Revision

**da8174e204c4b3c8aff0fa179a0b53656129ef8e**authored by Martin Schlather on**01 March 2004, 00:00:00 UTC**, committed by Gabor Csardi on**01 March 2004, 00:00:00 UTC****1 parent**3e1677b

mleRF.Rd

```
\name{fitvario}
\alias{mleRF}% obsolete
\alias{fitvario}
\alias{fitvario.default}
\title{LSQ and Maximum Likelihood Estimation
of Random Field Parameters}
\description{
This function estimates arbitrary parameters of
a random field specification.
}
\usage{
fitvario(x, y=NULL, z=NULL, T=NULL, data, model, param,
lower=NULL, upper=NULL, sill=NA, ...)
fitvario.default(x, y=NULL, z=NULL, T=NULL, data, model, param,
lower=NULL, upper=NULL, sill=NA, trend,
use.naturalscaling=TRUE, PrintLevel=RFparameters()$Print,
trace.optim=0, bins=20, nphi=1, ntheta=1, ntime=20,
distance.factor=0.5,
upperbound.scale.factor=10, lowerbound.scale.factor=20,
lowerbound.scale.LS.factor=5,
upperbound.var.factor=10, lowerbound.var.factor=100,
lowerbound.sill=1E-10, scale.max.relative.factor=1000,
minbounddistance=0.001, minboundreldist=0.02,
approximate.functioncalls=50, pch="*", var.name="X",
time.name="T", transform=NULL, standard.style=NULL)
}
\arguments{
\item{x}{(\eqn{n\times2}{n x 2})-matrix of coordinates, or vector of
x-coordinates}
\item{y}{vector of y coordinates}
\item{z}{vector of z coordinates}
\item{T}{vector of T coordinates; these coordinates are given in
triple notation, see \code{\link{GaussRF}}}
\item{data}{vector or matrix of values measured at \code{coord};
If also a time component is given, the in the data the indices for
the spatial components run the fastest.
}
\item{model}{string or list;
covariance model, see \code{\link{CovarianceFct}}, or
type \code{\link{PrintModelList}()} to get all options.
See also t
If \code{model} is a list, then the parameters with value \code{NA}
are estimated. Parameters that have value \code{NaN} should be
explicitely be defined by the function \code{transform}.
An alternative to define \code{NaN} values and
the function \code{transform}, is to replace the \code{NaN}
by a real-valued function with solely parameter a list defining
a covariance model. In case of the anisotropy matrix, the matrix
must be replaced by a list if functions are introduced.
Only the list elements
variance, scale or anisotropy, and kappas can be used, and
not the mean or the trend.
Further, the mean or the trend cannot be set by such a function.
See also \code{transform} below.
}
\item{param}{
vector or matrix or NULL.
If vector then
\code{param=c(mean, variance, nugget, scale,...)};
the parameters must be given
in this order. Further parameters are to be added in case of a
parametrised class of covariance functions, see
\code{\link{CovarianceFct}}.
\emph{Any} components set to \code{NA} are estimated; the others
are kept fix.
See also \code{model} above.
}
\item{lower}{list or vector. Lower bounds for the parameters.
If \code{lower} and \code{param} are vectors and \code{length(lower) <
length(param)}
then \code{lower} must match the number of additional parameters
\eqn{\kappa}{a,b,c,...}.
If \code{param} is matrix the length of \code{lower} must match the
number columns of param or being 2 elements smaller (then
\code{lower} is filled with \code{NA} from the left.
The bounds are equally applied to all rows.
If \code{lower} is a list, then elements that are not given are
considered as \code{NA}.
If \code{lower} is not given, or \code{lower} contains \code{NA}
then the missing bounds are generated automatically.
}
\item{upper}{list or vector. Upper bounds for the parameters.
See also lower.
}
\item{sill}{If not \code{NA} the sill is kept fix. Only used if the
standard format for the covariance model is given. See Details.}
\item{trend}{
Not programmed yet.
May only be set if \code{missing(param)};
linear formula : uses \code{X1}, \code{X2},... and \code{T} as internal
parameters for the coordinates; all parameters are estimated\cr
matrix : must have the same number of rows as \code{x}\cr
\emph{fixed} mean + matrix or linear formula : not possible within
this function (just subtract the mean from your data before calling
this function)
}
\item{...}{arguments as given in \code{mleRF.default} and listed in the
following.}
\item{use.naturalscaling}{
logical. Only used if model is given in standard (simple) way.
