HighDimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps
This article considers the challenging task of estimating geologic properties of interest using a suite of proxy measurements. The current work recast this task as a manifold learning problem. In this process, this article introduces a novel regression procedure for intrinsic variables constrained onto a manifold embedded in an ambient space. The procedure is meant to sharpen highdimensional interpolation by inferring nonlinear correlations from the data being interpolated. The proposed approach augments manifold learning procedures with a Gaussian process regression. It first identifies, using diffusion maps, a lowdimensional manifold embedded in an ambient highdimensional space associated with the data. It relies on the diffusion distance associated with this construction to define a distance function with which the data model is equipped. This distance metric function is then used to compute the correlation structure of a Gaussian process that describes the statistical dependence of quantities of interest in the highdimensional ambient space. The proposed method is applicable to arbitrarily highdimensional data sets. Here, it is applied to subsurface characterization using a suite of well log measurements. The predictions obtained in original, principal component, and diffusion space are compared using both qualitative and quantitative metrics. Considerable improvement in the prediction of themore »
 Authors:

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 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
 Univ. of Southern California, Los Angeles, CA (United States). Sonny Astani Dept. of Civil and Environmental Engineering
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Atmospheric, Earth and Energy Division
 Publication Date:
 Report Number(s):
 LLNLJRNL728760
Journal ID: ISSN 18748961; TRN: US1703098
 Grant/Contract Number:
 AC5207NA27344
 Type:
 Accepted Manuscript
 Journal Name:
 Mathematical Geosciences
 Additional Journal Information:
 Journal Volume: 50; Journal Issue: 1; Journal ID: ISSN 18748961
 Publisher:
 Springer
 Research Org:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org:
 USDOE Laboratory Directed Research and Development (LDRD) Program
 Country of Publication:
 United States
 Language:
 English
 Subject:
 58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING; Gaussian process regression; Kriging; Interpolation on manifold; Intrinsic interpolation; Intrinsic metrics; Diffusion distance
 OSTI Identifier:
 1400095
Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., and Chen, Xiao. HighDimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps. United States: N. p.,
Web. doi:10.1007/s110040179705y.
Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., & Chen, Xiao. HighDimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps. United States. doi:10.1007/s110040179705y.
Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., and Chen, Xiao. 2017.
"HighDimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps". United States.
doi:10.1007/s110040179705y. https://www.osti.gov/servlets/purl/1400095.
@article{osti_1400095,
title = {HighDimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps},
author = {Thimmisetty, Charanraj A. and Ghanem, Roger G. and White, Joshua A. and Chen, Xiao},
abstractNote = {This article considers the challenging task of estimating geologic properties of interest using a suite of proxy measurements. The current work recast this task as a manifold learning problem. In this process, this article introduces a novel regression procedure for intrinsic variables constrained onto a manifold embedded in an ambient space. The procedure is meant to sharpen highdimensional interpolation by inferring nonlinear correlations from the data being interpolated. The proposed approach augments manifold learning procedures with a Gaussian process regression. It first identifies, using diffusion maps, a lowdimensional manifold embedded in an ambient highdimensional space associated with the data. It relies on the diffusion distance associated with this construction to define a distance function with which the data model is equipped. This distance metric function is then used to compute the correlation structure of a Gaussian process that describes the statistical dependence of quantities of interest in the highdimensional ambient space. The proposed method is applicable to arbitrarily highdimensional data sets. Here, it is applied to subsurface characterization using a suite of well log measurements. The predictions obtained in original, principal component, and diffusion space are compared using both qualitative and quantitative metrics. Considerable improvement in the prediction of the geological structural properties is observed with the proposed method.},
doi = {10.1007/s110040179705y},
journal = {Mathematical Geosciences},
number = 1,
volume = 50,
place = {United States},
year = {2017},
month = {10}
}
Works referenced in this record:
Diffusion maps and coarsegraining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization
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 IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 28, Issue 9, p. 13931403