{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T03:34:07Z","timestamp":1764646447977,"version":"3.37.3"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T00:00:00Z","timestamp":1685404800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T00:00:00Z","timestamp":1685404800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100006192","name":"Advanced Scientific Computing Research","doi-asserted-by":"publisher","award":["DE-AC02-06CH11347"],"award-info":[{"award-number":["DE-AC02-06CH11347"]}],"id":[{"id":"10.13039\/100006192","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2023,8]]},"DOI":"10.1007\/s11222-023-10252-0","type":"journal-article","created":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T13:02:34Z","timestamp":1685451754000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Scalable computations for nonstationary Gaussian processes"],"prefix":"10.1007","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1650-5276","authenticated-orcid":false,"given":"Paul G.","family":"Beckman","sequence":"first","affiliation":[]},{"given":"Christopher J.","family":"Geoga","sequence":"additional","affiliation":[]},{"given":"Michael L.","family":"Stein","sequence":"additional","affiliation":[]},{"given":"Mihai","family":"Anitescu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,30]]},"reference":[{"key":"10252_CR1","volume-title":"Modis Atmosphere l2 Cloud Mask Product. NASA MODIS Adaptive Processing system","author":"S Ackerman","year":"2015","unstructured":"Ackerman, S., Frey, R.: Modis Atmosphere l2 Cloud Mask Product. NASA MODIS Adaptive Processing system. Goddard Space Flight Center (2015)"},{"key":"10252_CR2","unstructured":"Ambikasaran, S., O\u2019Neil, M., Singh, K.R.: Fast symmetric factorization of hierarchical matrices with applications (2014). arXiv preprint arXiv:1405.0223"},{"issue":"2","key":"10252_CR3","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1109\/TPAMI.2015.2448083","volume":"38","author":"S Ambikasaran","year":"2015","unstructured":"Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D.W., O\u2019Neil, M.: Fast direct methods for Gaussian processes. IEEE Trans. Pattern Anal. Mach. Intell. 38(2), 252\u2013265 (2015)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"3","key":"10252_CR4","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1016\/j.jmva.2010.10.010","volume":"102","author":"EB Anderes","year":"2011","unstructured":"Anderes, E.B., Stein, M.L.: Local likelihood estimation for nonstationary random fields. J. Multivar. Anal. 102(3), 506\u2013520 (2011)","journal-title":"J. Multivar. Anal."},{"issue":"4","key":"10252_CR5","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1111\/j.1467-9868.2008.00663.x","volume":"70","author":"S Banerjee","year":"2008","unstructured":"Banerjee, S., Gelfand, A.E., Finley, A.O., Sang, H.: Gaussian predictive process models for large spatial data sets. J. Roy. Stat. Soc. Ser. B (Stat. Methodol.) 70(4), 825\u2013848 (2008)","journal-title":"J. Roy. Stat. Soc. Ser. B (Stat. Methodol.)"},{"issue":"9","key":"10252_CR6","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1145\/361002.361007","volume":"18","author":"JL Bentley","year":"1975","unstructured":"Bentley, J.L.: Multidimensional binary search trees used for associative searching. Commun. ACM 18(9), 509\u2013517 (1975)","journal-title":"Commun. ACM"},{"key":"10252_CR7","doi-asserted-by":"crossref","unstructured":"B\u00f6rm, S., Garcke, J.: Approximating Gaussian processes with $${\\cal{H}}^2$$-matrices. In: European Conference on Machine Learning, pp. 42\u201353. Springer (2007)","DOI":"10.1007\/978-3-540-74958-5_8"},{"key":"10252_CR8","first-page":"1","volume":"118","author":"J Chen","year":"2021","unstructured":"Chen, J., Stein, M.L.: Linear-cost covariance functions for Gaussian random fields. J. Am. Stat. Assoc. 118, 1\u201318 (2021)","journal-title":"J. Am. Stat. Assoc."