Regression-kriging

In applied statistics and geostatistics, regression-kriging (RK) is a spatial prediction technique that combines a regression of the dependent variable on auxiliary variables (such as parameters derived from digital elevation modelling, remote sensing/imagery, and thematic maps) with interpolation (kriging) of the regression residuals. It is mathematically equivalent to the interpolation method variously called universal kriging and kriging with external drift, where auxiliary predictors are used directly to solve the kriging weights.

BLUP for spatial data

Regression-kriging is an implementation of the best linear unbiased predictor (BLUP) for spatial data, i.e. the best linear interpolator assuming the universal model of spatial variation. Matheron (1969) proposed that a value of a target variable at some location can be modeled as a sum of the deterministic and stochastic components:

Z ( s ) = m ( s ) + ε ′ ( s ) + ε ″ {\displaystyle Z(\mathbf {s} )=m(\mathbf {s} )+\varepsilon '(\mathbf {s} )+\varepsilon ''}

which he termed universal model of spatial variation. Both deterministic and stochastic components of spatial variation can be modeled separately. By combining the two approaches, we obtain:

z ^ ( s 0 ) = m ^ ( s 0 ) + e ^ ( s 0 ) = ∑ k = 0 p β ^ k ⋅ q k ( s 0 ) + ∑ i = 1 n λ i ⋅ e ( s i ) {\displaystyle {\hat {z}}(\mathbf {s} _{0})={\hat {m}}(\mathbf {s} _{0})+{\hat {e}}(\mathbf {s} _{0})=\sum \limits _{k=0}^{p}{{\hat {\beta }}_{k}\cdot q_{k}(\mathbf {s} _{0})}+\sum \limits _{i=1}^{n}\lambda _{i}\cdot e(\mathbf {s} _{i})}

where m ^ ( s 0 ) {\displaystyle {\hat {m}}(\mathbf {s} _{0})} is the fitted deterministic part, e ^ ( s 0 ) {\displaystyle {\hat {e}}(\mathbf {s} _{0})} is the interpolated residual, β ^ k {\displaystyle {\hat {\beta }}_{k}} are estimated deterministic model coefficients ( β ^ 0 {\displaystyle {\hat {\beta }}_{0}} is the estimated intercept), λ i {\displaystyle \lambda _{i}} are kriging weights determined by the spatial dependence structure of the residual and where e ( s i ) {\displaystyle e(\mathbf {s} _{i})} is the residual at location s i {\displaystyle {\mathbf {s} }_{i}} . The regression coefficients β ^ k {\displaystyle {\hat {\beta }}_{k}} can be estimated from the sample by some fitting method, e.g. ordinary least squares (OLS) or, optimally, using generalized least squares (GLS):

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