| constrainedKriging-package {constrainedKriging} | R Documentation |
Nonlinear Spatial Kriging Predictions under Change of Support
Description
The package constrainedKriging provides functions for
spatial interpolation by constrained,
covariance-matching constrained and universal
(external drift) Kriging for points or blocks of any shape in
a two-dimensional domain from data with a non-stationary mean function and
an isotropic weakly stationary variogram. The linear spatial interpolation
methods constrained and covariance-matching constrained Kriging provide
approximately unbiased predictions for non-linearly transformed target
values under change of support.
The package provides two main user functions, preCKrige and
CKrige, to calculate spatial predictions in two steps:
-
preCKrigecomputes the variance-covariance matrices for sets of prediction points or prediction blocks (polygons). -
CKrigecomputes from the output ofpreCKrigespatial predictions by one of three Kriging methods mentioned above.
Details
The constrained Kriging predictor introduced by Cressie (1993) and the covariance-matching constrained Kriging predictor proposed by Aldworth and Cressie (2003) are linear in the data like the universal Kriging predictor. However, the constrained Kriging predictor satisfies in addition to the unbiasedness constraint of universal Kriging a second constraint that matches the variances of the predictions to the variances of the prediction targets (either point values or block means). The covariance-matching constrained Kriging predictor matches for a set of points or blocks both the variances and covariances of predictions and prediction targets and is the extended version of the constrained Kriging predictor. Like constrained Kriging, covariance-matching constrained Kriging is less biased than universal Kriging for predictions of non-linearly transformed functionals of a spatial variable and exactly unbiased if the variable is Gaussian. We summarize the formulae of the three Kriging methods below for predicting block means from point observations, but analogous equations could be given for problems that do not involve change of support.
For a set of m blocks, B_1, \ldots, B_m, the covariance-matching
constrained Kriging predictor is given by
\widehat{\mathbf{Y}}_{\mathrm{CMCK}} =
\mathbf{X}_{m}\widehat{\boldsymbol{\beta}}_{\mathrm{GLS}} +
\mathbf{K}^{\prime}
\mathbf{C}^{\prime}\boldsymbol{\Sigma}^{-1} (\mathbf{Z} -
\mathbf{X}\widehat{\boldsymbol{\beta}}_{\mathrm{GLS}}),
where
\mathbf{Y} = (Y(B_1), \ldots, Y(B_m))^\prime
is the set of block means to be predicted,
with Y(B_i) the mean value of
Y averaged over the block B_i;
\mathbf{Z} = (Z(\mathbf{s}_{1}), \ldots, Z(\mathbf{s}_{n}))^{^\prime}
is the vector with data
Z(\mathbf{s}_{i}) = Y(\mathbf{s}_{i}) + \epsilon_i
where the response
Y(\mathbf{s}_{i}) is possibly contaminated by measurement error
\epsilon_i;
\mathbf{s} = (x,y)^{\prime} indicates a location in
the survey domain;
\mathbf{X} = (\mathbf{x}(\mathbf{s}_{1}), \ldots,
\mathbf{x}(\mathbf{s}_{n}))^{\prime}
is the design matrix of the data and
\mathbf{X}_m = (\mathbf{x}(B_{1}), \ldots, \mathbf{x}(B_{m}))^{\prime}
the design matrix of the target blocks;
\widehat{\boldsymbol{\boldsymbol{\beta}}}_{\mathrm{GLS}}
is the vector with the generalised least square estimate of the
linear regression coefficients;
\mathbf{C} = (\mathbf{c}_{1}, \ldots, \mathbf{c}_{m})
is a (n\times m)-matrix that
contains the covariances between the n data points and the m
prediction targets;
\boldsymbol{\Sigma} is the (n\times n)-covariance
matrix of the data;
and \mathbf{K} is the (m \times m)-matrix
\mathbf{K} = \mathbf{Q}_{1}^{-1}\mathbf{P_{1}},
where \mathbf{P}_{1} is the (m \times m)-matrix
\mathbf{P}_{1} = ( \mathrm{Cov}[\mathbf{Y}, \mathbf{Y}^{\prime}] -
\mathrm{Cov}[\mathbf{X}_{m}\widehat{\boldsymbol{\boldsymbol{\beta}}}_{\mathrm{GLS}},
(\mathbf{X}_{m}\widehat{\boldsymbol{\boldsymbol{\beta}}}_{\mathrm{GLS}})^{\prime}] )^{\frac{1}{2}}
and \mathbf{Q}_{1} the (m \times m)-matrix
\mathbf{Q}_{1} = (\mathrm{Cov}[\widehat{\mathbf{Y}}_{\mathrm{UK}},
\widehat{\mathbf{Y}}_{\mathrm{UK}}^{\prime}] -
\mathrm{Cov}[\mathbf{X}_{m}\widehat{\boldsymbol{\boldsymbol{\beta}}}_{\mathrm{GLS}},
(\mathbf{X}_{m}\widehat{\boldsymbol{\boldsymbol{\beta}}}_{\mathrm{GLS}})^{\prime}])^{\frac{1}{2}}
with
\widehat{\mathbf{Y}}_{\mathrm{UK}} =
\mathbf{X}_{m}\widehat{\boldsymbol{\beta}}_{\mathrm{GLS}} +
\mathbf{C}^{\prime}\boldsymbol{\Sigma}^{-1} (\mathbf{Z} -
\mathbf{X}\widehat{\boldsymbol{\beta}}_{\mathrm{GLS}}),
denoting the universal Kriging predictor of \mathbf{Y}.
