RFformula - syntax to design random field models with trend or linear mixed models
It is described how to create a formula, which, for example, can be used as an argument of RFsimulate and
RFfit to simulate and to fit data according to the
model described by the formula.
In general, the created formula serves two purposes:
to describe models in the “Linear Mixed Models”-framework
to define models for random fields including trend surfaces from a geostatistical point of view.
Thereby, fixed effects and trend surfaces can be adressed via
the expression RMfixed and the function
RMtrend. In simple cases, the trend can also
be given in a very simple, see the examples below.
The covariance structures of the zero-mean
multivariate normally distributed
random field
components are adressed by objects of class RMmodel, which
allow for a very flexible covariance specification.
See RFformulaAdvanced for rather complicated model definitions.
The formula should be of the type
response ~ fixed effects %+ random effects + error term
or
response ~ trend + zero-mean random field + nugget effect,
respectively.
Thereby:
response
optional; name of response variable
fixed effects/trend:
optional, should be a sum (using +)
of components either of the form X@RMfixed(beta) or
RMtrend(...) with X being a design matrix
and β being a vector of coefficients (see
RMfixed and RMtrend).
Note that a fixed effect of the form X is interpreted as
X@RMfixed(beta=NA) by default (and β is estimated
provided that the formula is used in RFfit).
error term/nugget effect
optional, should be of the form RMnugget(...).
RMnugget describes a vector
of iid Gaussian random variables.
Note that in formula constants are interpreted as part of a linear
model, i.e. the corresponding parameter has to be estimated
(e.g. ~ 1 + ...) whereas in models not given as formula the
parameters to be estimated must be given explicitly.
(additional) argument names should always start with a capital
letter. Small initial letters are reserved for RFoptions.
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
Chiles, J.-P. and P. Delfiner (1999) Geostatistics. Modeling Spatial Uncertainty. New York, Chichester: John Wiley & Sons.
McCulloch, C. E., Searle, S. R. and Neuhaus, J. M. (2008) Generalized, linear, and mixed models. Hoboken, NJ: John Wiley & Sons.
Ruppert, D. and Wand, M. P. and Carroll, R. J. (2003) Semiparametric regression. Cambridge: Cambridge University Press.
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
RFoptions(modus_operandi="sloppy")
##############################################################
#
# Example : Simulation and fitting of a two-dimensional
# Gaussian random field with exponential covariance function
#
###############################################################
V <- 10
S <- 0.3
M <- 3
model <- RMexp(var=V, scale=S) + M
x <- y <- seq(1, 3, 0.1)
simulated <- RFsimulate(model = model, x=x, y=y)
plot(simulated)
# an alternative code to the above code:
model <- ~ Mean + RMexp(var=Var, scale=Sc)
simulated2 <- RFsimulate(model = model,x=x, y=y, Var=V, Sc=S, Mean=M)
plot(simulated2)
# a third way of specifying the model using the argument 'param'
# the initials of the variables do not be captical letters
model <- ~ M + RMexp(var=var, scale=sc)
simulated3 <- RFsimulate(model = model,x=x, y=y,
param=list(var=V, sc=S, M=M))
plot(simulated3)
# Estimate parameters of underlying covariance function via
# maximum likelihood
model.na <- ~ NA + RMexp(var=NA, scale=NA)
fitted <- RFfit(model=model.na, data=simulated)
# compare sample mean of data with ML estimate, which is very similar:
mean(simulated@data[,1])
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