# "micro Krig" Spatial process estimate of a curve or surface, "kriging" with a known covariance function.

### Description

This is a simple version of the Krig function that is optimized for large data sets and a clear exposition of the computations. Lambda, the smoothing parameter must be fixed.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
mKrig(x, y, weights=rep(1, nrow(x)), cov.function="stationary.cov",
cov.args = NULL, Z = NULL, lambda = 0, m = 2,
chol.args = NULL, find.trA = TRUE, NtrA = 20,
iseed = 123, llambda = NULL, ...)
## S3 method for class 'mKrig'
predict( object, xnew=NULL,ynew=NULL, grid.list = NULL, derivative=0, Z=NULL,
drop.Z=FALSE,just.fixed=FALSE, ...)
## S3 method for class 'mKrig'
summary(object, ...)
## S3 method for class 'mKrig'
print( x, digits=4,... )
mKrig.coef(object, y)
mKrig.trace( object, iseed, NtrA)
``` |

### Arguments

`x` |
Matrix of unique spatial locations (or in print or surface the returned mKrig object.) |

`y` |
Vector or matrix of observations at spatial locations, missing values are not allowed! Or in mKrig.coef a new vector of observations. If y is a matrix the columns are assumed to be independent observations vectors generated from the same covariance and measurment error model. |

`weights` |
Precision ( 1/variance) of each observation |

`Z` |
Linear covariates to be included in fixed part of the
model that are distinct from the default low order
polynomial in |

`drop.Z` |
If true the fixed part will only be evaluated at the polynomial part of the fixed model. The contribution from the other covariates will be omitted. |

`lambda` |
Smoothing parameter or equivalently the ratio between the nugget and process varainces. |

`llambda` |
If not |

`cov.function` |
The name, a text string of the covariance function. |

`m` |
The degree of the polynomial used in teh fixed part is (m-1) |

`chol.args` |
A list of optional arguments (pivot, nnzR) that will be used with the call to the cholesky decomposition. Pivoting is done by default to make use of sparse matrices when they are generated. This argument is useful in some cases for sparse covariance functions to reset the memory parameter nnzR. (See example below.) |

`cov.args` |
A list of optional arguments that will be used in calls to the covariance function. |

`find.trA` |
If TRUE will estimate the effective degrees of freedom using a simple Monte Carlo method. This will add to the computational burden by approximately NtrA solutions of the linear system but the cholesky decomposition is reused. |

`grid.list` |
A grid.list to evaluate the surface in place of specifying arbitrary locations. |

`NtrA` |
Number of Monte Carlo samples for the trace. But if NtrA is greater than or equal to the number of observations the trace is computed exactly. |

`iseed` |
Random seed ( using |

`...` |
In mKrig and predict additional arguments that will be passed to the covariance function. |

`object` |
Object returned by mKrig. (Same as "x" in the print function.) |

`xnew` |
Locations for predictions. |

`ynew` |
New observation vector. |

`derivative` |
If zero the surface will be evaluated. If not zero the matrix of partial derivatives will be computed. |

`just.fixed` |
If TRUE only the predictions for the fixed part of the model will be evaluted. |

`digits` |
Number of significant digits used in printed output. |

### Details

This function is an abridged version of Krig. The m stand for micro and this function focuses on the computations in Krig.engine.fixed done for a fixed lambda parameter, for unique spatial locations and for data without missing values.

These restrictions simplify the code for reading. Note that also
little checking is done and the spatial locations are not transformed
before the estimation. Because most of the operations are linear
algebra this code has been written to handle multiple data
sets. Specifically if the spatial model is the same except for
different observed values (the y's), one can pass `y`

as a matrix
and the computations are done efficiently for each set. Note that
this is not a multivariate spatial model just an efficient computation
over several data vectors without explicit looping.A big difference in
the computations is that an exact expression for thetrace of the
smoothing matrix is (trace A(lambda)) is computationally expensive and
a Monte Carlo approximation is supplied instead.

See `predictSE.mKrig`

for prediction standard errors and
`sim.mKrig.approx`

to quantify the uncertainty in the estimated function using conditional
simulation.

