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

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
JournalMathematical Geosciences
DOIs
StatePublished - Jan 1 2019

Fingerprint

Covariance matrix
matrix
Evaluation
Prediction Variance
prediction
Scaling
Metric
Prediction
Euclidean
Semivariogram
method
evaluation
Distance Metric
Kriging
Covariance Structure
Spatial Structure
Euclidean Distance
kriging
Cross-validation
Positive definite

Keywords

  • Geostatistics
  • Kriging
  • Multi-dimensional scaling
  • Non-Euclidean distances
  • Positive-definite covariance matrices
  • Water salinity

ASJC Scopus subject areas

  • Mathematics (miscellaneous)
  • Earth and Planetary Sciences(all)

Cite this

@article{d4eec99a07234b5890d1c8ef76d60cb8,
title = "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.",
keywords = "Geostatistics, Kriging, Multi-dimensional scaling, Non-Euclidean distances, Positive-definite covariance matrices, Water salinity",
author = "Benjamin Davis and Curriero, {Frank C}",
year = "2019",
month = "1",
day = "1",
doi = "10.1007/s11004-019-09791-y",
language = "English (US)",
journal = "Mathematical Geosciences",
issn = "1874-8961",
publisher = "Springer Netherlands",

}

TY - JOUR

T1 - Development and Evaluation of Geostatistical Methods for Non-Euclidean-Based Spatial Covariance Matrices

AU - Davis, Benjamin

AU - Curriero, Frank C

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Geostatistics

KW - Kriging

KW - Multi-dimensional scaling

KW - Non-Euclidean distances

KW - Positive-definite covariance matrices

KW - Water salinity

UR - http://www.scopus.com/inward/record.url?scp=85063053130&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063053130&partnerID=8YFLogxK

U2 - 10.1007/s11004-019-09791-y

DO - 10.1007/s11004-019-09791-y

M3 - Article

C2 - 31827631

AN - SCOPUS:85063053130

JO - Mathematical Geosciences

JF - Mathematical Geosciences

SN - 1874-8961

ER -