Description Details IMPORTANT Note References See Also Examples

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 variablefixed 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`

.

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`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | ```
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])
fitted
``` |

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