Description Usage Arguments Details Value Author(s) References See Also Examples
Maximum likelihood (ML) or restricted maximum likelihood (REML) parameter estimation for (transformed) Gaussian random fields.
1 | likfitSANN(geodata, coords = geodata$coords, data = geodata$data, trend = "cte", ini.cov.pars, fix.nugget = FALSE, nugget = 0, fix.kappa = TRUE, kappa = 0.5, fix.lambda = TRUE, lambda = 1, fix.psiA = TRUE, psiA = 0, fix.psiR = TRUE, psiR = 1, cov.model, realisations, lik.method = "ML", components = TRUE, nospatial = TRUE, limits = pars.limits(), print.pars = FALSE, messages, SAcontrol = NULL, ...)
|
geodata |
a list containing elements |
coords |
an n x 2 matrix where each row has the 2-D
coordinates of the n data locations.
By default it takes the
component |
data |
a vector with n data values. By default it takes the
component |
trend |
specifies the mean part of the model. See documentation
of |
ini.cov.pars |
initial values for the covariance parameters:
sigma^2 (partial sill) and phi (range
parameter). Typically a vector with two components. However a
matrix can be used to provide several initial values. See
|
fix.nugget |
logical, indicating whether the parameter
tau^2 (nugget variance) should be regarded as fixed
( |
nugget |
value of the nugget parameter.
Regarded as a fixed value if |
fix.kappa |
logical, indicating whether the extra parameter
kappa should be regarded as fixed
( |
kappa |
value of the extra parameter kappa.
Regarded as a fixed value if |
fix.lambda |
logical, indicating whether the Box-Cox transformation parameter
lambda should be regarded as fixed
( |
lambda |
value of the Box-Cox transformation parameter
lambda.
Regarded as a fixed value if |
fix.psiA |
logical, indicating whether the anisotropy angle parameter
psi_R should be regarded as fixed
( |
psiA |
value (in radians) for the anisotropy angle parameter
psi_A.
Regarded as a fixed value if |
fix.psiR |
logical, indicating whether the anisotropy ratio parameter
psi_R should be regarded as fixed
( |
psiR |
value, always greater than 1, for the anisotropy ratio parameter
psi_R.
Regarded as a fixed value if |
cov.model |
a string specifying the model for the correlation
function. For further details see documentation for |
realisations |
optional. Logical or a vector indicating the number of replication
for each datum. For further information see |
lik.method |
(formely method.lik) options are |
components |
an n x 3 data-frame with fitted
values for the three model components: trend, spatial and residuals.
See the section |
nospatial |
logical. If |
limits |
values defining lower and upper limits for the model
parameters used in the numerical minimisation.
The auxiliary function |
print.pars |
logical. If |
messages |
logical. Indicates whether status messages should be printed on the screen (or output device) while the function is running. |
... |
Defunct. No longer used. |
SAcontrol |
control arguments used by the |
object |
an object with output of the function |
spatial |
logical, determines whether the spatial component of the model in included in the output. The geostatistical model components are: trend, spatial and residulas. See DETAILS. |
This function estimate the parameters of the Gaussian random field model, specified as:
Y(x) = mu(x) + S(x) + e
where
x defines a spatial location. Typically Euclidean coordinates on a plane.
Y is the variable been observed.
mu(x) = X %*% beta is the mean component of the model (trend).
S(x) is a stationary Gaussian process with variance sigma^2 (partial sill) and a correlation function parametrized in its simplest form by phi (the range parameter). Possible extra parameters for the correlation function are the smoothness parameter kappa and the anisotropy parameters phi_R and phi_A (anisotropy ratio and angle, respectively).
e is the error term with variance parameter tau^2 (nugget variance).
The additional parameter lambda allows for the Box-Cox transformation of the response variable. If used (i.e. if λ \neq 1) Y(x) above is replaced by g(Y(x)) such that
g(Y(x)) = ((Y^lambda(x)) - 1)/lambda .
Two particular cases are lambda = 1 which indicates no transformation and lambda = 0 indicating the log-transformation.
Numerical minimization
In general parameter estimation is performed numerically using the R
function optim
to minimise the
negative log-likelihood computed by the function negloglik.GRF
.
