CovarianceFct {RandomFields} | R Documentation |
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
CovarianceFct(x,model,param,dim=1) Variogram(x,model,param,dim=1) PrintModelList() GetModelNames()
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 |
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
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: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
)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.
power
(a=1
there).
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.
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.
stable
.
power
.
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
.
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
Martin Schlather, Martin.Schlather@uni-bayreuth.de http://www.geo.uni-bayreuth.de/~martin
Overviews:
Cauchy models, generalisations and extensions
Gneiting's models
Holeeffect model
Hyperbolic model
Power model
EmpiricalVariogram
,
RandomFields
,
RFparameters
,
ShowModels
.
PrintModelList() CovarianceFct(0:100, "bessel", c(NA,2,1,5,0.5))