CovarianceFct          package:RandomFields          R Documentation

_C_o_v_a_r_i_a_n_c_e _A_n_d _V_a_r_i_o_g_r_a_m _M_o_d_e_l_s

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

     `CovarianceFct' returns the values of an isotropic covariance
     function

     `Variogram' returns the values of an isotropic variogram model

     `PrintModelList' prints the list of currently implemented models

     `GetModelNames' returns a list of currently implemented models

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

     CovarianceFct(x,model,param,dim=1)
     Variogram(x,model,param,dim=1)
     PrintModelList()
     GetModelNames()

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

       x: a vector of distances at which the covariance function or
          variogram should be evaluated

   model: character; name of the covariance function or variogram
          model; see below, or type `PrintModelList()' for all options

   param: vector of parameters; 
          `param=c(NA,variance,nugget,scale,...)', in this order;
          The dots `...' stand for additional parameters of the model 

     dim: dimension of the space in which the model is applied

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

     The first component of param is reserved for the `mean' of a
     random field and thus ignored in the evaluation of the covariance
     function or variogram.  The parameters mean, variance, nugget, and
     scale must be given in this order; additional parameters have to
     be supplied in case of a parametrised class of models (e.g.
     `hyperbolic', see below), in the order a, b, c.

     The implemented models are in standard notation of a covariance
     function (variance 1, nugget 0, scale=1) and for positive real
     arguments x:

        *  `bessel'

                  C(x)= 2^a Gamma(a+1)x^(-a) J_a(x)

           The parameter a is greater than or equal to (d-2)/2, where d
           is the dimension of the random field.

        *  `cauchy'

                          C(x)=(1+x^2)^(-a)

           The parameter a is positive.  The model possesses two
           generalisations, the `gencauchy' model and the `hyperbolic'
           model.

        *  `cauchytbm'

                 C(x)= (1+(1-b/c)x^a)(1+x^a)^(-b/a-1)

           The parameter a is in (0,2],  b is positive, and c is an
           integer.  The model is valid for dimensions d<=c. 
           It allows for simulating random fields where  fractal
           dimension and Hurst coefficient can be chosen independently.
            It has negative correlations for b>c and large x.

        *  `circular'

    C(x)=1-2/pi*(x sqrt(1-x^2)+asin(x))   if 0<=x<=1, 0 otherwise

           This isotropic covariance function is valid only for
           dimensions less than or equal to 2.

        *  `cone'
           This model is used only for methods based on marked point
           processes (see `RFMethods'); it is defined only in two
           dimensions. The corresponding (boolean) function is a
           truncated cone with socle. The base has radius 1/2. The
           model has three parameters, a, b, and c:
           a gives the radius of the top circle of the cone, given as
           part of the socle radius; a in [0,1).
           b gives the height of the socle.
           c gives the height of the truncated cone.

        *  `cubic'

 C(x)= 1- 7 x^2 + 8.75 x^3 - 3.5 x^5 + 0.75 x^7  if 0<=x<=1, 0 otherwise

           This model is valid only for dimensions less than or equal
           to 3.  It is a 2 times differentiable covariance functions
           with compact support. 

        *  `exponential'

                             C(x)=exp(-x)

           This model is a special case of the `whittlematern' model
           (for a=1/2 there) and the `stable' class (for a=1).

        *  `gaussian'

                            C(x)=exp(-x^2)

           This model is a special case of the `stable' class (for a=2
           there). Note that the corresponding function for the random
           coins method (cf. the methods based on marked point
           processes in `RFMethods') is

                             exp(-2 x^2).

           See `gneiting' for an alternative model that does not have
           the disadvantages of the Gaussian model.

        *  `gencauchy' (generalised `cauchy')

                         C(x)= (1+x^a)^(-b/a)

           The parameter a is in (0,2], and b is positive. 
           This model allows for simulating random fields where fractal
           dimension and Hurst coefficient can be chosen independently.

