Permutation and Grouping Methods for Sharpening Gaussian Process Approximations
Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which has been cited as a deficiency. This article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations. Indeed, the first part of the article i...
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| Published in: | Technometrics Vol. 60; no. 4; pp. 415 - 429 |
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| Format: | Journal Article |
| Language: | English |
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United States
Taylor & Francis
01.01.2018
American Society for Quality and the American Statistical Association American Society for Quality |
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| ISSN: | 0040-1706, 1537-2723 |
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| Abstract | Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which has been cited as a deficiency. This article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations. Indeed, the first part of the article includes a systematic study of how ordering affects the accuracy of Vecchia's approximation. We demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate-based orderings. Additional ordering schemes are described and analyzed numerically, including orderings capable of improving on random orderings. The second contribution of this article is a new automatic method for grouping calculations of components of the approximation. The grouping methods simultaneously improve approximation accuracy and reduce computational burden. In common settings, reordering combined with grouping reduces Kullback-Leibler divergence from the target model by more than a factor of 60 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equations. Computational details are provided, including the use of the approximations for prediction and conditional simulation. An application to space-time satellite data is presented. |
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| AbstractList | Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which has been cited as a deficiency. This article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations. Indeed, the first part of the article includes a systematic study of how ordering affects the accuracy of Vecchia's approximation. We demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate-based orderings. Additional ordering schemes are described and analyzed numerically, including orderings capable of improving on random orderings. The second contribution of this article is a new automatic method for grouping calculations of components of the approximation. The grouping methods simultaneously improve approximation accuracy and reduce computational burden. In common settings, reordering combined with grouping reduces Kullback-Leibler divergence from the target model by more than a factor of 60 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equations. Computational details are provided, including the use of the approximations for prediction and conditional simulation. An application to space-time satellite data is presented. Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which has been cited as a deficiency. This article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations. Indeed, the first part of the paper includes a systematic study of how ordering affects the accuracy of Vecchia's approximation. We demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate-based orderings. Additional ordering schemes are described and analyzed numerically, including orderings capable of improving on random orderings. The second contribution of this paper is a new automatic method for grouping calculations of components of the approximation. The grouping methods simultaneously improve approximation accuracy and reduce computational burden. In common settings, reordering combined with grouping reduces Kullback-Leibler divergence from the target model by more than a factor of 60 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equations. Computational details are provided, including the use of the approximations for prediction and conditional simulation. An application to space-time satellite data is presented. Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which has been cited as a deficiency. This article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations. Indeed, the first part of the paper includes a systematic study of how ordering affects the accuracy of Vecchia's approximation. We demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate-based orderings. Additional ordering schemes are described and analyzed numerically, including orderings capable of improving on random orderings. The second contribution of this paper is a new automatic method for grouping calculations of components of the approximation. The grouping methods simultaneously improve approximation accuracy and reduce computational burden. In common settings, reordering combined with grouping reduces Kullback-Leibler divergence from the target model by more than a factor of 60 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equations. Computational details are provided, including the use of the approximations for prediction and conditional simulation. An application to space-time satellite data is presented.Vecchia's approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which has been cited as a deficiency. This article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations. Indeed, the first part of the paper includes a systematic study of how ordering affects the accuracy of Vecchia's approximation. We demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate-based orderings. Additional ordering schemes are described and analyzed numerically, including orderings capable of improving on random orderings. The second contribution of this paper is a new automatic method for grouping calculations of components of the approximation. The grouping methods simultaneously improve approximation accuracy and reduce computational burden. In common settings, reordering combined with grouping reduces Kullback-Leibler divergence from the target model by more than a factor of 60 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equations. Computational details are provided, including the use of the approximations for prediction and conditional simulation. An application to space-time satellite data is presented. |
| Author | Guinness, Joseph |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/31447491$$D View this record in MEDLINE/PubMed |
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| Cites_doi | 10.1137/S0895479894278952 10.1007/BF02187718 10.1111/j.1467-9868.2011.00777.x 10.1201/b17115 10.1093/biomet/86.3.677 10.1198/004017007000000155 10.1080/01621459.2015.1072541 10.1198/106186006X132178 10.1093/biomet/71.1.135 10.1198/016214502753479194 10.1016/S0098-3004(97)00040-X 10.1002/wics.1383 10.1080/01621459.2015.1044091 10.1111/j.1467-9868.2008.00700.x 10.1016/j.spasta.2013.06.003 10.1137/0602010 10.3150/12-BEJSP06 10.1007/978-1-4612-0699-6_25 10.1214/aos/1015362194 10.1016/j.jmva.2015.08.018 10.1198/016214508000000959 10.1093/biomet/93.4.989 10.1080/10618600.2014.975230 10.1046/j.1369-7412.2003.05512.x 10.1137/1.9780898718003 10.1111/j.2517-6161.1988.tb01729.x 10.1214/16-AOAS931 10.1214/ss/1177012413 10.1080/10618600.2014.914442 |
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| References | cit0011 cit0033 cit0012 cit0034 cit0010 cit0032 cit0030 Santner T. J. (cit0031) 2013 Nychka D. (cit0023) 2016 cit0019 cit0017 cit0018 cit0038 cit0013 cit0035 cit0014 cit0036 cit0022 cit0001 cit0020 cit0021 R Development Core Team (cit0027) 2008 Bates D. (cit0003) 2016 Vecchia A. V. (cit0037) 1988 Heyde C. C. (cit0015) 2008 cit0008 cit0009 Hung Y. (cit0016) 2016 cit0006 cit0028 cit0007 cit0029 cit0026 cit0005 cit0002 cit0024 Beygelzimer A. (cit0004) 2013 cit0025 |
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| SubjectTerms | Accuracy Approximation Comparative analysis Computation Computer simulation Conditional simulation Divergence Gaussian process Kriging Mathematical models Normal distribution Parallel computation Partial differential equations Permutations Process parameters Sharpening Spatial-temporal data Stochastic models Vecchia's approximation |
| Title | Permutation and Grouping Methods for Sharpening Gaussian Process Approximations |
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