# Simulation of Gaussian Random Fields `grf()` generates (unconditional) simulations of Gaussian random fields for given covariance parameters. ```r grf(n, grid = "irreg", nx, ny, xlims = c(0, 1), ylims = c(0, 1), borders, nsim = 1, cov.model = "matern", cov.pars = stop("missing covariance parameters sigmasq and phi"), kappa = 0.5, nugget = 0, lambda = 1, aniso.pars, mean = 0, method, messages) ``` ## Arguments - `n`: number of points (spatial locations) in each simulations. - `grid`: optional. An $n x 2$ matrix with coordinates of the simulated data. - `nx`: optional. Number of points in the X direction. - `ny`: optional. Number of points in the Y direction. - `xlims`: optional. Limits of the area in the X direction. Defaults to $[0,1]$. - `ylims`: optional. Limits of the area in the Y direction. Defaults to $[0,1]$. - `borders`: optional. Typically a two coluns matrix especifying a polygon. See DETAILS below. - `nsim`: Number of simulations. Defaults to 1. - `cov.model`: correlation function. See `cov.spatial` for further details. Defaults to the **exponential** model. - `cov.pars`: a vector with 2 elements or an $n x 2$ matrix with values of the covariance parameters $sigma^2$ (partial sill) and $phi$ (range parameter). If a vector, the elements are the values of $sigma^2$ and $phi$, respectively. If a matrix, corresponding to a model with several structures, the values of $sigma^2$ are in the first column and the values of $phi$ are in the second. - `kappa`: additional smoothness parameter required only for the following correlation functions: `"matern"`, `"powered.exponential"`, `"cauchy"` and `"gneiting.matern"`. More details on the documentation for the function `cov.spatial`. - `nugget`: the value of the nugget effect parameter $tau^2$. - `lambda`: value of the Box-Cox transformation parameter. The value $lambda = 1$ corresponds to no transformation, the default. For any other value of $lambda$ Gaussian data is simulated and then transformed. - `aniso.pars`: geometric anisotropy parameters. By default an isotropic field is assumed and this argument is ignored. If a vector with 2 values is provided, with values for the anisotropy angle $psi_A$ (in radians) and anisotropy ratio $psi_A$, the coordinates are transformed, the simulation is performed on the isotropic (transformed) space and then the coordinates are back-transformed such that the resulting field is anisotropic. Coordinates transformation is performed by the function `coords.aniso`. - `mean`: a numerical vector, scalar or the same length of the data to be simulated. Defaults to zero. - `method`: simulation method with options for `"cholesky"`, `"svd"`, `"eigen"`. Defaults to the **Cholesky** decomposition. See section `DETAILS` below. - `messages`: logical, indicating whether or not status messages are printed on the screen (or output device) while the function is running. Defaults to `TRUE`. ## Details For the methods `"cholesky"`, `"svd"` and `"eigen"` the simulation consists of multiplying a vector of standardized normal deviates by a square root of the covariance matrix. The square root of a matrix is not uniquely defined. These three methods differs in the way they compute the square root of the (positive definite) covariance matrix. The argument `borders`, if provides takes a polygon data set following argument `poly` for the `splancs`' function `csr`, in case of `grid="reg"` or `gridpts`, in case of `grid="irreg"`. For the latter the simulation will have **approximately** n points. ## Returns `grf` returns a list with the components: - **coords**: an $n x 2$ matrix with the coordinates of the simulated data. - **data**: a vector (if `nsim = 1`) or a matrix with the simulated values. For the latter each column corresponds to one simulation. - **cov.model**: a string with the name of the correlation function. - **nugget**: the value of the nugget parameter. - **cov.pars**: a vector with the values of $sigma^2$ and $phi$, respectively. - **kappa**: value of the parameter $kappa$. - **lambda**: value of the Box-Cox transformation parameter $lambda$. - **aniso.pars**: a vector with values of the anisotropy parameters, if provided in the function call. - **method**: a string with the name of the simulation method used. - **sim.dim**: a string "1d" or "2d" indicating the spatial dimension of the simulation. - **.Random.seed**: the random seed by the time the function was called. - **messages**: messages produced by the function describing the simulation. - **call**: the function call. ## References Wood, A.T.A. and Chan, G. (1994) Simulation of stationary Gaussian process in $[0,1]^d$. **Journal of Computatinal and Graphical Statistics**, 3 , 409--432. Schlather, M. (1999) **Introduction to positive definite functions and to unconditional simulation of random fields**. Tech. Report ST--99--10, Dept Maths and Stats, Lancaster University. Schlather, M. (2001) **Simulation and Analysis of Random Fields**. R-News 1 (2), p. 18-20. Further information on the package `geoR` can be found at: [http://www.leg.ufpr.br/geoR/](http://www.leg.ufpr.br/geoR/). ## Author(s) Paulo Justiniano Ribeiro Jr. paulojus@leg.ufpr.br , Peter J. Diggle p.diggle@lancaster.ac.uk . ## See Also `plot.grf` and `image.grf` for graphical output, `coords.aniso` for anisotropy coordinates transformation and `chol`, `svd` and `eigen` for methods of matrix decomposition. ## Examples ```r sim1 <- grf(100, cov.pars = c(1, .25)) # a display of simulated locations and values points(sim1) # empirical and theoretical variograms plot(sim1) ## alternative way plot(variog(sim1, max.dist=1)) lines.variomodel(sim1) # # a "smallish" simulation sim2 <- grf(441, grid = "reg", cov.pars = c(1, .25)) image(sim2) ## ## 1-D simulations using the same seed and different noise/signal ratios ## set.seed(234) sim11 <- grf(100, ny=1, cov.pars=c(1, 0.25), nug=0) set.seed(234) sim12 <- grf(100, ny=1, cov.pars=c(0.75, 0.25), nug=0.25) set.seed(234) sim13 <- grf(100, ny=1, cov.pars=c(0.5, 0.25), nug=0.5) ## par.ori <- par(no.readonly = TRUE) par(mfrow=c(3,1), mar=c(3,3,.5,.5)) yl <- range(c(sim11$data, sim12$data, sim13$data)) image(sim11, type="l", ylim=yl) image(sim12, type="l", ylim=yl) image(sim13, type="l", ylim=yl) par(par.ori) ## simulating within borders data(parana) pr1 <- grf(100, cov.pars=c(200, 40), borders=parana$borders, mean=500) points(pr1) pr1 <- grf(100, grid="reg", cov.pars=c(200, 40), borders=parana$borders) points(pr1) pr1 <- grf(100, grid="reg", nx=10, ny=5, cov.pars=c(200, 40), borders=parana$borders) points(pr1) ```

grf function