples
     below.

     If `interaction' is missing or `NULL', then the model to be fitted
     has no interpoint interactions, that is, it is a Poisson process
     (stationary or nonstationary according to `trend'). In this case
     the method of maximum pseudolikelihood coincides with maximum
     likelihood.

     The argument `data', if supplied, must be a data frame with as
     many rows as there are points in `Q'. The ith row of `data' should
     contain the values of spatial variables which have been observed
     at the ith point of `Q'. In this case `Q' must be a quadrature
     scheme, not merely a point pattern. Thus, it is not sufficient to
     have observed a spatial variable only at the points of the data
     point pattern;  the variable must also have been observed at
     certain other  locations in the window. The variable names `x',
     `y' and `marks' are reserved for the Cartesian  coordinates and
     the mark values, and these should not be used for variables in
     `data'.

     The argument `correction' is the name of an edge correction
     method. The default `correction="border"' specifies the border
     correction, in which the quadrature window (the domain of
     integration of the  pseudolikelihood) is obtained by trimming off
     a margin of width `rbord' from the observation window of the data
     pattern. Not all edge corrections are implemented (or
     implementable) for arbitrary windows. Other options depend on the
     argument `interaction', but these generally  include `"periodic"'
     (the periodic or toroidal edge correction in which opposite edges
     of a rectangular window are identified) and `"translation"' (the
     translation correction, see Baddeley 1998 and Baddeley and Turner
     2000).

     The fitted point process model returned by this function can be
     printed  (by the print method `print.ppm') to inspect the fitted
     parameter values. If a nonparametric spatial trend was fitted,
     this can be extracted using the predict method `predict.ppm'.

     This algorithm approximates the log pseudolikelihood by a sum over
     a finite set of quadrature points. Finer quadrature schemes (i.e.
     those with more quadrature points) generally yield a better
     approximation, at the expense of higher computational load. 
     Complete control over the quadrature scheme is possible. See
     `quadscheme' for an overview.

     Note that the method of maximum pseudolikelihood is believed to be
     inefficient and biased for point processes with strong interpoint
     interactions. In such cases, it is advisable to use iterative
     maximum likelihood methods such as Monte Carlo Maximum Likelihood
     (Geyer, 1999)  provided the appropriate simulation algorithm
     exists.  The maximum pseudolikelihood parameter estimate often
     serves as a good initial starting point for these iterative
     methods.  Maximum pseudolikelihood may also be used profitably for
     model selection in the initial phases of modelling.

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

     An object of class `"ppm"' describing a fitted point process
     model.

     The fitted parameters can be obtained just by printing this
     object. Fitted spatial trends can be extracted using the `predict'
     method for this object (see `predict.ppm').

     See `ppm.object' for details of the format of this object.

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

     See the comments above about the possible inefficiency and bias of
     the maximum pseudolikelihood estimator.

     The accuracy of the Berman-Turner-Baddeley approximation to the
     pseudolikelihood depends on the number of dummy points used in the
     quadrature scheme. The number of dummy points should  at least
     equal the number of data points.

     The parameter values of the fitted model do not necessarily
     determine a valid point process. Some of the point process models
     are only defined when the parameter values lie in a certain
     subset. For example the Strauss process only  exists when the
     interaction parameter gamma is less than or equal to 1,
     corresponding to a value of `mpl()$theta[2]' less than or equal to
     `0'. The current version of `mpl' maximises the pseudolikelihood
     without constraining the parameters, and does not apply any checks
     for sanity after fitting the model. 

     The `trend' formula should not use the names `Y', `V', `W', or
     `SUBSET', which are reserved for internal use. The data frame
     `data' should have as many rows as there are points in `Q'. It
     should not contain variables called `x', `y' or `marks' as these
     names are reserved for the Cartesian coordinates and the marks.

     If the model formula involves one of the functions `poly()',
     `bs()' or `ns()' (e.g. applied to spatial coordinates `x' and
     `y'), the fitted coefficients can be misleading. The resulting fit
     is not to the raw spatial variates (`x', `x^2', `x*y', etc.)  but
     to a transformation of these variates.  The transformation is
     implemented by `poly()' in order to achieve better numerical
     stability. However the resulting coefficients are appropriate for
     use with the transformed variates, not with the raw variates.  
     This affects the interpretation of the constant term in the fitted
     model, `logbeta'.  Conventionally, beta is the background
     intensity, i.e. the   value taken by the conditional intensity
     function when all predictors (including spatial or ``trend''
     predictors) are set equal to 0. However the coefficient actually
     produced is the value that the log conditional intensity takes
     when all the predictors,  including the transformed spatial
     predictors, are set equal to `0', which is not the same thing.