If \code{TRUE} then \emph{internally}, rescaled
covariance functions will be used for which
cov(1)\eqn{\approx}{~=}0.05.
\code{use.naturalscaling} has the advantage that \code{scale}
and the form parameters of the model get \sQuote{orthogonal},
but \code{use.naturalscaling} does not work for all models.
See Details.}
\item{PrintLevel}{level to which messages are shown. See Details.}
\item{trace.optim}{tracing of the function \code{\link{optim}}}
\item{bins}{number of bins of the empirical variogram. See Details.}
\item{nphi}{scalar or vector of 2 components.
If it is a vector then the first component gives the first angle
of the xy plane
and the second one gives the number of directions on the half circle.
If scalar then the first angle is assumed to be zero}
\item{ntheta}{scalar or vector of 2 components.
If it is a vector then the first component gives the first angle
in the third direction
and the second one gives the number of directions on the half circle.
If scalar then the first angle is assumed to be zero.}
\item{ntime}{scalar or vector of 2 components.
if \code{ntimes} is a vector, then the first component are the
maximum time distance (in units of the grid length \code{T[3]}) and the
second component gives the step size (in units of the grid length
\code{T[3]}). If scalar then the step size is assumed to 1 (in units
of the grid length \code{T[3]}.
}
\item{distance.factor}{relative right bound for the bins. See Details.}
\item{upperbound.scale.factor}{relative upper bound for scale in LSQ
and MLE.
See Details.}
\item{lowerbound.scale.factor}{relative lower bound for scale in MLE.
See Details.}
\item{lowerbound.scale.LS.factor}{relative lower bound for scale
in LSQ. See Details.}
\item{upperbound.var.factor}{relative upper bound for variance
and nugget. See Details.}
\item{lowerbound.var.factor}{relative lower bound for variance.
See Details.}
\item{lowerbound.sill}{absolute lower bound for variance
and nugget. See Details.}
\item{scale.max.relative.factor}{relative lower bound
for scale below
which an additional nugget effect is detected. See Details.}
\item{minbounddistance}{absolute distance to the bounds
below which a part of the algorithm is considered as
having failed. See Details.}
\item{minboundreldist}{relative distance to the bounds
below which a part of the algorithm is considered as
having failed. See Details.}
\item{approximate.functioncalls}{approximate evaluations
of the ML target function on a grid. See Details.}
\item{pch}{character shown before each step of calculation;
depending on the specification there are two to five steps.
Default: "*".}
\item{var.name}{basic name for the coordinates in the formula of the
trend. Default: \sQuote{X}}
\item{time.name}{basic name for the time component in the formula of the
trend. Default: \sQuote{X}}
\item{transform}{vector of strings.
Essentially, \code{transform} allows for the definition of a parameter as a
function of other estimated or fixed parameters.
All the parameters are supposed to be in a vector called \sQuote{param}
where the positions are given by \code{\link{parampositions}}.
An example of \code{transform} is
\code{function(param) \{param[3] <- 5 - param[1]; param\}}.
Any parameter that is set by \code{transform}, should be \code{NaN}
in the model definition. If it is \code{NA} a warning is given.
Note that the mean and the trend of the model can be neither set nor used in
\code{transform}. See also \code{standard.style}.
Instead of giving \code{transform}, in the model definition,
all NaN values are replaced by functions whose only parameter
is a bare model list, i.e., only the list elements
variance, scale or anisotropy, and kappas can be used, and
not the mean or the trend.
Further, the mean or the trend cannot be set by such a function.
Default: \code{NULL}}
\item{standard.style}{logical or \code{NULL}. This variable should only be
set by the advanced user. If \code{NULL} \code{standard.style} will be
\code{TRUE} if the covariance model allows for a \sQuote{standard}
definition (see \command{\link{convert.to.readable}} and
\command{\link{CovarianceFct}}) and \code{transform} is \code{NULL}.
Otherwise \code{standard.style} will be \code{FALSE}.
If a \\sQuote{standard} definition is given and both the variance and the
nugget are either not estimated or do not appear on the right hand
side of the \code{transform}, then \code{standard.style} might be set
to \code{TRUE} by the user. This accelerates the MLE algorithm.