        *  `gengneiting' (generalised `gneiting')
           if a=1 then

      C(x)=[1 + (b+1) * x] * (1-x)^(b+1) if 0<=x<=1, 0 otherwise

           if a=2 then

 C(x)= [1 + (b+2) * x + ((b+2)^2-1) * x^2 / 3] * (1-x)^(b+2) if 0<=x<=1, 0 otherwise

           if a=3 then

 C(x)=[1 + (b+3) * x +  (2 * (b+3)^2 - 3) * x^2 / 5 + ((b+3)^2 - 4) * (b+3) * x^3 / 15] * (1-x)^(b+3) if 0<=x<=1, 0 otherwise

           The parameter a is a positive integer; here only the cases
           a=1, 2, 3 are implemented. The parameter b is greater than
           or equal to (d + 2a +1)/2 where d is the dimension of the
           random field.

        *  `gneiting'

 C(x)= (1 + 8 s x + 25 s^2 x^2 + 32 s^3 x^3)*(1-s x)^8   if 0<=x<=1, 0 otherwise

           where s = 10 sqrt(2) / 47 ~= 0.3. This isotropic covariance
           function is valid only for dimensions less than or equal to
           3.  It is a 6 times differentiable covariance functions with
           compact support.
           It is an alternative to the `gaussian' model since its graph
           is visually hardly distinguishable from the graph of the
           Gaussian model, but possesses neither the mathematical and
           nor the numerical disadvantages of the Gaussian model.
           This model is a special case of `gengneiting' (for a=3 and
           b=5 there).

        *  `gneitingdiff'

 C(x)=(1 + 8 x/b + 25 (x/b)^2 + 32 (x/b)^3)*(1-x/b)^8 * 2^{1-a} Gamma(a)^{-1} x^a K_a(x)   if 0<=x<=b, 0 otherwise

           This isotropic covariance function is valid only for
           dimensions less than or equal to 3.  The parameters a and b
           are positive.
           This class of models with compact support allows for smooth
           parametrisation of the differentiability up to order 6.     

        *  `holeeffect'

                        C(x)= exp(-a x) cos(x)

           This model is valid for dimension 1 iff a>=0, for dimension
           2 iff a>=1, and for dimension 3 iff a >= sqrt(3).

        *  `hyperbolic'

 C(x)= c^(-b) (K_b(a*c))^(-1) * (c^2 +x^2)^(0.5 b) * K_b(sqrt(a(c^2 + x^2)))

           The parameters are such that
           c>=0,  a>0   and  b>0,    or
           c>0 ,  a>0   and  b=0,    or
           c>0 ,  a>=0, and  b<0.
           Note that this class is over-parametrised; always one of the
           three parameters  a, c, and scale can be eliminiated in the
           formula. Therefore, one of these parameters should be kept
           fixed in any simulation study. 
           The model contains as special cases the `whittlematern'
           model and the `cauchy' model, for  c=0 and a=0,
           respectively.

        *  matern
           See `whittlematern'.

        *  `nugget'

                               1(x==0)

           Here, either `param[2]', the `variance', or `param[3]', the
           `nugget', must be zero.

        *  `pentamodel'

 C(x)= 1 - 22/3 x^2 +33 x^4 - 77/2 x^5 + 33/2 x^7 - 11/2 x^9 + 5/6 x^11  if 0<=x<=1,   0 otherwise

           valid only for dimensions less than or equal to 3.  This is
           a 4 times differentiable covariance functions with compact
           support.

        *  `power'

               C(x)= (1-x)^a   if 0<=x<=1, 0 otherwise

           This covariance function is valid for dimension d if a >=
           (d+1)/2.  For a=1 we get the well-known triangle (or tent)
           model, which is valid on the real line, only.

        *  powered exponential
           See `stable'.

        *  `qexponential'

                 C(x) = (2 exp(-x)-a exp(-2x))/(2-a)

           The parameter a takes values in [0,1].   

        *  `spherical'

          C(x)= 1 - 1.5 x + 0.5 x^3  if 0<=x<=1, 0 otherwise

           This isotropic covariance function is valid only for
           dimensions less than or equal to 3.

