Development and Evaluation of Geostatistical Methods for Non-Euclidean-Based Spatial Covariance Matrices

Abstract

Customary and routine practice of geostatistical modeling assumes that inter-point distances are a Euclidean metric (i.e., as the crow flies) when characterizing spatial variation. There are many real-world settings, however, in which the use of a non-Euclidean distance is more appropriate, for example, in complex bodies of water. However, if such a distance is used with current semivariogram functions, the resulting spatial covariance matrices are no longer guaranteed to be positive-definite. Previous attempts to address this issue for geostatistical prediction (i.e., kriging) models transform the non-Euclidean space into a Euclidean metric, such as through multi-dimensional scaling (MDS). However, these attempts estimate spatial covariances only after distances are scaled. An alternative method is proposed to re-estimate a spatial covariance structure originally based on a non-Euclidean distance metric to ensure validity. This method is compared to the standard use of Euclidean distance, as well as a previously utilized MDS method. All methods are evaluated using cross-validation assessments on both simulated and real-world experiments. Results show a high level of bias in prediction variance for the previously developed MDS method that has not been highlighted previously. Conversely, the proposed method offers a preferred tradeoff between prediction accuracy and prediction variance and at times outperforms the existing methods for both sets of metrics. Overall results indicate that this proposed method can provide improved geostatistical predictions while ensuring valid results when the use of non-Euclidean distances is warranted.

Appendix A: List of Equations for Cross-Validation Metrics

Equations for the cross-validation metrics are provided below

$$ {\text{CV-}}R^{2} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left({y_{i} - \widehat{y}_{i} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{n} \left({y_{i} - \overline{y} } \right)^{2} }} ;\quad {\text{where }}\overline{y} = \frac{1}{n} \mathop \sum \limits_{i = 1}^{n} y_{i} , $$

(9)

$$ {\text{ME }} = \frac{1}{n} \mathop \sum \limits_{i = 1}^{n} y_{i} - \widehat{y}_{i} , $$

(10)

$$ {\text{RMSE }} = \sqrt {\frac{1}{n} \mathop \sum \limits_{i = 1}^{n} \left({y_{i} - \widehat{y}_{i} } \right)^{2} } , $$

(11)

$$ {\text{MPSE }} = \frac{1}{n} \mathop \sum \limits_{i = 1}^{n} \widehat{\sigma }_{i} , $$

(12)

$$ {\text{RMSSE }} = \sqrt {\frac{1}{n} \mathop \sum \limits_{i = 1}^{n} \left({\frac{{y_{i} - \widehat{y}_{i} }}{{\widehat{\sigma }_{i} }}} \right)^{2} ,} $$

(13)

$$ {\text{PI95 }} = \frac{1}{n} \mathop \sum \limits_{i = 1}^{n} I_{i} ;\quad {\text{where }}I_{i} = \left\{ {\begin{array}{*{20}l} 1 &\quad {{\text{if}}\quad \widehat{y}_{i} - 1.96\widehat{\sigma }_{i} \le y_{i} \le \widehat{y}_{i} + 1.96\widehat{\sigma }_{i} } \\ 0 &\quad {{\text{if }} \quad y_{i} < {\widehat{y}_{i} - 1.96\widehat{\sigma }_{i} \quad{\text{or }}y_{i} }> \widehat{y}_{i} + 1.96\widehat{\sigma }_{i} } \\ \end{array} ,} \right. $$

(14)

where $y_{i}$ is an observed outcome for location i, $\widehat{y}_{i}$ is the predicted (i.e., kriged) value, and $\widehat{\sigma }_{i}$ is the kriging standard error.