`predict.mKrig`

will evaluate the derivatives of the estimated
function if derivatives are supported in the covariance function. For
example the wendland.cov function supports derivatives.

`print.mKrig`

is a simple summary function for the object.

`mKrig.coef`

finds the "d" and "c" coefficients represent the
solution using the previous cholesky decomposition for a new data
vector. This is used in computing the prediction standard error in
predictSE.mKrig and can also be used to evalute the estimate
efficiently at new vectors of observations provided the locations and
covariance remain fixed.

Sparse matrix methods are handled through overloading the usual linear
algebra functions with sparse versions. But to take advantage of some
additional options in the sparse methods the list argument chol.args
is a device for changing some default values. The most important of
these is `nnzR`

, the number of nonzero elements anticipated in
the Cholesky factorization of the postive definite linear system used
to solve for the basis coefficients. The sparse of this system is
essentially the same as the covariance matrix evalauted at the
observed locations. As an example of resetting `nzR`

to 450000
one would use the following argument for chol.args in mKrig:

` chol.args=list(pivot=TRUE,memory=list(nnzR= 450000))`

`mKrig.trace`

This is an internal function called by `mKrig`

to estimate the effective degrees of freedom. The Kriging surface
estimate at the data locations is a linear function of the data and
can be represented as A(lambda)y. The trace of A is one useful
measure of the effective degrees of freedom used in the surface
representation. In particular this figures into the GCV estimate of
the smoothing parameter. It is computationally intensive to find the
trace explicitly but there is a simple Monte Carlo estimate that is
often very useful. If E is a vector of iid N(0,1) random variables
then the trace of A is the expected value of t(E)AE. Note that AE is
simply predicting a surface at the data location using the synthetic
observation vector E. This is done for `NtrA`

independent N(0,1)
vectors and the mean and standard deviation are reported in the
`mKrig`

summary. Typically as the number of observations is
increased this estimate becomse more accurate. If NtrA is as large as
the number of observations (`np`

) then the algorithm switches to
finding the trace exactly based on applying A to `np`

unit
vectors.

### Value

`d` |
Coefficients of the polynomial fixed part and if present the covariates (Z).To determine which is which the logical vector ind.drift also part of this object is TRUE for the polynomial part. |

`c` |
Coefficients of the nonparametric part. |

`nt` |
Dimension of fixed part. |

`np` |
Dimension of c. |

`nZ` |
Number of columns of Z covariate matrix (can be zero). |

`ind.drift` |
Logical vector that indicates polynomial
coefficients in the |

`lambda.fixed` |
The fixed lambda value |

`x` |
Spatial locations used for fitting. |

`knots` |
The same as x |

`cov.function.name` |
Name of covariance function used. |

`args` |
A list with all the covariance arguments that were specified in the call. |

`m` |
Order of fixed part polynomial. |

`chol.args` |
A list with all the cholesky arguments that were specified in the call. |

`call` |
A copy of the call to mKrig. |

`non.zero.entries` |
Number of nonzero entries in the covariance matrix for the process at the observation locations. |

`shat.MLE` |
MLE of sigma. |

`rho.MLE` |
MLE or rho. |

`rhohat` |
Estimate for rho adjusted for fixed model degrees of freedom (ala REML). |

`lnProfileLike` |
log Profile likelihood for lambda |

`lnDetCov` |
Log determinant of the covariance matrix for the observations having factored out rho. |

`Omega` |
GLS covariance for the estimated parameters in the fixed part of the model (d coefficients0. |

`qr.VT, Mc` |
QR and cholesky matrix decompositions needed to recompute the estimate for new observation vectors. |

`fitted.values, residuals` |
Usual predictions from fit. |

`eff.df` |
Estimate of effective degrees of freedom. Either the mean of the Monte Carlo sample or the exact value. |

`trA.info` |
If NtrA ids less than |

`GCV` |
Estimated value of the GCV function. |

`GCV.info` |
Monte Carlo sample of GCV functions |

### Author(s)

Doug Nychka, Reinhard Furrer, John Paige

### References

http://cran.r-project.org/web/packages/fields/fields.pdf http://www.image.ucar.edu/~nychka/Fields/

### See Also

Krig, surface.mKrig, Tps, fastTps, predictSurface, predictSE.mKrig, sim.mKrig.approx,
` mKrig.grid`

### 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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 | ```
#
# Midwest ozone data 'day 16' stripped of missings
data( ozone2)
y<- ozone2$y[16,]
good<- !is.na( y)
y<-y[good]
x<- ozone2$lon.lat[good,]
# nearly interpolate using defaults (Exponential covariance range = 2.0)
# see also mKrig.MLE to choose lambda by maxmimum likelihood
out<- mKrig( x,y, theta = 2.0, lambda=.01)
out.p<- predictSurface( out)
surface( out.p)
