constrainedKriging-package {constrainedKriging} R Documentation

## constrainedKriging a package for nonlinear spatial predictions with local change of support

### Description

The constrainedKriging package provides functions for tow-dimensional spatial interpolation by constrained, covariance-matching constrained and universal (external drift) kriging for points or block of any shape for data with a nonstationary mean function and an isotropic weakly stationary variogram. The linear spatial interpolation methods, constrained and covariance-matching constrained kriging, provide approximately unbiased prediction for nonlinear target values under change of support.

In principle, the package provides two user functions, preCKrige and CKrige to calculate spatial prediction in two steps:

1. Call of `preCKrige` to calculate the variance-covariance matrices for defined sets of points or polygons (blocks).

2. Call of `CKrige` by using the output of preCKrige to calculate the spatial interpolation by one of the three kriging methods.

### Details

The constrained kriging predictor proposed 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 target (either a point value or a block mean). The covariance-matching constrained kriging predictor matches for a set of blocks both the variances and covariances of predictions and prediction targets ( points or block means) and is the extended version of the constrained kriging predictor. Like constrained kriging, covariance-matching constrained kriging is less biased than universal kriging for nonlinear predictions and exactly unbiased if the spatial variable is Gaussian.

The formulas of the three kriging predictors are given below:

• universal kriging predictor for 1 block:

Y_UK(B) = x(B)'beta.hat_(GLS) + c'Sigma( Z - Xbeta.hat_(GLS) ),

• constrained kriging predictor for 1 block:

Y_CK(B) = x(B)'beta.hat_(GLS) + K_(CK)c'Sigma( Z - Xbeta.hat_(GLS) ),

• covariance-matching constrained kriging predictor for m blocks:

Y_CMCK = X_m'beta.hat_(GLS) + K_(CMCK)'C'Sigma( Z - X'beta.hat_(GLS) ),

where Z = (Z(s_1), ..., Z(s_n) )' is the vector with the data; s = (x,y)' indicates a location in the survey domain and Y(B) is the block mean value of the block area B; X = (x(B_1), ..., x(B_m))' is the design matrix of the data and X_m is the design matrix of the target blocks; beta.hat_(GLS) is the vector with the generalised least square estimate of the linear regression coefficients; C = (c_1, ..., c_m) is a (n,m)-matrix that contains the covariances between the m prediction targets (point or blocks) and the n data points; Sigma is the covariance matrix of the data; the scalar

K_(CK) = ( Var[ Y(B) ] - Var[ x(B)'beta.hat_(GLS) ] )^0.5 / ( Var[ Y(B)_UK ] - Var[ x(B)'beta.hat_(GLS) ] )^0.5 = P^0.5 / Q^0.5

and the (m, m)-matrix

K_(CMCK) = Q1^(-1)P1,

where the (m,m)-matrix

P1 = Cov[ Y, Y' ] - Cov[ X_mbeta.hat_(GLS), X_mbeta.hat_(GLS)' ]

and the (m, m)-matrix

Q1 = Cov[ Y_(UK), Y_(UK)' ] - Cov[ X_mbeta.hat_(GLS), (X_mbeta.hat_(GLS))' ]

The mean square prediction error (MSEP) of the three predictors are:

• MSPE of the universal kriging predictor for 1 block:

MSPE[ Y(B)_(UK) ] = Var[ ( Y(B)_(UK) - Y(B) ), ( Y(B)_(UK) - Y(B) )' ],

• MSPE of the constrained kriging predictor for 1 block:

MSPE[ Y(B)_(CK) ] = MSPE[ Y(B)_(UK) ] +( P^0.5 + Q^0.5 )^2,

• MSPE of the covariance-matching constrained kriging predictor for m blocks:

MSPE[ Y_(CMCK) ] = MSPE[ Y_(UK) ] +( P1 + Q1 ) (P1 + Q1 ).

### Author(s)

Christoph Hofer christoph.hofer@allumni.ethz.ch

### References

Aldworth, J. and Cressie, N. (2003). Prediction of nonlinear spatial functionals. Journal of Statistical Planning and Inference, 112, 3–41

Cressie, N. (1993). Aggregation in geostatistical problems. In A. Soares, editor, Geostatistics Troia 92, 1, pages 25–36, Dordrecht. Kluwer Academic Publishers.

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. (in preparation). constrainedKriging: an R-package for customary, constrained and covariance-matching constrained point or block kriging. Computers & Geosciences.

[Package constrainedKriging version 0.2.4 Index]