Spatial distributional estimation via ensemble spatial analysis

Alvaro F. Egaña; Gonzalo Díaz; Felipe Navarro; Mohammad Maleki; Juan F. Sánchez-Pérez; Alvaro F. Egaña; Gonzalo Díaz; Felipe Navarro; Mohammad Maleki; Juan F. Sánchez-Pérez

doi:10.3934/math.20251159

AIMS Mathematics

2025, Volume 10, Issue 11: 26351-26388. doi: 10.3934/math.20251159

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Spatial distributional estimation via ensemble spatial analysis

1.
Advanced Laboratory of Geostatistical Supercomputing (ALGES), Advanced Mining Technology Center (AMTC), Universidad de Chile, Santiago, Chile
2.
Department of Metallurgical and Mining Engineering, Universidad Católica del Norte, Antofagasta, Chile
3.
Department of Applied Physics and Naval Technology, Universidad Politécnica de Cartagena, Murcia, Spain

Received: 10 August 2025 Revised: 29 September 2025 Accepted: 17 October 2025 Published: 14 November 2025
MSC : 60G60, 62M30, 62M40, 86A32

This paper introduces a novel data-driven methodology for estimating the full random field (random function) of a regionalized variable, building upon and extending the concepts of ensemble spatial interpolation and adaptive ensemble spatial analysis. The core contribution is the development of a rigorous theoretical framework that proves how the method implicitly learns the underlying spatial dependence structure. By inferring local predictive distributions through adaptively defined, overlapping spatial partitions, the approach ensures that spatial coherence is governed by a learned statistical copula. Formal proof is provided demonstrating that this dependency structure converges to the actual structure of the regionalized variable. Therefore, the resulting spatial distributional estimator is consistent, captures all the uncertainty of the random field, and naturally allows for the generation of multiple equally probable realizations. This non-parametric strategy offers a flexible alternative, inherently preserving spatial patterns and capturing fine-scale variability without requiring prior model specification. Experiments conducted in a detailed case study applied to geostatistical simulation, using both synthetic and real datasets, confirm the effectiveness and computational efficiency of the method, demonstrating its ability to recover local statistics and spatial structure with greater robustness compared to conventional techniques.
- geostatistics,
- computational geostatistics,
- generative geostatistics,
- non-linear geostatistics,
- distributional geostatistics,
- geostatistical simulation,
- empirical copula,
- data-driven methods
Citation: Alvaro F. Egaña, Gonzalo Díaz, Felipe Navarro, Mohammad Maleki, Juan F. Sánchez-Pérez. Spatial distributional estimation via ensemble spatial analysis[J]. AIMS Mathematics, 2025, 10(11): 26351-26388. doi: 10.3934/math.20251159

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Abstract

This paper introduces a novel data-driven methodology for estimating the full random field (random function) of a regionalized variable, building upon and extending the concepts of ensemble spatial interpolation and adaptive ensemble spatial analysis. The core contribution is the development of a rigorous theoretical framework that proves how the method implicitly learns the underlying spatial dependence structure. By inferring local predictive distributions through adaptively defined, overlapping spatial partitions, the approach ensures that spatial coherence is governed by a learned statistical copula. Formal proof is provided demonstrating that this dependency structure converges to the actual structure of the regionalized variable. Therefore, the resulting spatial distributional estimator is consistent, captures all the uncertainty of the random field, and naturally allows for the generation of multiple equally probable realizations. This non-parametric strategy offers a flexible alternative, inherently preserving spatial patterns and capturing fine-scale variability without requiring prior model specification. Experiments conducted in a detailed case study applied to geostatistical simulation, using both synthetic and real datasets, confirm the effectiveness and computational efficiency of the method, demonstrating its ability to recover local statistics and spatial structure with greater robustness compared to conventional techniques.

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