},{"issue":"1","key":"10252_CR9","first-page":"2214","volume":"18","author":"J Chen","year":"2017","unstructured":"Chen, J., Avron, H., Sindhwani, V.: Hierarchically compositional kernels for scalable nonparametric learning. J. Mach. Learn. Res. 18(1), 2214\u20132255 (2017)","journal-title":"J. Mach. Learn. Res."},{"issue":"1","key":"10252_CR10","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1111\/j.1467-9868.2007.00633.x","volume":"70","author":"N Cressie","year":"2008","unstructured":"Cressie, N., Johannesson, G.: Fixed rank kriging for very large spatial data sets. J. Roy. Stat. Soc. Ser. B (Stat. Methodol.) 70(1), 209\u2013226 (2008)","journal-title":"J. Roy. Stat. Soc. Ser. B (Stat. Methodol.)"},{"issue":"6","key":"10252_CR11","doi-asserted-by":"publisher","first-page":"1362","DOI":"10.1016\/j.csda.2011.10.022","volume":"56","author":"J Eidsvik","year":"2012","unstructured":"Eidsvik, J., Finley, A.O., Banerjee, S., Rue, H.: Approximate Bayesian inference for large spatial datasets using predictive process models. Comput. Stat. Data Anal. 56(6), 1362\u20131380 (2012)","journal-title":"Comput. Stat. Data Anal."},{"issue":"3","key":"10252_CR12","doi-asserted-by":"publisher","first-page":"502","DOI":"10.1198\/106186006X132178","volume":"15","author":"R Furrer","year":"2006","unstructured":"Furrer, R., Genton, M.G., Nychka, D.: Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Stat. 15(3), 502\u2013523 (2006)","journal-title":"J. Comput. Graph. Stat."},{"key":"10252_CR13","doi-asserted-by":"crossref","unstructured":"Geoga, C.J., Marin, O., Schanen, M., Stein, M.L.: Fitting Mat\u00e9rn smoothness parameters using automatic differentiation (2022). arXiv preprint arXiv:2201.00090","DOI":"10.1007\/s11222-022-10127-w"},{"key":"10252_CR14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11222-017-9790-2","volume":"29","author":"CJ Geoga","year":"2019","unstructured":"Geoga, C.J., Anitescu, M., Stein, M.L.: Scalable Gaussian process computations using hierarchical matrices. J. Comput. Graph. Stat. 29, 1\u201311 (2019)","journal-title":"J. Comput. Graph. Stat."},{"issue":"3","key":"10252_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11222-021-09999-1","volume":"31","author":"J Guinness","year":"2021","unstructured":"Guinness, J.: Gaussian process learning via fisher scoring of Vecchia\u2019s approximation. Stat. Comput. 31(3), 1\u20138 (2021)","journal-title":"Stat. Comput."},{"key":"10252_CR16","unstructured":"Huang, H., Blake, L.R., Katzfuss, M., Hammerling, D.M.: Nonstationary spatial modeling of massive global satellite data (2021). arXiv preprint arXiv:2111.13428"},{"issue":"3","key":"10252_CR17","doi-asserted-by":"publisher","first-page":"1059","DOI":"10.1080\/03610918908812806","volume":"18","author":"MF Hutchinson","year":"1989","unstructured":"Hutchinson, M.F.: A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat. Simul. Comput. 18(3), 1059\u20131076 (1989)","journal-title":"Commun. Stat. Simul. Comput."},{"issue":"517","key":"10252_CR18","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1080\/01621459.2015.1123632","volume":"112","author":"M Katzfuss","year":"2017","unstructured":"Katzfuss, M.: A multi-resolution approximation for massive spatial datasets. J. Am. Stat. Assoc. 112(517), 201\u2013214 (2017)","journal-title":"J. Am. Stat. Assoc."},{"issue":"1","key":"10252_CR19","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1002\/env.1147","volume":"23","author":"M Katzfuss","year":"2012","unstructured":"Katzfuss, M., Cressie, N.: Bayesian hierarchical spatio-temporal smoothing for very large datasets. Environmetrics 23(1), 94\u2013107 (2012)","journal-title":"Environmetrics"},{"issue":"4","key":"10252_CR20","first-page":"2203","volume":"30","author":"M Katzfuss","year":"2020","unstructured":"Katzfuss, M., Gong, W.: A class of multi-resolution approximations for large spatial datasets. Stat. Sin. 30(4), 2203\u20132226 (2020)","journal-title":"Stat. Sin."