For m = 1 \widehat{\mathbf{Y}}_{\mathrm{CMCK}} reduces
to the constrained Kriging predictor
\widehat{Y}_{\mathrm{CK}}(B_1) =
\mathbf{x}(B_1)^{\prime}\widehat{\boldsymbol{\beta}}_{\mathrm{GLS}} + K
\mathbf{c_1}^{\prime} \boldsymbol{\Sigma}^{-1} ( \mathbf{Z} -
\mathbf{X}\widehat{\boldsymbol{\beta}}_{\mathrm{GLS}}) ,
with the scalar
K = (\mathrm{Var}[Y(B_1)] -
\mathrm{Var}[\mathbf{x}(B_1)^{\prime}\widehat{\boldsymbol{\boldsymbol{\beta}}}_{\mathrm{GLS}}] )^{\frac{1}{2}} /
(\mathrm{Var}[\widehat{Y}_{\mathrm{UK}}(B_1)] - \mathrm{Var}[
\mathbf{x}(B_1)^{\prime}\widehat{\boldsymbol{\boldsymbol{\beta}}}_{\mathrm{GLS}}] )^{\frac{1}{2}} =
(P/Q)^{\frac{1}{2}}.
The mean square prediction error (MSEP) of
\widehat{\mathbf{Y}}_{\mathrm{CMCK}} is given by
\mathrm{MSPE}[\widehat{\mathbf{Y}}_{\mathrm{CMCK}}] = \mathrm{MSPE}[
\widehat{\mathbf{Y}}_{\mathrm{UK}} ] +
(\mathbf{P}_{1}-\mathbf{Q}_{1})(\mathbf{P}_{1}-\mathbf{Q}_{1}).
and of \widehat{Y}_{\mathrm{CK}}(B_1) by
\mathrm{MSPE}[\widehat{Y}_{\mathrm{CK}}(B_{1})] = \mathrm{MSPE}[
\widehat{Y}_{\mathrm{UK}}(B_{1})] + (P^{\frac{1}{2}} - Q^{\frac{1}{2}})^{2},
where the MSEP of universal Kriging is given by
\mathrm{MSPE}[\widehat{\mathbf{Y}}_\mathrm{UK}] =
\mathrm{Cov}[\mathbf{Y}, \mathbf{Y}^{\prime}] -
\mathbf{C}^{\prime}\boldsymbol{\Sigma}^{-1}\mathbf{C} +
(\mathbf{X}_{m}^{\prime} - \mathbf{X}^{\prime}\boldsymbol{\Sigma}^{-1}\mathbf{C})^{\prime}
(\mathbf{X}^{\prime}\boldsymbol{\Sigma}^{-1}\mathbf{X})^{-1}
(\mathbf{X}_{m}^{\prime} - \mathbf{X}^{\prime}\boldsymbol{\Sigma}^{-1}\mathbf{C}).
Author(s)
Christoph Hofer, christoph.hofer@alumni.ethz.ch
References
Aldworth, J. and Cressie, N. (2003). Prediction of non-linear spatial functionals. Journal of Statistical Planning and Inference, 112, 3–41, doi:10.1016/S0378-3758(02)00321-X.
Cressie, N. (1993). Aggregation in geostatistical problems. In A. Soares, editor, Geostatistics Troia 92, 1, pages 25–36, Dordrecht. Kluwer Academic Publishers, doi:10.1007/978-94-011-1739-5_3.
Hofer, C. and Papritz, A. (2010). Predicting threshold exceedance by local block means in soil pollution surveys. Mathematical Geosciences. 42, 631–656, doi:10.1007/s11004-010-9287-4
Hofer, C. and Papritz, A. (2011). constrainedKriging: an R-package for customary, constrained and covariance-matching constrained point or block Kriging. Computers & Geosciences. 37, 1562–1569, doi:10.1016/j.cageo.2011.02.009