The responsibility is completely left to the user, then.
Currently \command{mleRF} is only implemented for the
\\sQuote{standard} definition of the covariance model.
Hence \code{standard.style} must always be \code{TRUE} and consequently,
neither the variance nor the nugget may appear on either side
of the \code{transform}
}
}
\details{
The maximisation is performed using \code{\link{optim}}. Since
\code{\link{optim}}
needs as input parameter an initial vector of parameters, \code{mleRF}
takes the initial parameter from the LSQ estimation.
If the best parameter vector of the MLE found so far is too close
to some given bounds, see the specific parameters below, it is
assumed that
\code{\link{optim}} ran into a local minimum because of a bad starting
value.
In this case the MLE target function is calculated on a grid, the
best parameter vector is taken, and the optimisation is restarted with
this parameter vector.
Comments on specific parameters:
\itemize{
\item \code{lower}\cr
The lower bounds are technical bounds that should not
restrict really restrict the domaine of the value.
However, if these values are too small the optimisation
algorithm will frequently run into local minima or get
stuck close the border of the parameter domain.
It is advised to limit seriously the domain of the additional
parameters of the covariance model and/or the total number of
parameters to be estimated, if \dQuote{many} parameters
of the covariance model are estimated.
If the model is given in standard form, the user may supply
the lower bounds for the whole parameter vector, or only for
the additional form parameters of the model.
The lower bound for the mean will be ignored.
\code{lower} may contain \code{NA}s, then these values
are generated by the
If a nested model is given, the bounds may again be supplied
for all parameters or only for the additional form parameters
of the model. The bounds given apply uniformely to all
submodels of the nested model.
If the \code{model} is given in list format, then
\code{lower} is a list, where components may be missing
or \code{NA}. These are generated by the algorithm, then.
If \code{lower} is \code{NULL} all lower bounds are generated
automatically.
\item \code{upper.kappa}\cr
See \code{lower.kappa}.
\item \code{sill}\cr
Additionally to estimating \code{nugget} and \code{variance}
separately, they may also be estimated together under the
condition that \code{nugget} + \code{variance} = \code{sill}.
For the latter a finite value for \code{sill} has to be supplied,
and \code{nugget} and \code{variance} are set to \code{NA}.
\code{sill} is only used for the standard model.
}\itemize{
\item \code{use.naturalscaling}\cr
logical. If \code{TRUE} then internally, rescaled
covariance functions will be used for which
cov(1)\eqn{\approx}{~=}0.05. However this parameter
does not influence
the output of \code{mleRF}: the parameter vector
returned by \code{mleRF} refers
\emph{always} to the standard covariance model as given in
\code{\link{CovarianceFct}}. (In contrast to \code{PracticalRange}
in \code{\link{RFparameters}}.)\cr
Advantages if \code{use.naturalscaling==TRUE}:
\itemize{
\item \code{scale} and the shape parameter of a parameterised
covariance model can be estimated better if they are estimated
simultaneously.
\item The estimated bounds calculated by means of
\code{upperbound.scale.factor} and \code{lowerbound.scale.factor},
etc. might be more realistic.
\item in case of anisotropic models, the inverse of the elements
of the anisotropy matrix should be in the above bounds.
}
Disadvantages if \code{use.naturalscaling==TRUE}:
\itemize{
\item For some covariance models with additional parameters, the
rescaling factor has to be determined numerically.
Then, more time is needed to perform \code{mleRF}.
}
Default: \code{TRUE}.
\item \code{PrintLevel}\cr
0 : no message\cr
1 : error messages\cr
2 : warnings\cr
3 : minimum debugging information\cr
5 : extended debugging information, including graphics\cr
Default: \code{0}.
}\itemize{
\item \code{trace.optim}\cr
see control parameter \code{trace} of
\code{\link{optim}}. Default: \code{0}.
\item \code{bins}\cr
vector of explicit boundaries for the bins or the
number of bins for the empirical variogram (used in the
LSQ target function, which is described at the beginning
of the Details). Default: \code{20}.
\item \code{distance.factor}\cr
right boundary of the last
bin is calculated as \code{distance.factor} * (maximum distance
between all pairs of points). Only used if \code{bins} is a scalar.
