| RFformula {RandomFields} | R Documentation |
It is described how to create a formula, which can
e.g. be used as an argument of RFsimulate and
RFfit to simulate and to fit data accordingly to the
model described by the formula.
In general, the created formula serves two purposes:
to describe models in the “Linear Mixed Models”-framework including fixed and random effects
to define models for random fields including trend surfaces from a geostatistical point of view.
Thereby, fixed effects and trend surfaces are adressed via
the expression RMfixed and the function
RMtrend; the covariance structures of the zero-mean
multivariate normally distributed random effects and random field
components are adressed by objects of class RMmodel, which
allow for a very flexible covariance specification.
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).
random effects/zero-mean random field:
optional, should be a sum (using +)
of components of the form Z@model
where Z is a design matrix and model is an object of
class RMmodel.
Z@model describes a vector of random effects which is
normally distributed with zero mean and covariance matrix Z
Σ Z^T where Z^T is the transpose of Z and
Σ is the covariance matrix according to model.
Note that a random effect/random fluctuation of the form
model is viewed as I@model where I is the
identity matrix of corresponding dimension.
error term/nugget effect
optional, should be of the form RMnugget(...).
RMnugget describes a vector
of iid Gaussian random variables.
Please note that the character “@” in the RFformula-context can only be used to multiply design-matrices with corresponding vectors of fixed or random effects, whereas in the context of S4-classes “@” is used to access slots of corresponding objects.
Note that in formula constants are interpreted as part of a mixed
model, i.e. the corresponding parameter has to be estimated
(e.g. ~ 1 + ...) whereas models not given as formula the
parameters to be estimated must be given explicitely.
(additional) argument names should always start with a capital
letter. Small initial letters are reserved for RFoptions.
Martin Schlather, schlather@math.uni-mannheim.de
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.
RMmodel,
RFsimulate,
RFfit,
RandomFields.
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:
simulated2 <- RFsimulate(model = ~ 1@RMfixed(beta=M) +
RMexp(var=V, scale=S),x=x, y=y, V=V, S=S, M=M)
plot(simulated2)
# Estimate parameters of underlying covariance function via
# maximum likelihood
model.na <- ~ 1@RMfixed(beta=NA) + RMexp(var=NA, scale=NA)
fitted <- RFfit(model=model.na, data=simulated)
# compare sample mean of data with ML estimate:
mean(simulated@data[,1])
fitted