If the nugget, anisotropy (psiA, psiR),
smoothness (kappa) and transformation (lambda) parameters
are held fixed then the numerical minimisation can be reduced to
one-dimension and the function optimize
is used instead
of optim
. In this case initial values are irrelevant.
Limits
Lower and upper limits for parameter values can be
individually specified using the function link{pars.limits}
.
For example, including the following in the function call:
limits = pars.limits(phi=c(0, 10), lambda=c(-2.5, 2.5))
,
will change the limits for the parameters phi and lambda.
Default values are used if the argument limits
is not provided.
There are internal reparametrisation depending on the options for
parameters to be estimated.
For instance for the common situation when fix.nugget=FALSE
the
minimisation is performed in a reduced
parameter space using
tau^2_{rel} = tau^2/sigma^2.
In this case values of sigma^2 and beta
are then given by
analytical expressions which are function of the two parameters
remaining parameters and limits for these two parameters will be ignored.
Transformation
If the fix.lambda = FALSE
and nospatial = FALSE
the
Box-Cox parameter for the model without the spatial component is
obtained numerically, with log-likelihood computed by the function
boxcox.ns
.
Multiple initial values can be specified providing a n x 2 matrix for the argument ini.cov.pars
and/or
providing a vector for the values of the remaining model parameters.
In this case the log-likelihood is computed for all combinations of
the model parameters. The parameter set which maximises the
value of the log-likelihood is then used to start the
minimisation algorithm.
Alternatively the argument ini.cov.pars
can take an object of
the class eyefit
or variomodel
. This allows the usage
of an output of the functions eyefit
, variofit
or
likfit
be used as initial value.
The argument realisations allows sets of data assumed to be
independent replications of the same process.
Data on different realisations may or may not be co-located.
For instance, data collected at different times
can be pooled together in the parameter estimation assuming
time independence.
The argument realisations
takes a vector indicating the
replication number (e.g. the times).
If realisations = TRUE
the code looks for an element
named realisations
in the geodata
object.
The log-likelihoods are computed for each replication and added together.
An object of the classes "likGRF"
and "variomodel"
.
The function summary.likGRF
is used to print a summary
of the fitted model.
The object is a list with the following components:
cov.model |
a string with the name of the correlation function. |
nugget |
value of the nugget parameter tau^2.
This is an estimate if |
cov.pars |
a vector with the estimates of the parameters sigma^2 and phi, respectively. |
kappa |
value of the smoothness parameter. Valid only if
the correlation function is one of: |
beta |
estimate of mean parameter beta. This can be a scalar or vector depending on the trend (covariates) specified in the model. |
beta.var |
estimated variance (or covariance matrix) for the mean parameter beta. |
lambda |
values of the Box-Cox transformation parameter. A fixed value if
|
aniso.pars |
fixed values or estimates of the anisotropy parameters, according to the function call. |
method.lik |
estimation method used, |
loglik |
the value of the maximized likelihood. |
npars |
number of estimated parameters. |
AIC |
value of the Akaike Information Criteria, AIC=-2 ln(L) + 2 p where L is the maximised likelihood and p is the number of parameters in the model. |
BIC |
value of the Bayesian Information Criteria, BIC=-2ln(L) + p log(n), where n is the number of data, L,p as for AIC above. |
parameters.summary |
a data-frame with all model parameters, their status (estimated or fixed) and values. |
info.minimisation |
results returned by the minimisation function. |
max.dist |
maximum distance between 2 data points. This
information relevant for other functions which use outputs from
|
trend |
the trend (covariates) matrix X. |
log.jacobian |
numerical value of the logarithm of the Jacobian of the transformation. |
nospatial |
estimates for the model without the spatial component. |
call |
the function call. |
Please contact Jason Lessels jlessels@gmail.com about this package. All code was originally written by;
Paulo Justiniano Ribeiro Jr. paulojus@leg.ufpr.br,
Peter J. Diggle p.diggle@lancaster.ac.uk.
Further information on the package
geoR
can be found at:
http://www.leg.ufpr.br/geoR.
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