Tucker Tensor Analysis of Matérn Functions in Spatial Statistics
In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. C...
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| Published in: | Journal of computational methods in applied mathematics Vol. 19; no. 1; pp. 101 - 122 |
|---|---|
| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Minsk
De Gruyter
01.01.2019
Walter de Gruyter GmbH |
| Subjects: | |
| ISSN: | 1609-4840, 1609-9389 |
| Online Access: | Get full text |
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| Abstract | In this work, we describe advanced numerical tools for working with multivariate functions and for the
analysis of large data sets. These tools will drastically reduce the required computing time and the
storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes.
Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to
compute and store, especially in three dimensions. Therefore, we approximate covariance functions by cheap surrogates
in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of
Matérn- and Slater-type functions with varying parameters and demonstrate numerically
that their approximations exhibit exponentially fast convergence.
We prove the exponential convergence of the Tucker and canonical approximations in tensor
rank parameters.
Several statistical operations are performed in this low-rank tensor format, including evaluating the
conditional covariance matrix, spatially averaged
estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse,
and Cholesky decomposition of a large covariance matrix.
Low-rank tensor approximations reduce the computing and storage costs essentially.
For example, the storage cost
is reduced from an exponential
to a linear scaling
,
where
is the spatial dimension,
is the number of mesh points in one direction,
and
is the tensor rank.
Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations,
and measurements lie on a tensor (axes-parallel) grid and that the covariance
function depends on a distance, |
|---|---|
| AbstractList | In this work, we describe advanced numerical tools for working with multivariate functions and for the
analysis of large data sets. These tools will drastically reduce the required computing time and the
storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes.
Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to
compute and store, especially in three dimensions. Therefore, we approximate covariance functions by cheap surrogates
in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of
Matérn- and Slater-type functions with varying parameters and demonstrate numerically
that their approximations exhibit exponentially fast convergence.
We prove the exponential convergence of the Tucker and canonical approximations in tensor
rank parameters.
Several statistical operations are performed in this low-rank tensor format, including evaluating the
conditional covariance matrix, spatially averaged
estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse,
and Cholesky decomposition of a large covariance matrix.
Low-rank tensor approximations reduce the computing and storage costs essentially.
For example, the storage cost
is reduced from an exponential
(
n
d
)
{\mathcal{O}(n^{d})}
to a linear scaling
(
d
r
n
)
{\mathcal{O}(drn)}
,
where
d
is the spatial dimension,
n
is the number of mesh points in one direction,
and
r
is the tensor rank.
Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations,
and measurements lie on a tensor (axes-parallel) grid and that the covariance
function depends on a distance,
∥
x
-
y
∥
{\|x-y\|}
. In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in three dimensions. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matérn- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential to a linear scaling , where is the spatial dimension, is the number of mesh points in one direction, and is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axes-parallel) grid and that the covariance function depends on a distance, In this work, we describe advanced numerical tools for working with multivariate functions and for theanalysis of large data sets. These tools will drastically reduce the required computing time and thestorage cost, and, therefore, will allow us to consider much larger data sets or finer meshes.Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive tocompute and store, especially in three dimensions. Therefore, we approximate covariance functions by cheap surrogatesin a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family ofMatérn- and Slater-type functions with varying parameters and demonstrate numericallythat their approximations exhibit exponentially fast convergence.We prove the exponential convergence of the Tucker and canonical approximations in tensorrank parameters.Several statistical operations are performed in this low-rank tensor format, including evaluating theconditional covariance matrix, spatially averagedestimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse,and Cholesky decomposition of a large covariance matrix.Low-rank tensor approximations reduce the computing and storage costs essentially.For example, the storage costis reduced from an exponential ð'ª(nd){\mathcal{O}(n^{d})} to a linear scaling ð'ª(drn){\mathcal{O}(drn)},where d is the spatial dimension, n is the number of mesh points in one direction,and r is the tensor rank.Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations,and measurements lie on a tensor (axes-parallel) grid and that the covariancefunction depends on a distance, ∥x-y∥{\|x-y\|}. |
| Author | Khoromskaia, Venera Matthies, Hermann G. Khoromskij, Boris N. Litvinenko, Alexander Keyes, David |
| Author_xml | – sequence: 1 givenname: Alexander orcidid: 0000-0001-5427-3598 surname: Litvinenko fullname: Litvinenko, Alexander email: alexander.litvinenko@kaust.edu.sa organization: Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology, Thuwal-Jeddah, Saudi Arabia – sequence: 2 givenname: David surname: Keyes fullname: Keyes, David email: david.keyes@kaust.edu.sa organization: Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology, Thuwal-Jeddah, Saudi Arabia – sequence: 3 givenname: Venera surname: Khoromskaia fullname: Khoromskaia, Venera email: vekh@mis.mpg.de organization: Max-Planck Institute for Mathematics in the Sciences, 4103Leipzig; and Max-Planck Institute for Dynamics of Complex Technical Systems, 9106 Magdeburg, Germany – sequence: 4 givenname: Boris N. surname: Khoromskij fullname: Khoromskij, Boris N. email: bokh@mis.mpg.de organization: Max-Planck Institute for Mathematics in the Sciences, 4103Leipzig, Germany – sequence: 5 givenname: Hermann G. surname: Matthies fullname: Matthies, Hermann G. email: wire@tu-braunschweig.de organization: Institute of Scientific Computing, TU Braunschweig, 8106Braunschweig, Germany |
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| Copyright | 2018 Walter de Gruyter GmbH, Berlin/Boston |
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| Snippet | In this work, we describe advanced numerical tools for working with multivariate functions and for the
analysis of large data sets. These tools will... In this work, we describe advanced numerical tools for working with multivariate functions and for theanalysis of large data sets. These tools will drastically... |
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| SubjectTerms | 60H15 60H35 65N25 Bayesian Update Computing costs Computing time Convergence Covariance matrix Datasets Decomposition Economic models Format Fourier Transform Geostatistical Optimal Design Hilbert Tensor Kalman Filter Kriging Loglikelihood Surrogate Low-Rank Tensor Approximation Matérn Covariance Parameters Quadratic forms Tensor analysis Tensors |
| Title | Tucker Tensor Analysis of Matérn Functions in Spatial Statistics |
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