Assessing local and spatial uncertainty with nonparametric geostatistics

Uncertainty quantification is an important topic for many environmental studies, such as identifying zones where potentially toxic materials exist in the soil. In this work, the nonparametric geostatistical framework of histogram via entropy reduction (HER) is adapted to address local and spatial uncertainty in the context of risk of soil contamination. HER works with empirical probability distributions, coupling information theory and probability aggregation methods to estimate conditional distributions, which gives it the flexibility to be tailored for different data and application purposes. To explore how HER can be used for estimating threshold-exceeding probabilities, it is applied to map the risk of soil contamination by lead in the well-known dataset of the region of Swiss Jura. Its results are compared to indicator kriging (IK) and to an ordinary kriging (OK) model available in the literature. For the analyzed dataset, IK and HER predictions achieve the best performance and exhibit comparable accuracy and precision. Compared to IK, advantages of HER for uncertainty estimation in a fine resolution are that it does not require modeling of multiple indicator variograms, correcting order-relation violations, or defining interpolation/extrapolation of distributions. Finally, to avoid the well-known smoothing effect when using point estimations (as is the case with both kriging and HER), and to provide maps that reflect the spatial fluctuation of the observed reality, we demonstrate how HER can be used in combination with sequential simulation to assess spatial uncertainty (uncertainty jointly over several locations).

Assessing local and spatial uncertainty with nonparametric geostatistics

Uncertainty quantification is an important topic for many environmental studies, such as identifying zones where potentially toxic materials exist in the soil. In this work, the nonparametric geostatistical framework of histogram via entropy reduction (HER) is adapted to address local and spatial uncertainty in the context of risk of soil contamination. HER works with empirical probability distributions, coupling information theory and probability aggregation methods to estimate conditional distributions, which gives it the flexibility to be tailored for different data and application purposes. To explore the method adaptation for handling estimations of threshold-exceeding probabilities, it is used to map the risk of soil contamination by lead in the well-known dataset of the region of Swiss Jura. Its results are compared to indicator kriging (IK) and to an ordinary kriging (OK) model available in literature. For the analyzed dataset, IK and HER achieved the best performance and exhibited comparable accuracy and precision of their predictions. When compared to IK, HER has shown to be a unique approach for dealing with uncertainty estimation in a fine resolution, without the need of modeling multiple indicator variograms, correcting order-relation violations, or defining interpolation/extrapolation of distribution. Finally, to avoid the well-known smoothing effect when using point estimations (this is the case with kriging, but also with HER) and to provide maps that reflect the spatial fluctuation of the revealed reality, we demonstrate how HER can be used in combination with sequential simulation to assess spatial uncertainty (uncertainty jointly over several locations).

Histogram via entropy reduction (HER): an information-theoretic alternative for geostatistics

Interpolation of spatial data has been regarded in many different forms, varying from deterministic to stochastic, parametric to nonparametric, and purely data-driven to geostatistical methods. In this study, we propose a nonparametric interpolator, which combines information theory with probability aggregation methods in a geostatistical framework for the stochastic estimation of unsampled points. Histogram via entropy reduction (HER) predicts conditional distributions based on empirical probabilities, relaxing parameterizations and, therefore, avoiding the risk of adding information not present in data. By construction, it provides a proper framework for uncertainty estimation since it accounts for both spatial configuration and data values, while allowing one to introduce or infer properties of the field through the aggregation method. We investigate the framework using synthetically generated data sets and demonstrate its efficacy in ascertaining the underlying field with varying sample densities and data properties. HER shows a comparable performance to popular benchmark models, with the additional advantage of higher generality. The novel method brings a new perspective of spatial interpolation and uncertainty analysis to geostatistics and statistical learning, using the lens of information theory.

SUPPLIER SELECTION FOR HOUSING DEVELOPMENT BY AN INTEGRATED METHOD WITH INTERVAL ROUGH BOUNDARIES

Residential whole-decoration is an important initiative for housing industrialization in China. Selecting the most suitable component supplier for housing development is of great significance for both property developers and buyers in the implementation of such a strategy. To address such a problem, this study uses hesitant fuzzy linguistic term sets to express the inaccurate judgments of individuals and then introduces a novel probability aggregation approach based on interval rough boundaries to enable a realistic presentation of the collective evaluations of a group. Then, we propose a hybrid multi-expert multiple criteria decision-making model by integrating the Best Worst Method (BWM) and Combined Compromise Solution (CoCoSo) method based on the interval rough boundaries. A case study about the supplier selection for housing development is carried out, which demonstrates the feasibility and applicability of our proposed hybrid model. A comparison study is also performed to further validate the robustness of the model.

Salient Slices: Improved Neural Network Training and Performance with Image Entropy

As a training and analysis strategy for convolutional neural networks (CNNs), we slice images into tiled segments and use, for training and prediction, segments that both satisfy an information criterion and contain sufficient content to support classification. In particular, we use image entropy as the information criterion. This ensures that each tile carries as much information diversity as the original image and, for many applications, serves as an indicator of usefulness in classification. To make predictions, a probability aggregation framework is applied to probabilities assigned by the CNN to the input image tiles. This technique, which we call Salient Slices, facilitates the use of large, high-resolution images that would be impractical to analyze unmodified; provides data augmentation for training, which is particularly valuable when image availability is limited; and the ensemble nature of the input for prediction enhances its accuracy.

