These are internal functions to Krig that compute the basic matrix decompositions or solve the linear systems needed to evaluate the Krig/Tps estimate. Others listed below do some simple housekeeping and formatting. Typically they are called from within Krig but can also be used directly if passed a Krig object list.
Krig.engine.default(out, verbose = FALSE)
Krig.engine.fixed( out, verbose=FALSE, lambda=NA)Krig.coef(out, lambda = out$lambda, y = NULL, yM = NULL, verbose = FALSE)
Krig.make.u(out, y = NULL, yM = NULL, verbose = FALSE)
Krig.check.xY(x, Y,Z, weights, na.rm, verbose = FALSE)
Krig.transform.xY(obj, knots, verbose = FALSE)
Krig.make.W( out, verbose=FALSE)
Krig.make.Wi ( out, verbose=FALSE)
A complete or partial Krig object. If partial it must have all the information accumulated to this calling point within the Krig function.
Same as out.
If TRUE prints out intermediate results for debugging.
Value of smoothing parameter "hard wired" into decompositions.
Default is NA, i.e. use the value in out\$lambda.
New y vector for recomputing coefficients. OR for %d*% a vector or matrix.
New y vector for recomputing coefficients but the values have already been collapsed into replicate group means.
raw data Y vector
raw x matrix of spatial locations OR In the case of %d*%, y is either a matrix or a vector. As a vector, y, is interpreted to be the elements of a digaonal matrix.
Raw weights vector passed to Krig
Raw vector or matrix of additional covariates.
NA action logical values passed to Krig
Raw knots matrix passed to Krig
ENGINES:
The returned value is a list with the matrix decompositions and other information. These are incorporated into the complete Krig object.
Common to all engines:
Type of decomposition
dimension of T matrix
number of knots
Krig.engine.default:
Transformed data using eigenvectors.
Eigenvalues
Reduced and weighted matrix of the eigenvectors
QR decomposition of fixed regression matrix
The eigenvectors
Krig.engine.fixed:
estimated coefficients for the fixed part of model
estimated coefficients for the basis functions derived from the covariance function.
Using all data locations
QR decomposition of the inverse Cholesky factor times the T matrix.
Cholesky factor
Using knot locations
QR decomposition of regression matrix modified by the estimate of the nonparametric ( or spatial) component.
Value of lambda used in the decompositions
OTHER FUNCTIONS:
Krig.coef
Y values as replicate group means
Sample standard deviation of replicates
Same as tauHat.rep
Pure error sums of squares based on replicates
The "c" basis coefficients associated with the covariance or radial basis functions.
The "d" regression type coefficients that are from the fixed part of the model or the linear null space.
When the default decomposition is used the data vector transformed by the orthogonal matrices. This facilitates evaluating the GCV function at different values of the smoothing parameter.
Krig.make.W
The weight matrix
Symmetric square root of weight matrix
Krig.make.Wi
The inverse weight matrix
Symmetric square root of inverse weight matrix
Doug Nychka
ENGINES:
The engines are the code modules that handle the basic linear algebra needed to computed the estimated curve or surface coefficients. All the engine work on the data that has been reduced to unique locations and possibly replicate group means with the weights adjusted accordingly. All information needed for the decomposition are components in the Krig object passed to these functions.
Krig.engine.default finds the decompositions for a Universal
Kriging estimator by simultaneously diagonalizing the linear system
system for the coefficients of the estimator. The main advantage of this
form is that it is fairly stable numerically, even with ill-conditioned
covariance matrices with lambda > 0. (i.e. provided there is a "nugget"
or measure measurement error. Also the eigendecomposition allows for
rapid evaluation of the likelihood, GCV and coefficients for new data
vectors under different values of the smoothing parameter, lambda.
Krig.engine.knots
This code has been omitted from verisions >= 12.0. See 11.6 too recover this functionality.
Finds the decompositions in the case that the covariance is evaluated at arbitrary locations possibly different than the data locations (called knots). The intent of these decompositions is to facilitate the evaluation at different values for lambda.
Krig.engine.fixed are specific decomposition based on the Cholesky
factorization assuming that the smoothing parameter is fixed. This
is the only case that works in the sparse matrix.
Both knots and the full set of locations can be handled by this case.
The difference between the "knots" engine above is that only a single value
of lambda is considered in the fixed engine.
OTHER FUNCTIONS:
Krig.coef Computes the "c" and "d" coefficients to represent the
estimated curve. These coefficients are used by the predict functions for
evaluations. Krig.coef can be used outside of the call to Krig to
recompute the fit with different Y values and possibly with different
lambda values. If new y values are not passed to this function then the yM
vector in the Krig object is used. The internal function
Krig.ynew sorts out the logic of what to do and use based on the
passed arguments.
Krig.make.u Computes the "u" vector, a transformation of the collapsed
observations that allows for rapid evaluation of the GCV function and
prediction. This only makes sense when the decomposition is WBW or DR, i.e.
an eigen decomposition. If the decompostion is the Cholesky based then this
function returns NA for the u component in the list.
Krig.check.xY Checks for removes missing values (NAs).
Krig.cor.Y This code has been omitted from verisions >= 12.0.
See 11.6 too recover this functionality.
Krig.transform.xY Finds all replicates and collapse to unique
locations and mean response and pooled variances and weights. These are
the xM, yM and weightsM used in the engines. Also scales the x locations
and the knots according to the transformation.
Krig.make.W and Krig.make.Wi These functions create an
off-diagonal weight matrix and its symmetric square root or the inverse
of the weight matrix based on the information passed to Krig. If
out$nondiag is TRUE W is constructed based on a call to the
passed
function wght.function along with additional arguments. If this flag
is FALSE then W is just diag(out$weightsM) and the square root
and inverse are computed directly.
%d*% Is a simple way to implement efficient diagonal
multiplications. x%d*%y is interpreted to mean diag(x)%*% y
if x is a vector. If x is a matrix then this becomes the same as the
usual matrix multiplication.
Krig, Tps
Krig( ChicagoO3$x, ChicagoO3$y, aRange=100)-> out
Krig.engine.default( out)-> stuff
# compare "stuff" to components in out$matrices
look1<- Krig.coef( out)
look1$c
# compare to out$c
look2<- Krig.coef( out, yM = ChicagoO3$y)
look2$c
# better be the same even though we pass as new data!
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