Default: \code{0.5}.
\item \code{upperbound.scale.factor}\cr
The upper bound for the scale is determined
as \code{upperbound.scale.factor} * (maximum distance
between all pairs of points).
Default: \code{10}.
\item \code{lowerbound.scale.factor}\cr
The lower bound for the scale is determined
as
\eqn{\hbox{(minimum distance
between different pairs of points)} /
\code{lowerbound.scale.factor}}{(minimum distance
between different pairs of points) /
\code{lowerbound.scale.factor}}.
Default: \code{20}.\cr
}\itemize{
\item \code{lowerbound.scale.LS.factor}\cr
For the LSQ target function a different lower bound
for the scale is used. It is determined as
\eqn{\hbox{(minimum distance
between different pairs of points)} /
\code{lowerbound.scale.LS.factor}}{(minimum distance
between different pairs of points) /
\code{lowerbound.scale.LS.factor}}.
Default: \code{5}.\cr
\item \code{upperbound.var.factor}\cr
The upper bound for the variance and the nugget is determined
as
\deqn{\code{upperbound.var.factor} *
{\hbox{var}}(\code{data}).}{\code{upperbound.var.factor} *
var(\code{data}).}
Default: \code{10}.
}\itemize{
\item \code{lowerbound.var.factor}\cr
The lower bound for the variance and the nugget is determined
as
\deqn{\hbox{var}(\code{data}) /
\code{lowerbound.var.factor}.}{var(\code{data}) /
\code{lowerbound.var.factor}.}
If a standard model definition is given and
either the nugget or the variance is fixed,
the parameter to be estimated
must also be greater than \code{lowerbound.sill}.
If a non-standard model definition is given
then \code{lowerbound.var.factor} is only used
for the first model; the other lower bounds for the
variance are zero.
Default: \code{100}.
\item \code{lowerbound.sill}\cr
See \code{lowerbound.var.factor}.
Default: \code{1E-10}.
\item \code{scale.max.relative.factor}\cr
If the initial scale value for the ML estimation
obtained by the LSQ target function is
less than
\eqn{\hbox{(minimum distance
between different pairs of points)} /
\code{scale.max.relative.factor}}{ (minimum distance
between different pairs of points) /
\code{scale.max.relative.factor}}
it is assumed that a nugget effect
is present. In case the user set \code{nugget=0},
the ML estimation is automatically performed
for \code{nugget=NA} instead of \code{nugget=0}.
Note: if \code{scale.max.relative.factor} is greater
than \code{lowerbound.scale.LS.factor} then
\code{nugget} is never set to \code{NA} as
the scale has the lower bound (minimum distance
between different pairs of points) /
\code{lowerbound.scale.LS.factor}.
Default: \code{1000}.
}\itemize{
\item \code{minbounddistance}\cr
If any value of the parameter vector
returned from the ML estimation
is closer than \code{minbounddistance}
to any of the bounds or if any value
has a relative distance smaller than
\code{minboundreldist}, then it is assumed that
the MLE algorithm has dropped into a local minimum,
and it will be continued with evaluating the
ML target function on a grid, cf. the beginning paragraphs
of the Details.
Default: \code{0.001}.
\item \code{minboundreldist}\cr
See \code{minbounddistance}.
Default: \code{0.02}.
\item \code{approximate.functioncalls}\cr
In case the parameter vector is too close to the given
bounds, the ML target function is evaluated on a grid
to get a new initial value for the ML estimation.
The number of points of the grid is approximately
\code{approximate.functioncalls}.
Default: \code{50}.
}
Another maximum likelihood estimator for random fields
exists as part of the package \code{geoR} whose homepage
is at \url{http://www.maths.lancs.ac.uk/~ribeiro/geoR.html},
with a different philosophy behind.
