Random seed used to generate the Monte Carlo
samples. Keep the same to compare results with mKrig and also for
multiple values of lambda.
lambda
The ratio of the nugget variance (sigma squared) to the
parameter controlling the marginal variance of the process (called
rho in fields).
LKinfo
The LKinfo object. See help(LKinfo)
Mc
Cholesky decomposition of regression matrix.
NtrA
Number of Monte Carlo samples to estimate trace. Default
is 20 in LKrig.
nObs
Number of observations.
nReps
Number of replicate fields.
object
The LKrig object.
Q
Precision matrix for coefficients.
quad.form
The part of the log likelihood that is a quadratic form.
(This is typically found in LKrig.coef.)
residuals
Residuals from fitting spatial process.
U
The matrix that maps the d.coef coefficients of the fixed component
(typically a low order polynomial) part of the observation model.
verbose
If TRUE intermediate debugging information is printed.
weights
A vector that is proportional to the reciprocal
variances of the errors. I.e. errors are assumed to be uncorrelated
with variances sigma^2/weights.
wU
Weighted U matrix the fixed part of the model.
wX
Weighted X matrix (in spam format) related to nonparametric (stochastic) part of
model. Here weights refer to the sqrt(weights).
NOTE: predicted values are U%*%d.coef + X%*%c.coef
wy
Weighted observations.
X
The matrix that maps the c.coef coefficients into the nonparametric component
(spatial process) part of the observation model.
x
Matrix of spatial locations passed to LKrig.
y
Vector or matrix of observations passed to LKrig.
Z
A matrix of covariates.
ZGrid
A list or array with the covariates on the same grid as that specified by
the grid.list argument.
Details
The LatticeKrig article can be used as a reference for the matrix computations
and the G matrix from those formulas figures prominently. The GCholesky object
in these functions is the cholesky decompoistion of this matrix. For
compatibility with older version of this package this object may also be named
as Mc ( Cholesky of the M matrix) but the user should not identify this M with that in the article. Ideally all coding using Mc should be changed to GCholesky.
createLKrigObject Based on the arguments passed into LKrig forms the
prototype LKrig object. This object is added to as one computes additional steps in the LKrig function.
LKrigMakewU and LKrigMakewX construct the weighted U and X matrices from what is passed. In the case of observations that are point locations wU is found the weights and using the fixedFunction and wX is found from the weights and the multiresolution basis functions. Note that X and wX are assumed to be in spam
sparse matrix format.
LKrig.coef and LKrig.lnPlike are two low level functions
to find the basis function coefficients and to evaluate the
likelihood. The coefficients (c.mKrig) are also found because
they provide for shortcut formulas for the standard errors and MLE
estimates. These coefficients are identical to the basis coefficients
(c.coef) found for usual Kriging in the mKrig
function. LKrig.lnPlike also finds the profile MLE of sigma and
rho given a fixed value for lambda (and alpha and
a.wght). See the source for LKrig and also MLE.LKrig to see
how these functions are used.
LKrig.traceA finds an estimate of the effective degrees of
freedom of the smoothing matrix based a simple Monte Carlo scheme. The
smoothing matrix A is the matrix for fixed covariance parameters so
that y.hat = A y, where y.hat are the predicted values at the data
locations. trace(A) is the effective degrees of freedom. If e are
iid N(0,1) then the expected value of t(e)% * % A % * % e is equal
to the trace of A. This is the basis for estimating the trace and the
standard error for this estimate is based on NtrA independent
samples.
dfind2d is a fast FORTRAN subroutine to find nearest neighbors
within a fixed distance and is called by Wendland.basis. The
function dfind3d is currently not used but is intended for
future use to determine chordal distance between points on a sphere or
cylinder.
LKrigDefaultFixedFunction Is called to construct the fixed part of the
spatial model. The default is a polynomial of degree (m-1).
Value
LKrig.coef
a list with components d.coef the coefficients of
the spatial dirft and for covariates (Z) and c.coef the basis function
coefficients. The logical vector ind.drift from the LKrig object
indicates with components of d.coef are associated with the polynomial
spatial drift and which are other fixed spatial covariates.
LKrig.lnPlike
has the components:
lnProfileLike
the log likelihood profiled for lambda, alpha
and a.wght
rho.MLE
the MLE of rho given lambda, alpha and a.wght
shat.MLE
the MLE of sigma given lambda, alpha and a.wght
quad.form
the quadratic form in the exponent of the
multivariate normal likelihood
lnDetCov
the log determinant of the covariance matrix in the
likelihood
LKrigDefaultFixedFunction
A matrix with dimension nrow(x) and columns
of the number of polynomial terms and the number of columns of Z if given.
Author(s)
Doug Nychka
References
Nychka, D., Bandyopadhyay, S., Hammerling, D., Lindgren, F., & Sain, S. (2015). A multiresolution gaussian process model for the analysis of large spatial datasets.Journal of Computational and Graphical Statistics, 24(2), 579-599.
providing 
.