# RFformula: RFformula - syntax to design random field models with trend... In RandomFields: Simulation and Analysis of Random Fields

## Description

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.

## Details

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.

## IMPORTANT

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.

## Note

(additional) argument names should always start with a capital letter. Small initial letters are reserved for `RFoptions`.

## References

• 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 ```