}
\value{
the function returns a list with the following elements
\item{mle.value}: the log-likelihood for the fitted parameters
\item{trend.coff}{parameters for linear trend (optional)}
\item{mle}{fitted model}
\item{ev}{list returned by \code{\link{EmpiricalVariogram}}}
\item{lsq}{model fitted by least squares; trends are never taken into account}
\item{nlsq}{weighted lsq. Weight is the square root of the number
of points in the bin}
\item{slsq}{weighted lsq. Weight is the inverse the standard deviation of the
variogram cloud within the bin}
\item{flsq}{weighted lsq. Weights are the values of the fitted variogram
to the power of -2}
\code{mle.lower}{lower bounds for the parameters used in the
optimisation algorithm}
\code{mle.upper}{upper bounds for the parameters used in the
optimisation algorithm}
}
\references{
Ribeiro, P. and Diggle, P. (2001) Software for geostatistical analysis
using R and S-PLUS: geoR and geoS, version 0.6.15.
\url{http://www.maths.lancs.ac.uk/~ribeiro/geoR.html}.
}
\author{Martin Schlather, \email{martin.schlather@cu.lu}
\url{http://www.cu.lu/~schlathe}}
\note{This function does not depend on the value of
\code{\link{RFparameters}()$PracticalRange}.
The function \code{mleRF} always uses the standard specification
of the covariance model as given in \code{\link{CovarianceFct}}.
% The function is still in an experimental stage (and will probably
% never leave it). The function works well if
% only a few components of the parameter vector are to be estimated.
}
\section{Acknowledgement}{
Thanks to Paulo Ribeiro for hints and comparing \code{mleRF} to
\code{\link[geoR]{likfit}} of the package \code{geoR} whose homepage is at
\url{http://www.est.ufpr.br/geoR/}.
}
\seealso{
\code{\link{CovarianceFct}},
\code{\link{GetPracticalRange}},
\command{\link{parampositions}}
\code{\link{RandomFields}},
% \code{\link{winddata}}.
}
\examples{
% library(RandomFields, lib="~/TMP"); source("~/article/R/NEW.RF/RandomFields/R/MLE.R");
model <-"gencauchy"
param <- c(0, 1, 0, 1, 1, 2)
estparam <- c(0, NA, 0, NA, NA, 2) ## NA means: "to be estimated"
## sequence in `estparam' is
## mean, variance, nugget, scale, (+ further model parameters)
## So, mean, variance, and scale will be estimated here.
## Nugget is fixed and equals zero.
points <- 100
x <- runif(points,0,3)
y <- runif(points,0,3) ## 100 random points in square [0, 3]^2
d <- GaussRF(x=x, y=y, grid=FALSE, model=model, param=param, n=10)
str(fitvario(x=cbind(x,y), data=d, model=model, param=estparam,
lower=c(0.1, 0.1), upper=c(1.9, 5)))
## The next two estimations give about the same result.
## For the first the sill is fixed to 1.5. For the second the sill
## is reached if the estimated variance is smaller than 1.5
estparam <- c(0, NA, NA, NA, NA, NA)
str(fitvario(x=cbind(x,y), data=d, model=model, param=estparam, sill=1.5))
estparam <- c(0, NA, NaN, NA, NA, NA)
parampositions(model=model, param=estparam)
f <- function(param) {
param[5] <- max(0, 1.5 - param[1])
return(param)
}
str(fitvario(x=cbind(x,y), data=d, model=model, param=estparam,
sill=1, transform=f))
## the next call gives a warning, since the user may programme
## strange things in this setup, and the program cannot check it.
estparam <- c(0, NA, NA, NA, NA, NaN)
parampositions(model=model, param=estparam)
f <- function(param) {param[3] <- param[2]; param}
unix.time(str(fitvario(x=cbind(x,y), data=d, model=model,
param=estparam, transform=f, standard.style=TRUE), vec.len=6))
## much better programmed, but also much slower:
estmodel <- list(list(model="gencauchy", var=NA, scale=NA,
kappa=list(NA, function(m) m[[1]]$kappa[1])))
unix.time(str(fitvario(x=cbind(x,y), data=d, model=estmodel),
vec.len=6))
%#################################################################
%## ##
%## for a further, more complicated example, see data(winddata) ##
%## ##
%#################################################################
%
% check space time extension in MLE\n add examples\n add test
}
\keyword{spatial}
% LocalWords: fitvario mleRF LSQ param NA naturalscaling PrintLevel optim pch
% LocalWords: RFparameters nphi ntheta ntime lowerbound minbounddistance var
% LocalWords: minboundreldist functioncalls eqn coord CovarianceFct emph cr
% LocalWords: PrintModelList cov variogram uote parampositions NaN kappas MLE
```

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