# Summary of the LKRectangle geometry for a standard two dimensional spatial domain. The basic LatticeKrig model is for a 2-d spatial domain and is identified by the geometry class LKRectangle. The lattices used in this case are equally spaced regular grids that are nested and the default distance function is standard Euclidean distance. The number of lattice points in the first level is NC inside the spatial domain and the subsequent levels decrease the lattice spacing by factors of 2. The spatial autoregressive coefficients, referred to as the a.wght parameter(s) have several levels of detail depending on the stationarity and isotropy. In the simplest case a.wght is the central value, and should be greater than 4. Also the four nearest neighbors are take to be -1. In the most complicated `a.wght` can include all 8 nearest neighbors can be specified differently at every lattice node. To be stable, the sum of the `a.wght` parameters for a node should be greater than zero. ## Details Here is a simple and small three level example that sets up the LatticeKrig model for spatial estimation and prediction. It assumes a spatial domain with extent [-1,1] in the horizontal and [-1,1] in vertical and beginning with 4 grid points in lowest level and with a default of 5 extra points outside the domain for boundary corrections. The assumed SAR model defaults to a central value of 4.1 and the 4 nearest weights are -1. Defining the extent of the spatial domain explicitly was done to simplify the example. Typically one just defines the domain to the minimum and maximum x and y locations of the spatial data (and the locations can passed as the first argument in the example below instead of finding ranges.) Note that does not mean that the spatial locations fill out a rectangle and they do not need to regularly spaced. The `nu` parameter is a handy way to specify the relative weights given each level. For a larger numbers of levels this parameter is equivalent to the Matern smoothness parameter. Setting up the LKinfo object. ``` sDomain<- cbind( c(-1,1), c( -1, 1)) LKinfo<- LKrigSetup(sDomain, nlevel=3, NC=4, a.wght=4.1, nu=1.0) ``` with the result ``` print(LKinfo) Classes for this object are: LKinfo LKRectangle The second class usually will indicate the geometry e.g. 2-d rectangle is LKRectangle Ranges of locations in raw scale: [,1] [,2] [1,] -1 -1 [2,] 1 1 Number of levels: 3 delta scalings: 0.6666667 0.3333333 0.1666667 with an overlap parameter of 2.5 alpha: 0.7619048 0.1904762 0.04761905 based on smoothness nu = 1 a.wght: 4.1 4.1 4.1 Basis type: Radial using WendlandFunction and Euclidean distance. Basis functions will be normalized Total number of basis functions 1014 Level Basis size 1 196 14 14 2 289 17 17 3 529 23 23 Lambda value: NA ``` About the lattice The number of nodes define the number of basis functions and at first may seem a bit mysterious. However at the first level we get 4 lattice points with 5 extra boundary points added in the x direction amounting to 14 total and similarly 14 in the y direction because it is exactly of the same size. The default is the the parameter NC determines the number of lattice points in the larger dimension and the smaller dimension is divided according to the spacing from the longer dimension. For example if the y extent was [-1,.5] the number of lattice points would be spaced according to ``` delta scalings: 0.6666667 0.3333333 0.1666667 ``` Thus `seq( -1,.5, 0.6666667 )` are the (three) generated points within the spatial domain, and of course 5 would also added beyond each endpoint. To query the LKinfo object this information is in the `latticeInfo` component. E.g. the first set of lattice locations. ``` LKinfo$latticeInfo$grid[[1]] $x [1] -4.3333 -3.6667 -3.0000 -2.3333 -1.6667 [6] -1.0000 -0.3333 0.3333 1.0000 1.6667 [11] 2.3333 3.0000 3.6667 4.3333 $y [1] -4.3333 -3.6667 -3.0000 -2.3333 -1.6667 [6] -1.0000 -0.3333 0.3333 1.0000 1.6667 [11] 2.3333 3.0000 3.6667 4.3333 attr(,"class") [1] "gridList" ``` By default equal spacing of the lattice is assumed in the x and y directions. To change this use the optional `V` argument to scale the coordinates. For example, to set 4 lattice points in both dimension for the above example: ``` sDomain2<- cbind( c(-1,1), c( -1, .5)) LKinfo<- LKrigSetup(sDomain2, nlevel=3, NC=4, a.wght=4.1, nu=1.0, V= diag(c(2, 1.5)) ) print(LKinfo$latticeInfo$mx) [,1] [,2] [1,] 14 14 [2,] 17 17 [3,] 23 23 ``` About the basis functions With the lattice points defined the default basis functions are radial Wendland functions centered at each point. The scaling of the basis functions is determined by the `overlap` and the `delta` values. In R code, if the lattice point is x0 at level l then the basis function evaluated at location x1 is ``` basisFunctionValue <- Wendland( rdist( x1,x0)/( delta[l]*overlap)) ``` If V is included then x1 is transformed as `x1%*%solve(V)` before evaluating in the basis function. About ‘a.wght’ For this geometry the basic form of awght is as a list with as many components as levels. However, the LKrigSetup function will reshape a scalar or vector argument in this this format. `a.wght` can take the following forms: Scalar value: In this case the value is used at all lattice points and at all levels in the SAR. Four nearest neighbors are set to -1. List/vector of length ‘nlevel’ In this case separate values for the a.wght will be used for the central SAR value at each level. List of vectors With a list of length `nlevel` and each component is a 9 element vector, the values in the vector correspond to the central lattice point and 8 nearest neighbors with the indexing: ``` 1 4 7 2 5 8 3 6 9 ``` E.g. the 5 element is the center, 3 is the lower left hand corner etc. Non-stationary models This is the nuclear option to handle a completely non-stationary correlation structure! In this case one specifies the models by passing prediction objects for one or more of `sigma2.object`, `alphaObject` or `a.wghtObject`. ## Author(s) Doug Nychka ## See Also `LatticeKrig` `LKGeometry` `LKrig` ## Examples ```r # the grid with only 2 extra boundary points sDomain<- cbind( c(-1,1), c( -1, 1)) LKinfo<- LKrigSetup(sDomain, nlevel=3, NC=4, a.wght=4.1, NC.buffer=2, alpha=c(1,.5,.125) ) LKgrid<- LKinfo$latticeInfo$grid plot( make.surface.grid(LKgrid[[1]]), pch=16, cex=1.5) points( make.surface.grid(LKgrid[[2]]), pch=15, cex=.8, col="red" ) points( make.surface.grid(LKgrid[[3]]), pch="+", col="green" ) rect(sDomain[1,1],sDomain[1,2], sDomain[2,1],sDomain[2,2], lwd=3 ) # basis functions on a grid # this function actually evaluates all of them on the grid. xg<- make.surface.grid( list(x=seq( -2,2,,80), y=seq( -2,2,,80)) ) out<- LKrig.basis( xg, LKinfo) # basis functions 20, 26, 100 and 200 plot( make.surface.grid( LKgrid[[1]] ) , pch=16, cex=.5) rect(sDomain[1,1],sDomain[1,2], sDomain[2,1],sDomain[2,2], lwd=3,border="grey" ) contour( as.surface(xg, out[,20]), col="red1", add=TRUE) contour( as.surface(xg, out[,36]), col="red4", add=TRUE) contour( as.surface(xg, out[,100]), col="blue1", add=TRUE) contour( as.surface(xg, out[,200]), col="blue4", add=TRUE) title( "basis functions 20, 26, 100, 200") ```

LKRectangle function