PeriodicGeometry: Geometries to represent 2-d and 3-d spherical data.
In LatticeKrig: Multi-Resolution Kriging Based on Markov Random Fields
Description
Details
Author(s)
See Also
Examples
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.
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
1
2
3
4
5
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7
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9
10
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.
Author(s)
Doug Nychka and Zachary Thomas
See Also
LatticeKrig,
LKrigSetup,
LKrigSAR,
LKrigLatticeCenters
Examples
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#
# 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$y,LKinfo=LKinfo0, lambda=0.01)
## Not run:
# Surface plot of estimate
surface(obj0, nx=288, ny=165)
world( add=TRUE)
## End(Not run)
## Not run:
data(CO2)
# estimate lambda ( should be around 0.003)
# NOTE: lambda values will tend to be sensitive to the model choice
LKinfo0<- LKrigSetup( CO2$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$y,LKinfo=LKinfo0)
# takes about 1 minute on a Macbook air
# setting findAwght = TRUE takes about 8 minutes with
# lambda = 1.737 and a.wght = 15.8
## End(Not run)
#####################################
# The ring geometry
#####################################
## Not run:
data(CO2)
LKinfo1<- LKrigSetup(CO2$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)
## End(Not run)
# compare to fitting without wrapping:
## Not run:
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)
## End(Not run)
# a synthetic example and larger example
## Not run:
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)
## End(Not run)
LatticeKrig documentation built on Nov. 9, 2019, 5:07 p.m.
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Description Details Author(s) See Also Examples
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.
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
1 2 3 4 5 6 7 8 9 10 | 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.
Author(s)
Doug Nychka and Zachary Thomas
See Also
LatticeKrig,
LKrigSetup,
LKrigSAR,
LKrigLatticeCenters
Examples
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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | #
# 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)
## Not run:
# Surface plot of estimate
surface(obj0, nx=288, ny=165)
world( add=TRUE)
## End(Not run)
## Not run:
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
## End(Not run)
#####################################
# The ring geometry
#####################################
## Not run:
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$y,LKinfo=LKinfo1)
# take a look:
surface( obj1)
world( add=TRUE)
## End(Not run)
# compare to fitting without wrapping:
## Not run:
LKinfo2<- LKrigSetup(CO2$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)
## End(Not run)
# a synthetic example and larger example
## Not run:
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)
## End(Not run)
|
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