        *  `stable'

                            C(x)=exp(-x^a)

           The parameter a is in [0,2]. See `exponential' and
           `gaussian' for special cases.

        *  symmetric stable
           See `stable'.

        *  tent model
           See `power'.

        *  triangle
           See `power'.

        *  `wave'

                         C(x)=sin(x)/x if x>0

           This isotropic covariance function is valid only for
           dimensions less than or equal to 3. It is a special case of
           the `bessel' model (for a=3).

        *  `whittlematern'

                C(x)=2^{1-a} Gamma(a)^{-1} x^a K_a(x),

           The parameter a is positive. 
           This is the model of choice if the smoothness of a random
           field is to be parametrised. It is a special case of the
           `hyperbolic' model (for c=0 there).

     Let cov be a model given in standard notation.  Then the
     covariance model applied with arbitrary variance, nugget, and
     scale equals

                 nugget + variance * cov( (.)/scale).


     For a given covariance function cov the variogram gamma equals 

                     gamma(x) = cov(0) - cov(x).


     Note that the value of the covariance function or variogram
     depends also on `RFparameters()$PracticalRange'.  If the latter is
     `TRUE' then the covariance function is internally rescaled such
     that cov(1)~=0.05 for standard parameters (`scale==1').

     Some models allow certain parameter combinations only for certain
     dimensions.  As any model in d dimensions is also valid in 1
     dimension, the default in `CovarianceFct' and `Variogram' is
     `dim=1'.

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

     `CovarianceFct' returns a vector of values of the covariance
     function.

     `Variogram' returns a vector of values of the variogram model.

     `PrintModelList' prints a table of the currently implemented
     covariance functions and the matching methods. `PrintModelList'
     returns `NULL'.

     `GetModelNames' returns a list of implemented models

_A_u_t_h_o_r(_s):

     Martin Schlather, Martin.Schlather@uni-bayreuth.de <URL:
     http://www.geo.uni-bayreuth.de/~martin>

_R_e_f_e_r_e_n_c_e_s:

     Overviews:

        *  Chiles, J.-P. and Delfiner, P. (1999) Geostatistics.
           Modeling Spatial Uncertainty. New York: Wiley.

        *  Gneiting, T. and Schlather, M. (2001) Statistical modeling
           with covariance functions. In preparation.

        *  Schlather, M. (1999) An introduction to positive definite
           functions and to unconditional simulation of random fields.
           Technical report ST 99-10, Dept. of Maths and Statistics,
           Lancaster University.

        *  Schlather, M. (2001) Models for stationary max-stable random
           fields. Submitted.

        *  Yaglom, A.M. (1987) Correlation Theory of Stationary and
           Related Random Functions I, Basic Results. New York:
           Springer.

        *  Wackernagel, H. (1998) Multivariate Geostatistics. Berlin:
           Springer, 2nd edition.

     Cauchy models, generalisations and extensions

        *  Gneiting, T. and Schlather, M. (2001) Stochastic models
           which separate fractal dimension and Hurst effect.
           Submitted.

     Gneiting's models

        *  Gneiting, T. (1999) Correlation functions for atmospheric
           data analysis Q. J. Roy. Meteor. Soc., Part A 125,
           2449-2464. 

     Holeeffect model

        *  Zastavnyi, V.P. (1993) Positive definite functions depending
           on a norm, Russian Acad. Sci. Dokl. Math. 46, 112-114. 

     Hyperbolic model

        *  Shkarofsky, I.P. (1968) Generalized turbulence
           space-correlation and wave-number spectrum-function pairs.
           Can. J. Phys. 46, 2133-2153.

     Power model

        *  Golubov, B.I. (1981) On Abel-Poisson type and Riesz means,
           Analysis Mathematica 7, 161-184.

        *  Zastavnyi, V.P. (2000) On positive definiteness of some
           functions, J. Multiv. Analys. 73, 55-81.

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

     `EmpiricalVariogram', `RandomFields', `RFparameters',
     `ShowModels'.

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

      PrintModelList()
      CovarianceFct(0:100, "bessel", c(NA,2,1,5,0.5))