     Worse still, the result of `predict.ppm' can be completely wrong
     if the trend formula contains one of the functions `poly()',
     `bs()' or `ns()'. This is a weakness of the underlying function
     `predict.glm'. 

     If you wish to fit a polynomial trend,  we offer an alternative to
     `poly()', namely `polynom()', which avoids the difficulty induced
     by transformations.  It is completely analogous to `poly' except
     that it does not orthonormalise. The resulting coefficient
     estimates then have their natural interpretation and can be
     predicted correctly.  Numerical stability may be compromised.

_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. and Turner, R. Practical maximum pseudolikelihood for
     spatial point patterns. Australian and New Zealand Journal of
     Statistics 42 (2000) 283-322.

     Berman, M. and Turner, T.R.  Approximating point process
     likelihoods with GLIM. Applied Statistics 41 (1992) 31-38.

     Besag, J. Statistical analysis of non-lattice data. The
     Statistician 24 (1975) 179-195.

     Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and
     Tanemura, M. On parameter estimation for pairwise interaction
     processes. International Statistical Review 62 (1994) 99-117.

     Jensen, J.L. and Moeller, M. Pseudolikelihood for exponential
     family models of spatial point processes. Annals of Applied
     Probability 1 (1991) 445-461.

     Jensen, J.L. and Kuensch, H.R.  On asymptotic normality of pseudo
     likelihood estimates for pairwise interaction processes, Annals of
     the Institute of Statistical Mathematics 46 (1994) 475-486.

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

     `ppp',  `quadscheme',  `ppm.object', `Poisson', `Strauss',
     `StraussHard', `Softcore', `Pairwise', `PairPiece', `Geyer',
     `Saturated', `OrdThresh', `Ord'

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

      library(spatstat)
      data(nztrees)
      Q <- quadscheme(nztrees) # default quadrature scheme
      mpl(Q)
      # fit the stationary Poisson process
      # to point pattern or data/dummy quadrature scheme Q

      mpl(Q, ~ x)
      # fit the nonstationary Poisson process 
      # with intensity function lambda(x,y) = exp(a + bx)
      # where x,y are the Cartesian coordinates
      # and a,b are parameters to be estimated

      mpl(Q, ~ polynom(x,2))
      # fit the nonstationary Poisson process 
      # with intensity function lambda(x,y) = exp(a + bx + cx^2)

      library(splines)
      mpl(Q, ~ bs(x,df=3))
      #       WARNING: do not use predict.ppm() on this result
      # Fits the nonstationary Poisson process 
      # with intensity function lambda(x,y) = exp(B(x))
      # where B is a B-spline with df = 3

      mpl(Q, ~1, Strauss(r=0.1), rbord=0.1)
      # Fit the stationary Strauss process with interaction range 0.1
      # using the border method with margin rbord=0.1

      mpl(Q, ~ x, Strauss(0.1), correction="periodic")
      # Fit the nonstationary Strauss process with interaction range 0.07
      # and exp(first order potential) =  activity = beta(x,y) = exp(a+bx)
      # using the periodic correction.


      data(soilsurvey)
      mpl(soilsurvey, ~ bs(pH,3), Strauss(0.1), rbord=0.1, data=soilchem)
      #       WARNING: do not use predict.ppm() on this result
      # Fit the nonstationary Strauss process 
      # with intensity modelled as a third order spline function of the 
      # spatial variable "pH" in data frame 'soilchem'


      ## MULTITYPE POINT PROCESSES ### 
      data(lansing)
      # Multitype point pattern --- trees marked by species
      mpl(lansing, ~ marks, Poisson())
      # fit stationary marked Poisson process
      # with different intensity for each species


      mpl(lansing, ~ marks * polynom(x,y,3), Poisson())

      # fit nonstationary marked Poisson process
      # with different log-cubic trend for each species