#
# NOTE this should be identical to
# Krig( x,y, theta=2.0, lambda=.01)
##############################################################################
# an example using a "Z" covariate and the Matern family
# again see mKrig.MLE to choose parameters by MLE.
data(COmonthlyMet)
yCO<- CO.tmin.MAM.climate
good<- !is.na( yCO)
yCO<-yCO[good]
xCO<- CO.loc[good,]
Z<- CO.elev[good]
out<- mKrig( xCO,yCO, Z=Z, cov.function="stationary.cov", Covariance="Matern",
theta=4.0, smoothness=1.0, lambda=.1)
set.panel(2,1)
# quilt.plot with elevations
quilt.plot( xCO, predict(out))
# Smooth surface without elevation linear term included
surface( out)
set.panel()
#########################################################################
# Interpolate using tapered version of the exponential,
# the taper scale is set to 1.5 default taper covariance is the Wendland.
# Tapering will done at a scale of 1.5 relative to the scaling
# done through the theta passed to the covariance function.
data( ozone2)
y<- ozone2$y[16,]
good<- !is.na( y)
y<-y[good]
x<- ozone2$lon.lat[good,]
mKrig( x,y,cov.function="stationary.taper.cov",
theta = 2.0, lambda=.01,
Taper="Wendland", Taper.args=list(theta = 1.5, k=2, dimension=2)
) -> out2
# Try out GCV on a grid of lambda's.
# For this small data set
# one should really just use Krig or Tps but this is an example of
# approximate GCV that will work for much larger data sets using sparse
# covariances and the Monte Carlo trace estimate
#
# a grid of lambdas:
lgrid<- 10**seq(-1,1,,15)
GCV<- matrix( NA, 15,20)
trA<- matrix( NA, 15,20)
GCV.est<- rep( NA, 15)
eff.df<- rep( NA, 15)
logPL<- rep( NA, 15)
# loop over lambda's
for( k in 1:15){
out<- mKrig( x,y,cov.function="stationary.taper.cov",
theta = 2.0, lambda=lgrid[k],
Taper="Wendland", Taper.args=list(theta = 1.5, k=2, dimension=2) )
GCV[k,]<- out$GCV.info
trA[k,]<- out$trA.info
eff.df[k]<- out$eff.df
GCV.est[k]<- out$GCV
logPL[k]<- out$lnProfileLike
}
#
# plot the results different curves are for individual estimates
# the two lines are whether one averages first the traces or the GCV criterion.
#
par( mar=c(5,4,4,6))
matplot( trA, GCV, type="l", col=1, lty=2,
xlab="effective degrees of freedom", ylab="GCV")
lines( eff.df, GCV.est, lwd=2, col=2)
lines( eff.df, rowMeans(GCV), lwd=2)
# add exact GCV computed by Krig
out0<- Krig( x,y,cov.function="stationary.taper.cov",
theta = 2.0,
Taper="Wendland", Taper.args=list(theta = 1.5, k=2, dimension=2),
spam.format=FALSE)
lines( out0$gcv.grid[,2:3], lwd=4, col="darkgreen")
# add profile likelihood
utemp<- par()$usr
utemp[3:4] <- range( -logPL)
par( usr=utemp)
lines( eff.df, -logPL, lwd=2, col="blue", lty=2)
axis( 4)
mtext( side=4,line=3, "-ln profile likelihood", col="blue")
title( "GCV ( green = exact) and -ln profile likelihood", cex=2)
#########################################################################
# here is a series of examples with bigger datasets
# using a compactly supported covariance directly
set.seed( 334)
N<- 1000
x<- matrix( 2*(runif(2*N)-.5),ncol=2)
y<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( 1000)*.1
look2<-mKrig( x,y, cov.function="wendland.cov",k=2, theta=.2,
lambda=.1)
# take a look at fitted surface
predictSurface(look2)-> out.p
surface( out.p)