},{"issue":"1","key":"10252_CR21","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1214\/19-STS755","volume":"36","author":"M Katzfuss","year":"2021","unstructured":"Katzfuss, M., Guinness, J.: A general framework for Vecchia approximations of gaussian processes. Stat. Sci. 36(1), 124\u2013141 (2021)","journal-title":"Stat. Sci."},{"issue":"1","key":"10252_CR22","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1109\/TIP.2004.838703","volume":"14","author":"F Khellah","year":"2004","unstructured":"Khellah, F., Fieguth, P., Murray, M.J., Allen, M.: Statistical processing of large image sequences. IEEE Trans. Image Process. 14(1), 80\u201393 (2004)","journal-title":"IEEE Trans. Image Process."},{"issue":"3","key":"10252_CR23","first-page":"1209","volume":"29","author":"Y Li","year":"2019","unstructured":"Li, Y., Sun, Y.: Efficient estimation of nonstationary spatial covariance functions with application to high-resolution climate model emulation. Stat. Sin. 29(3), 1209\u20131231 (2019)","journal-title":"Stat. Sin."},{"issue":"4","key":"10252_CR24","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1111\/j.1467-9868.2011.00777.x","volume":"73","author":"F Lindgren","year":"2011","unstructured":"Lindgren, F., Rue, H., Lindstr\u00f6m, J.: An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J. Roy. Stat. Soc. Ser. B (Stat. Methodol.) 73(4), 423\u2013498 (2011)","journal-title":"J. Roy. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"10252_CR25","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/j.csda.2019.02.002","volume":"137","author":"A Litvinenko","year":"2019","unstructured":"Litvinenko, A., Sun, Y., Genton, M.G., Keyes, D.E.: Likelihood approximation with hierarchical matrices for large spatial datasets. Comput. Stat. Data Anal. 137, 115\u2013132 (2019)","journal-title":"Comput. Stat. Data Anal."},{"issue":"5","key":"10252_CR26","doi-asserted-by":"publisher","first-page":"1015","DOI":"10.1175\/BAMS-D-15-00002.1","volume":"98","author":"E Maturi","year":"2017","unstructured":"Maturi, E., Harris, A., Mittaz, J., Sapper, J., Wick, G., Zhu, X., Dash, P., Koner, P.: A new high-resolution sea surface temperature blended analysis. Bull. Am. Meteor. Soc. 98(5), 1015\u20131026 (2017)","journal-title":"Bull. Am. Meteor. Soc."},{"issue":"4","key":"10252_CR27","doi-asserted-by":"publisher","first-page":"1584","DOI":"10.1137\/17M1116477","volume":"15","author":"V Minden","year":"2017","unstructured":"Minden, V., Damle, A., Ho, K.L., Ying, L.: Fast spatial Gaussian process maximum likelihood estimation via skeletonization factorizations. Multiscale Model. Simul. 15(4), 1584\u20131611 (2017)","journal-title":"Multiscale Model. Simul."},{"key":"10252_CR28","doi-asserted-by":"crossref","unstructured":"Neal, R.M., et\u00a0al.: MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo, vol. 2(11), p. 2 (2011)","DOI":"10.1201\/b10905-6"},{"issue":"5","key":"10252_CR29","first-page":"483","volume":"17","author":"CJ Paciorek","year":"2006","unstructured":"Paciorek, C.J., Schervish, M.J.: Spatial modelling using a new class of nonstationary covariance functions. Environ. Off. J. Int. Environ. Soc. 17(5), 483\u2013506 (2006)","journal-title":"Environ. Off. J. Int. Environ. Soc."},{"key":"10252_CR30","unstructured":"Risser, M.D., Calder, C.A.: Local likelihood estimation for covariance functions with spatially-varying parameters: the convospat package for r. arXiv preprint (2015). arXiv:1507.08613"},{"issue":"1","key":"10252_CR31","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1111\/j.1467-9868.2011.01007.x","volume":"74","author":"H Sang","year":"2012","unstructured":"Sang, H., Huang, J.Z.: A full scale approximation of covariance functions for large spatial data sets. J. Roy. Stat. Soc. Ser. B (Stat. Methodol.) 74(1), 111\u2013132 (2012)","journal-title":"J. Roy. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"10252_CR32","first-page":"2519","volume":"20","author":"H Sang","year":"2011","unstructured":"Sang, H., Jun, M., Huang, J.Z.: Covariance approximation for large multivariate spatial data sets with an application to multiple climate model errors. Ann. Appl. Stat. 20, 2519\u20132548 (2011)","journal-title":"Ann. Appl. Stat."},{"key":"10252_CR33","unstructured":"Snelson, E., Ghahramani, Z.: Local and global sparse Gaussian process approximations. Artif. Intell. Stat. 524\u2013531 (2007)"},{"issue":"2","key":"10252_CR34","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1007\/s11222-019-09886-w","volume":"30","author":"A Solin","year":"2020","unstructured":"Solin, A., S\u00e4rkk\u00e4, S.: Hilbert space methods for reduced-rank Gaussian process regression. Stat. Comput. 30(2), 419\u2013446 (2020)","journal-title":"Stat. Comput."},{"key":"10252_CR35","doi-asserted-by":"crossref","unstructured":"Stein, M.L., Chen, J., Anitescu, M.: Stochastic approximation of score functions for Gaussian processes. Ann. Appl. Stat. 1162\u20131191 (2013)","DOI":"10.1214\/13-AOAS627"},{"key":"10252_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1494-6","volume-title":"Interpolation of Spatial Data: Some Theory for Kriging","author":"ML Stein","year":"1999","unstructured":"Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer (1999)"},{"key":"10252_CR37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.spasta.2013.06.003","volume":"8","author":"ML Stein","year":"2014","unstructured":"Stein, M.L.: Limitations on low rank approximations for covariance matrices of spatial data. Spat. Stat. 8, 1\u201319 (2014)","journal-title":"Spat. Stat."},{"issue":"2","key":"10252_CR38","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1046\/j.1369-7412.2003.05512.x","volume":"66","author":"ML Stein","year":"2004","unstructured":"Stein, M.L., Chi, Z., Welty, L.J.: Approximating likelihoods for large spatial data sets. J. Roy. Stat. Soc. Ser. B (Stat. Methodol.) 66(2), 275\u2013296 (2004)","journal-title":"J. Roy. Stat. Soc. Ser. B (Stat. Methodol.)"},{"issue":"2","key":"10252_CR39","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1111\/j.2517-6161.1988.tb01729.x","volume":"50","author":"AV Vecchia","year":"1988","unstructured":"Vecchia, A.V.: Estimation and model identification for continuous spatial processes. J. Roy. Stat. Soc. Ser. B (Stat. Methodol.) 50(2), 297\u2013312 (1988)","journal-title":"J. Roy. Stat. Soc. Ser. B (Stat. Methodol.)"},{"issue":"67\u201368","key":"10252_CR40","first-page":"7","volume":"35","author":"S Wright","year":"1999","unstructured":"Wright, S., Nocedal, J.: Numerical optimization. Science 35(67\u201368), 7 (1999)","journal-title":"Science"},{"issue":"465","key":"10252_CR41","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1198\/016214504000000241","volume":"99","author":"H Zhang","year":"2004","unstructured":"Zhang, H.: Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Am. Stat. Assoc. 99(465), 250\u2013261 (2004)","journal-title":"J. Am. Stat. Assoc."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-023-10252-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-023-10252-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-023-10252-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,21]],"date-time":"2024-10-21T13:38:16Z","timestamp":1729517896000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-023-10252-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,30]]},"references-count":41,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["10252"],"URL":"https:\/\/doi.org\/10.1007\/s11222-023-10252-0","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"type":"print","value":"0960-3174"},{"type":"electronic","value":"1573-1375"}],"subject":[],"published":{"date-parts":[[2023,5,30]]},"assertion":[{"value":"10 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 May 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"84"}}