Spatial models for data on spherical regions. : Geometries to represent 2-d and 3-d spherical data.

Description

These models include a completely spherical geometry based on a nearly regular node layout on the sphere. Also a simpler and perhaps more efficient versions are implemented where the first coordinate is periodic in the interval [0,360] and the remaining coordinates are regular (Euclidean). This might be used to approximate a section of spherical data that excludes the polar caps. These approximations are useful because one can take advantage of faster methods based on rectangular grids rather the more complex grids on a sphere. The disadvantage is that the mapping from these coordinates to the sphere is distorted as one gets close to the poles.

Arguments

Author

Doug Nychka and Zachary Thomas

Details

These geometries are specified with the LKGeometry argument either in LKrigSetup or LatticeKrig. The first coordinate is longitude either from [0,360] or [-180,180] and the second is latitude [-90, 90].

They each have the four specific methods: LKrigLatticeCenters, LKrigSAR, LKrigSetupLattice, setDefaultsLKinfo and the source code is consolidated in the source files ModelRing.R and ModelCylinder.R in the R sub-directory of this package source. For the spherical grid the a.wght is handled a but differently please read the note below.

LKSphere Recall that the core of the LatticeKrig model is the placement of the basis function centers and how the spatial autoregression is constructed from these node locations. Here the centers are generated according to a multi-resolution based on the triangles from an icosahedron. IcosahedronGrid creates a grid by taking the first level as the 12 points from a regular icosahedron. The subsequent levels generate a finer set of points by subdividing each triangular face into 4 new triangles. The three new mid points from the subdivision are added to the previous nodes to give the new level of resolution. The triangles tend to roughly equilateral and so the nodes will tend to be roughly equally spaced. This regularly improves for the finer generations of triangles, however, the 12 original nodes from the icosahedron will have 5 nearest neighbors rather than 6.

The setup argument startingLevel specifies the first level of the lattice and nlevel the total number.

NOTE: Because the distances between nodes are not perfectly equally spaced and nearest neighbors can be either 5 or 6 some adjustment is made to the weights when forming the SAR matrix. The net result is that it makes more sense to have the off diagonal rows sum to 1 and so a.wght must be greater than 1.0 for a stationary model.

See the help file on link{IcosahedronFaces} for code on visualizing these. The first 12 centers will have 5 close neighbors and remaining centers will have 6. Currently the SAR weights are roughly equal among all the neighbors but are adjusted so that a locally linear function will be in the "null" space of these weights. For each node the neighbors are projected onto the tangent plane to the sphere at this node location. Now consider a linear function on the coordinates in this tangent plane. The idea is to find weights applied to the neighbors that will give a perfect linear prediction for the value at the node. The negatives of these values are used as the SAR weights. Specifically let $w_k$ be the weights and $Y_k$ the values of the field at the these locations, and $Y_0$ the value at the node. Then by construction $$ Y_0 - sum w_k Y_k = 0 $$ for any field that is linear in the tangent plane to the node. Specifically in the function LKrigSAR.LKSphere the weights follow the code fragment


		   x1<- grid3d[J,]
       x0<- grid3d[I,]
       u<- projectionSphere( x0,x1) 
    # u are local 2 d coordinates on tangent plane to sphere at x0
    # x0 projects to (0,0)
      X<- cbind( rep( 1,nJ), u )
    # find weights to predict a linear function
    # at node from the nearest neighbors. 
      W<- c( (X)

I is the index of the node at lattice point x0, J is the index of the neighbors at lattice points x1, and -W the weights used in the SAR.

The basis functions have their default as using great circle distance to determine the values between the center and other points. Because the distances between centers are not equal some adjustment is made to the delta parameter for the basis function support. Currently the number of basis functions in each level are

Level	1	2	3	4	5	6	7	8
Number Basis functions	12	42	162	642	2562	10242	40962	163842

So if startingLevel=2 and nlevel=3 there will be a total of 42 + 162 + 642 = 846 basis functions in the model if the spatial domain includes the entire sphere.

LKRing This model follows the Mercator projection for a sphere where longitude and latitude are treated as Euclidean coordinates except that longitude is periodic. So the actual coordinates represent the surface of cylinder which is one way of visualizing the Mercator projection. To keep things simple the first coordinate is essentially hardwired to be in the scale of degrees (sorry for all you fans of radians) and wrapping 0 to 360 ( or periodic in [-180,180]). It is important to scale the second coordinate in this geometry to be comparable in spatial scale to degrees (use the V argument in LKrigSetup). However, if the second coordinate can be interpreted as a latitude it is often reasonable to assume the spatial scales are the same in these two coordinates.

Note this geometry can also be used to represent an equatorial section of a spherical volume. Here the first coordinate is longitude but the second can be interpreted as a radius from the sphere's center. This is a case where care needs to taken to scale the radial component to make sense with the degrees in the first.

