# High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps

## Abstract

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 high-dimensional interpolation by inferring non-linear 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 low-dimensional manifold embedded in an ambient high-dimensional 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 high-dimensional ambient space. The proposed method is applicable to arbitrarily high-dimensional 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:

- 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:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1400095

- Report Number(s):
- LLNL-JRNL-728760

Journal ID: ISSN 1874-8961; TRN: US1703098

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Mathematical Geosciences

- Additional Journal Information:
- Journal Volume: 50; Journal Issue: 1; Journal ID: ISSN 1874-8961

- Publisher:
- Springer

- 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

### Citation Formats

```
Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., and Chen, Xiao.
```*High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps*. United States: N. p., 2017.
Web. doi:10.1007/s11004-017-9705-y.

```
Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., & Chen, Xiao.
```*High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps*. United States. doi:10.1007/s11004-017-9705-y.

```
Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., and Chen, Xiao. Tue .
"High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps". United States.
doi:10.1007/s11004-017-9705-y.
```

```
@article{osti_1400095,
```

title = {High-Dimensional 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 high-dimensional interpolation by inferring non-linear 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 low-dimensional manifold embedded in an ambient high-dimensional 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 high-dimensional ambient space. The proposed method is applicable to arbitrarily high-dimensional 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/s11004-017-9705-y},

journal = {Mathematical Geosciences},

number = 1,

volume = 50,

place = {United States},

year = {Tue Oct 10 00:00:00 EDT 2017},

month = {Tue Oct 10 00:00:00 EDT 2017}

}