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.