LKCylinder This is just the three dimensional extension of LKRing with the first coordinate being periodic in (0,360) and the remaining two treated as Euclidean

The periodicity in the first coordinate is implemented in two places. First in setting up the spatial autoregression (SAR) weights, the weights reflect the wrapping. Second in finding distances between coordinates the nodes in the lattice has the first coordinate tagged as being periodic. Specifically in LKrigSetupLattice the gridList for each lattice has an attribute vector that indicates which coordinates are periodic. This information is used in the distance function LKrigDistance when called with arguments of a matrix and a gridList.

Why is this so complicated? This structure is designed around the fact that one can find the pairwise distance matrix quickly between an arbitrary set of locations and a rectangular grid (a gridList object in this package). The grid points within a delta radius of an arbitrary point can be found by simple arithmetic and indexing. Because these two geometries have regular lattice spacings is it useful to exploit this. See LKrigDistance for more details about the distance function.

Finally, we note that for just patches of the sphere one can use the usual LKRectangle geometry and change the distance function to either chordal or great circle distance. This gives a different approach to dealing with the inherent curvature but will be awkward as the domain is close to the poles.

Examples

Run this code

# 	
# fit the CO2 satellite data with a fixed lambda
# (but use a very small, unrealistic number of basis functions and levels so example
#  runs quickly)
 data(CO2)
# to look at raw data: quilt.plot(CO2$lon.lat, CO2$y, nx=288, ny=165)
# Use two levels starting at the second generation of lattice points from the triangulation
  LKinfo0<- LKrigSetup( CO2$lon.lat, startingLevel=1 ,nlevel=2,
                       a.wght=1.1, alpha=c(1,.25),
                       LKGeometry="LKSphere" )
# Take a look at Model summary
  print( LKinfo0)
  
# Use arbitrary  lambda  (.01) 
  obj0<- LKrig( CO2$lon.lat,CO2$y,LKinfo=LKinfo0, lambda=0.01)
if (FALSE) {
# Surface plot of estimate
  surface(obj0, nx=288, ny=165)
  world( add=TRUE)
}  
if (FALSE) {
data(CO2)
# estimate lambda ( should be around 0.003)
# NOTE: lambda values will tend to be sensitive to the model choice
  LKinfo0<- LKrigSetup( CO2$lon.lat, startingLevel=2 ,nlevel=2,
                       a.wght=1.1, alpha=c(1,.25),
                       LKGeometry="LKSphere") 
  obj0B<-  LatticeKrig( CO2$lon.lat,CO2$y,LKinfo=LKinfo0)
# a more serious model this uses about 3300 basis functions
LKinfo0<- LKrigSetup( CO2$lon.lat, startingLevel=3, ,nlevel=3,
                       a.wght=1.1, alpha=c(1, .5, .25),
                       LKGeometry="LKSphere" )
                       
obj0B<-  LatticeKrig( CO2$lon.lat,CO2$y,LKinfo=LKinfo0)
# takes about 1 minute on a Macbook air
# setting findAwght = TRUE  takes about 8 minutes with 
# lambda = 1.737 and a.wght = 15.8
} 
#####################################
# The ring geometry
#####################################
if (FALSE) {
  data(CO2)
  LKinfo1<- LKrigSetup(CO2$lon.lat, NC=8 ,nlevel=1, lambda=.2,
                       a.wght=5, alpha=1, 
                       LKGeometry="LKRing" )                                         
  obj1<- LatticeKrig( CO2$lon.lat,CO2$y,LKinfo=LKinfo1)	
# take a look: 
surface( obj1)
world( add=TRUE) 
}
# compare to fitting without wrapping:
if (FALSE) {
  LKinfo2<- LKrigSetup(CO2$lon.lat, NC=8 ,nlevel=1,
                   lambda=.2, a.wght=5, alpha=1 )                                         
  obj2<- LKrig( CO2$lon.lat,CO2$y,LKinfo=LKinfo2)	
 # NOTE: not periodic in longitude:
 surface(obj2)  
}

# a synthetic example and larger example
if (FALSE) {
 set.seed(124)
 N<- 1e4
  x0<- matrix( rnorm(3*N), ncol=3)
  x0<- x0/ sqrt( rowSums( x0^2))
  
  x<-  toSphere( x0 )
  
# the true function for testing -- a bump at the direction alpha
  fun<- function(X){
    alpha<-  c( .1,.1,1)
    alpha<- alpha/ sqrt( sum( alpha^2))
    4*( 1 + c(( X)%*%alpha) )^2 
  }
  
  ytrue <- fun(x0)
  y<- ytrue + .05*rnorm( length(ytrue))
# this defines about 3300 basis functions
  LKinfo1<- LKrigSetup( x,
                        startingLevel=3,
                        LKGeometry="LKSphere",
                        a.wght=1.01,
                        nlevel=3, alpha = c(1.0,.5,.25)^2,
                        choleskyMemory=list(nnzR= 20E6),
                        normalize=TRUE)
  out<- LatticeKrig( x,y, LKinfo=LKinfo1, lambda=.01)                      
surface( out)                        
}

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Description

Arguments

Author

Details

See Also

Examples