fitvario {RandomFields}R Documentation

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

x (n x 2)-matrix of coordinates, or vector of x-coordinates
y vector of y coordinates
z vector of z coordinates
T vector of T coordinates; these coordinates are given in triple notation, see GaussRF
data vector or matrix of values measured at coord; If also a time component is given, the in the data the indices for the spatial components run the fastest.
model string or list; covariance model, see CovarianceFct, or type PrintModelList() to get all options. See also t
If model is a list, then the parameters with value NA are estimated. Parameters that have value NaN should be explicitely be defined by the function transform. An alternative to define NaN values and the function transform, is to replace the 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 transform below.
param vector or matrix or NULL. If vector then 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 CovarianceFct. Any components set to NA are estimated; the others are kept fix. See also model above.
lower list or vector. Lower bounds for the parameters. If lower and param are vectors and length(lower) < length(param) then lower must match the number of additional parameters a,b,c,....
If param is matrix the length of lower must match the number columns of param or being 2 elements smaller (then lower is filled with NA from the left. The bounds are equally applied to all rows.
If lower is a list, then elements that are not given are considered as NA.
If lower is not given, or lower contains NA then the missing bounds are generated automatically.
upper list or vector. Upper bounds for the parameters. See also lower.
sill If not NA the sill is kept fix. Only used if the standard format for the covariance model is given. See Details.
trend Not programmed yet. May only be set if missing(param); linear formula : uses X1, X2,... and T as internal parameters for the coordinates; all parameters are estimated
matrix : must have the same number of rows as x
fixed mean + matrix or linear formula : not possible within this function (just subtract the mean from your data before calling this function)
... arguments as given in mleRF.default and listed in the following.
use.naturalscaling logical. Only used if model is given in standard (simple) way. If TRUE then internally, rescaled covariance functions will be used for which cov(1)~=0.05. use.naturalscaling has the advantage that scale and the form parameters of the model get ‘orthogonal’, but use.naturalscaling does not work for all models. See Details.
PrintLevel level to which messages are shown. See Details.
trace.optim tracing of the function optim
bins number of bins of the empirical variogram. See Details.
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
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.
ntime scalar or vector of 2 components. if ntimes is a vector, then the first component are the maximum time distance (in units of the grid length T[3]) and the second component gives the step size (in units of the grid length T[3]). If scalar then the step size is assumed to 1 (in units of the grid length T[3].
distance.factor relative right bound for the bins. See Details.
upperbound.scale.factor relative upper bound for scale in LSQ and MLE. See Details.
lowerbound.scale.factor relative lower bound for scale in MLE. See Details.
lowerbound.scale.LS.factor relative lower bound for scale in LSQ. See Details.
upperbound.var.factor relative upper bound for variance and nugget. See Details.
lowerbound.var.factor relative lower bound for variance. See Details.
lowerbound.sill absolute lower bound for variance and nugget. See Details.
scale.max.relative.factor relative lower bound for scale below which an additional nugget effect is detected. See Details.
minbounddistance absolute distance to the bounds below which a part of the algorithm is considered as having failed. See Details.
minboundreldist relative distance to the bounds below which a part of the algorithm is considered as having failed. See Details.
approximate.functioncalls approximate evaluations of the ML target function on a grid. See Details.
pch character shown before each step of calculation; depending on the specification there are two to five steps. Default: "*".
var.name basic name for the coordinates in the formula of the trend. Default: ‘X’
time.name basic name for the time component in the formula of the trend. Default: ‘X’
transform vector of strings. Essentially, 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 ‘param’ where the positions are given by parampositions. An example of transform is function(param) {param[3] <- 5 - param[1]; param}. Any parameter that is set by transform, should be NaN in the model definition. If it is NA a warning is given. Note that the mean and the trend of the model can be neither set nor used in transform. See also standard.style.
Instead of giving 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: NULL
standard.style logical or NULL. This variable should only be set by the advanced user. If NULL standard.style will be TRUE if the covariance model allows for a ‘standard’ definition (see convert.to.readable and CovarianceFct) and transform is NULL. Otherwise standard.style will be FALSE. If a ‘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 transform, then standard.style might be set to TRUE by the user. This accelerates the MLE algorithm. The responsibility is completely left to the user, then.
Currently mleRF is only implemented for the ‘standard’ definition of the covariance model. Hence standard.style must always be TRUE and consequently, neither the variance nor the nugget may appear on either side of the transform

Details

The maximisation is performed using optim. Since optim needs as input parameter an initial vector of parameters, 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 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:

Another maximum likelihood estimator for random fields exists as part of the package geoR whose homepage is at http://www.maths.lancs.ac.uk/~ribeiro/geoR.html, with a different philosophy behind.

Value

the function returns a list with the following elements

mle.value
trend.coff parameters for linear trend (optional)
mle fitted model
ev list returned by EmpiricalVariogram
lsq model fitted by least squares; trends are never taken into account
nlsq weighted lsq. Weight is the square root of the number of points in the bin
slsq weighted lsq. Weight is the inverse the standard deviation of the variogram cloud within the bin
flsq weighted lsq. Weights are the values of the fitted variogram to the power of -2

mle.lower{lower bounds for the parameters used in the optimisation algorithm} mle.upper{upper bounds for the parameters used in the optimisation algorithm}

Acknowledgement

Thanks to Paulo Ribeiro for hints and comparing mleRF to likfit of the package geoR whose homepage is at http://www.est.ufpr.br/geoR/.

Note

This function does not depend on the value of RFparameters()$PracticalRange. The function mleRF always uses the standard specification of the covariance model as given in CovarianceFct.

Author(s)

Martin Schlather, martin.schlather@cu.lu http://www.cu.lu/~schlathe

References

Ribeiro, P. and Diggle, P. (2001) Software for geostatistical analysis using R and S-PLUS: geoR and geoS, version 0.6.15. http://www.maths.lancs.ac.uk/~ribeiro/geoR.html.

See Also

CovarianceFct, GetPracticalRange, parampositions RandomFields,

Examples


 RFparameters(Print=10)

 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))



[Package RandomFields version 1.1.24 Index]