# this works because the number of nonzero elements within distance theta
# are less than the default maximum allocated size of the
# sparse covariance matrix.
# see spam.options() for the default values
# The following will give a warning for theta=.9 because
# allocation for the covariance matirx storage is too small.
# Here theta controls the support of the covariance and so
# indirectly the number of nonzero elements in the sparse matrix
## Not run:
look2<- mKrig( x,y, cov.function="wendland.cov",k=2, theta=.9, lambda=.1)
## End(Not run)
# The warning resets the memory allocation for the covariance matrix
# according the to values 'spam.options(nearestdistnnz=c(416052,400))'
# this is inefficient becuase the preliminary pass failed.
# the following call completes the computation in "one pass"
# without a warning and without having to reallocate more memory.
spam.options(nearestdistnnz=c(416052,400))
look2<- mKrig( x,y, cov.function="wendland.cov",k=2,
theta=.9, lambda=1e-2)
# as a check notice that
# print( look2)
# reports the number of nonzero elements consistent with the specifc allocation
# increase in spam.options
# new data set of 1500 locations
set.seed( 234)
N<- 1500
x<- matrix( 2*(runif(2*N)-.5),ncol=2)
y<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
## Not run:
# the following is an example of where the allocation (for nnzR)
# for the cholesky factor is too small. A warning is issued and
# the allocation is increased by 25
#
look2<- mKrig( x,y,
cov.function="wendland.cov",k=2, theta=.1, lambda=1e2 )
## End(Not run)
# to avoid the warning
look2<-mKrig( x,y,
cov.function="wendland.cov", k=2, theta=.1,
lambda=1e2, chol.args=list(pivot=TRUE, memory=list(nnzR= 450000)))
###############################################################################
# fiting multiple data sets
#
#\dontrun{
y1<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
y2<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
Y<- cbind(y1,y2)
look3<- mKrig( x,Y,cov.function="wendland.cov",k=2, theta=.1,
lambda=1e2 )
# note slight difference in summary because two data sets have been fit.
print( look3)
#}
##################################################################
# finding a good choice for theta as a taper
# Suppose the target is a spatial prediction using roughly 50 nearest neighbors
# (tapering covariances is effective for roughly 20 or more in the situation of
# interpolation) see Furrer, Genton and Nychka (2006).
# take a look at a random set of 100 points to get idea of scale
set.seed(223)
ind<- sample( 1:N,100)
hold<- rdist( x[ind,], x)
dd<- (apply( hold, 1, sort))[65,]
dguess<- max(dd)
# dguess is now a reasonable guess at finding cutoff distance for
# 50 or so neighbors
# full distance matrix excluding distances greater than dguess
# but omit the diagonal elements -- we know these are zero!
hold<- nearest.dist( x, delta= dguess,upper=TRUE)
# exploit spam format to get quick of number of nonzero elements in each row
hold2<- diff( hold@rowpointers)
# min( hold2) = 55 which we declare close enough
# now the following will use no less than 55 nearest neighbors
# due to the tapering.
## Not run:
mKrig( x,y, cov.function="wendland.cov",k=2, theta=dguess,
lambda=1e2) -> look2
## End(Not run)
###############################################################################
# use precomputed distance matrix
#
## Not run:
y1<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
y2<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
Y<- cbind(y1,y2)
#precompute distance matrix in compact form
distMat = rdist(x, compact=TRUE)
look3<- mKrig( x,Y,cov.function="stationary.cov", theta=.1,
lambda=1e2, distMat=distMat )
#precompute distance matrix in standard form
distMat = rdist(x)
look3<- mKrig( x,Y,cov.function="stationary.cov", theta=.1,
lambda=1e2, distMat=distMat )
## End(Not run)
``` |