Assessment of a continuous spatial variable based on a set of m observations is usually performed in a Gaussian random field framework. The optimal predictor under this model can be presented either as a linear kriging predictor or as a dual kriging predictor. The spatial variable predictor is usually stored in a kriging grid representation of size n. Alternatively, one may define a kernel function representation based on the dual kriging formulation. The latter can be efficiently reduced to the former, but not vice versa. To provide a prediction at an arbitrary location, a piecewise planar interpolation in the actual grid unit is typically required. For the functional representation, the functional value in the actual location must be calculated. The computational challenge of both representations is primarily related to the inversion of the observation covariance <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ ( m \times m ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>×</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-matrix. In large spatial studies with huge sets of observations, and thus huge m, this inversion may not be computationally feasible. Localized kriging predictors are then frequently used to generate the grid representation of the spatial variable. This approach has computational demands proportional to the grid size n. We present a localized kernel predictor to provide a functional representation of the spatial variable. The specification of this localized kernel predictor constitutes the major contribution of this paper. This predictor has computational demands proportional to the number of observations m. This is particularly beneficial in three-dimensional models and spatiotemporal studies where one typically has <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ n \gg m $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≫</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>. The characteristics of the kernel predictor are demonstrated in an example. A study on real observations indicates that the localized kernel function representation has substantial computational advantages over the localized kriging grid representation. Even generating the grid representation from a kernel function representation appears more computationally efficient than generating it directly using a localized kriging predictor.

The Grid-Free Spatial Kernel Predictor for Huge Observation Sets

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2026年1月13日

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Henning Omre,
Mina Spremic

摘要

Assessment of a continuous spatial variable based on a set of m observations is usually performed in a Gaussian random field framework. The optimal predictor under this model can be presented either as a linear kriging predictor or as a dual kriging predictor. The spatial variable predictor is usually stored in a kriging grid representation of size n. Alternatively, one may define a kernel function representation based on the dual kriging formulation. The latter can be efficiently reduced to the former, but not vice versa. To provide a prediction at an arbitrary location, a piecewise planar interpolation in the actual grid unit is typically required. For the functional representation, the functional value in the actual location must be calculated. The computational challenge of both representations is primarily related to the inversion of the observation covariance $ ( m \times m ) $ $(m \times m)$ -matrix. In large spatial studies with huge sets of observations, and thus huge m, this inversion may not be computationally feasible. Localized kriging predictors are then frequently used to generate the grid representation of the spatial variable. This approach has computational demands proportional to the grid size n. We present a localized kernel predictor to provide a functional representation of the spatial variable. The specification of this localized kernel predictor constitutes the major contribution of this paper. This predictor has computational demands proportional to the number of observations m. This is particularly beneficial in three-dimensional models and spatiotemporal studies where one typically has $ n \gg m $ $n ≫ m$ . The characteristics of the kernel predictor are demonstrated in an example. A study on real observations indicates that the localized kernel function representation has substantial computational advantages over the localized kriging grid representation. Even generating the grid representation from a kernel function representation appears more computationally efficient than generating it directly using a localized kriging predictor.