HER: an information theoretic alternative for geostatistics

<p>Interpolation of spatial data has been considered in many different forms. This study proposes a stochastic, non-parametric, geostatistical estimator that combines measures of information theory with probability aggregation method. Histogram via entropy reduction (HER) can be used to analyze the data spatial correlation and for predicting distributions at unobserved locations directly based on empirical probability. The method minimizes estimation uncertainty, relaxes normality assumptions and therefore avoids the risk of adding information not available in data (or losing available information). In particular, the applied probability aggregation method provides a proper framework for uncertainty estimation that reflects both the spatial configuration of the data as well as data values, while allowing to infer (or introduce) physical properties (continuous or discontinuous characteristics) from the field under study. Three different aggregation methods were explored in terms of uncertainty, resulting in predictions ranging from conservative to more confident ones. We investigate the performance of the framework using four synthetically generated datasets from known Gaussian processes and demonstrate the efficacy of the method in ascertaining the underlying true field with varying sample sizes. By comparing the method performance to popular benchmark models, namely nearest neighbors (NN), inverse distance weighting (IDW) and ordinary kriging (OK), we were able to obtain competitive results with respect to OK, with the advantage of presenting generalization properties. The novel method brings a new perspective of spatial and uncertainty analysis to geostatistics and statistical learning, using the lens of information theory.</p>

HER: an information theoretic alternative for geostatistics

Interpolation of spatial data has been regarded in many different forms, varying from deterministic to stochastic, purely data-driven to geostatistical, and parametric to non-parametric methods. In this study, we propose a stochastic, geostatistical estimator which combines information theory with probability aggregation methods for minimizing predictive uncertainty, and predicting distributions directly based on empirical probability. Histogram via entropy reduction (HER) relaxes parametrizations, avoiding the risk of adding information not present in data (or losing available information). It provides a proper framework for uncertainty estimation that takes into account both spatial configuration and data values, while allowing to infer (or introduce) physical properties (continuous or discontinuous characteristics) of the field. We investigate the framework utility using synthetically generated datasets and demonstrate its efficacy in ascertaining the underlying field with varying sample densities and data properties (different spatial correlation distances and addition of noise). HER shows comparable performance with popular benchmark models and the additional advantage of higher generality. The novel method brings a new perspective of spatial interpolation and uncertainty analysis to geostatistics and statistical learning, using the lens of information theory.

Locality-based 3-D multiple-point statistics reconstruction using 2-D geological cross sections

Multiple-point statistics (MPS) has shown promise in representing complicated subsurface structures. For a practical three-dimensional (3-D) application, however, one of the critical issues is the difficulty in obtaining a credible 3-D training image. However, bidimensional (2-D) training images are often available because established workflows exist to derive 2-D sections from scattered boreholes and/or other samples. In this work, we propose a locality-based MPS approach to reconstruct 3-D geological models on the basis of such 2-D cross sections (3DRCS), making 3-D training images unnecessary. Only several local training subsections closer to the central uninformed node are used in the MPS simulation. The main advantages of this partitioned search strategy are the high computational efficiency and a relaxation of the stationarity assumption. We embed this strategy into a standard MPS framework. Two probability aggregation formulas and their combinations are used to assemble the probability density functions (PDFs) from different subsections. Moreover, a novel strategy is adopted to capture more stable PDFs, where the distances between patterns and flexible neighborhoods are integrated on multiple grids. A series of sensitivity analyses demonstrate the stability of the proposed approach. Several hydrogeological 3-D application examples illustrate the applicability of the 3DRCS approach in reproducing complex geological features. The results, in comparison with previous MPS methods, show better performance in portraying anisotropy characteristics and in CPU cost.

Locality-based 3-D multiple-point statistics reconstruction using 2-D geological cross-sections

Multiple-point statistics (MPS) has shown promise in representing complicated subsurface structures. For a practical three-dimensional (3-D) application, however, one of the critical issues the difficulty to obtain a credible 3-D training image. However, bidimensional (2-D) training images are often available because established workflows exist to derive 2-D sections from scattered boreholes and/or other samples. In this work, we propose a locality-based MPS approach to reconstruct 3-D geological models on the basis of such 2-D cross-sections, making 3-D training images unnecessary. Only several local training sub-sections closer to the central uninformed node are used in the MPS simulation. The main advantages of this partitioned search strategy are the high computational efficiency and a relaxation of the stationarity assumption. We embed this strategy into a standard MPS framework. Two probability aggregation formulas and their combinations are used to assemble the probability density functions (pdfs) from different sub-sections. Moreover, a novel strategy is adopted to capture more stable pdfs, where the distances between patterns and flexible neighborhoods are integrated on several multiple grids. A series of sensitivity analyses demonstrate the stability of the proposed approach. Several hydrogeological 3-D application examples illustrate the applicability of our approach in reproducing complex geological features. The results, in comparison with previous MPS methods, show better performance in portraying anisotropy characteristics and in CPU cost.