{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:00:04Z","timestamp":1760151604828,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T00:00:00Z","timestamp":1648684800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Ozone concentrations are key indicators of air quality. Modeling ozone concentrations is challenging because they change both spatially and temporally with complicated structures. Missing data bring even more difficulties. One of our interests in this paper is to model ozone concentrations in a region in the presence of missing data. We propose a method without any assumptions on the correlation structure to estimate the covariance matrix through a dimension expansion method for modeling the semivariograms in nonstationary fields based on the estimations from the hierarchical Bayesian spatio-temporal modeling technique (Le and Zidek). Further, we apply an entropy criterion (Jin et al.) based on a predictive model to decide if new stations need to be added. This entropy criterion helps to solve the environmental network design problem. For demonstration, we apply the method to the ozone concentrations at 25 stations in the Pittsburgh region studied. The comparison of the proposed method and the one is provided through leave-one-out cross-validation, which shows that the proposed method is more general and applicable.<\/jats:p>","DOI":"10.3390\/e24040492","type":"journal-article","created":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T21:33:03Z","timestamp":1648762383000},"page":"492","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Estimation of the Covariance Matrix in Hierarchical Bayesian Spatio-Temporal Modeling via Dimension Expansion"],"prefix":"10.3390","volume":"24","author":[{"given":"Bin","family":"Sun","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuehua","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1023\/A:1009662704779","article-title":"Hierachichical Bayesian space-time models","volume":"5","author":"Wikle","year":"1998","journal-title":"Environ. 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Assoc."},{"unstructured":"Journel, A.G., and Huijbregts, C.J. (1978). Mining Geostatistics, Academic.","key":"ref_9"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1122","DOI":"10.1175\/1520-0493(1980)108<1122:SNMMFV>2.0.CO;2","article-title":"Some new mathematical methods for variational objective analysis using splines and cross-validation","volume":"108","author":"Wabba","year":"1980","journal-title":"Mon. Weather Rev."},{"unstructured":"Ford, K.W. (1963). Information Theory and Statistical Mechanics, Statistical Physics, Benjamin. [3rd ed.].","key":"ref_11"},{"unstructured":"Le, N.D., and Zidek, J.V. (2006). Statistical Analysis of Environmental Space-Time Processes, Springer.","key":"ref_12"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/492\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:47:29Z","timestamp":1760136449000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/492"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,31]]},"references-count":12,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["e24040492"],"URL":"https:\/\/doi.org\/10.3390\/e24040492","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,3,31]]}}}