{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:36:57Z","timestamp":1760060217200,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T00:00:00Z","timestamp":1754438400000},"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>Multivariate space\u2013time datasets are often collected at discrete, regularly monitored time intervals and are typically treated as components of time series in environmental science and other applied fields. To effectively characterize such data in geostatistical frameworks, valid and practical covariance models are essential. In this work, we propose several classes of multivariate spatio-temporal covariance matrix functions to model underlying stochastic processes whose discrete temporal margins correspond to well-known autoregressive and moving average (ARMA) models. We derive sufficient and\/or necessary conditions under which these functions yield valid covariance matrices. By leveraging established methodologies from time series analysis and spatial statistics, the proposed models are straightforward to identify and fit in practice. Finally, we demonstrate the utility of these multivariate covariance functions through an application to Kansas weather data, using co-kriging for prediction and comparing the results to those obtained from traditional spatio-temporal models.<\/jats:p>","DOI":"10.3390\/e27080837","type":"journal-article","created":{"date-parts":[[2025,8,7]],"date-time":"2025-08-07T08:33:06Z","timestamp":1754555586000},"page":"837","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multivariate Modeling of Some Datasets in Continuous Space and Discrete Time"],"prefix":"10.3390","volume":"27","author":[{"given":"Rigele","family":"Te","sequence":"first","affiliation":[{"name":"Department of Statistics, Kansas State University, Manhattan, KS 66506, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Statistics, Kansas State University, Manhattan, KS 66506, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1002\/qj.49712555417","article-title":"Construction of correlation functions in two and three dimensions","volume":"125","author":"Gaspari","year":"1999","journal-title":"Q. J. R. Meteorol. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1214\/10-AOAS369","article-title":"A spatial analysis of multivariate output from regional climate models","volume":"5","author":"Sain","year":"2011","journal-title":"Ann. Appl. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"L08705","DOI":"10.1029\/2008GL033423","article-title":"Towards probabilistic projections of climate change impacts on global crop yields","volume":"35","author":"Tebaldi","year":"2008","journal-title":"Geophys. Res. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1330","DOI":"10.1080\/01621459.1999.10473885","article-title":"Classes of nonseparable, spatio-temporal stationary covariance functions","volume":"94","author":"Cressie","year":"1999","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/S0378-3758(02)00353-1","article-title":"Families of spatio-temporal stationary covariance models","volume":"116","author":"Ma","year":"2003","journal-title":"J. Stat. Plann. Inference"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1593","DOI":"10.1214\/13-AOAS656","article-title":"Global space-time models for climate ensembles","volume":"7","author":"Castruccio","year":"2013","journal-title":"Ann. Appl. Stat."},{"key":"ref_7","unstructured":"Cressie, N., and Wikle, C.K. (2015). Statistics for Spatio-Temporal Data, John Wiley & Sons."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"e2648","DOI":"10.1002\/env.2648","article-title":"A spatio-temporal model for the analysis and prediction of fine particulate matter concentration in Beijing","volume":"32","author":"Wan","year":"2021","journal-title":"Environmetrics"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.atmosenv.2019.04.011","article-title":"An advanced spatio-temporal model for particulate matter and gaseous pollutants in Beijing, China","volume":"211","author":"Xu","year":"2019","journal-title":"Atmos. Environ."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Medeiros, E.S., de Lima, R.R., Olinda, R.A., Dantas, L.G., and Santos, C.A.C. (2019). Space\u2013time kriging of precipitation: Modeling the large-scale variation with model GAMLSS. Water, 11.","DOI":"10.3390\/w11112368"},{"key":"ref_11","first-page":"263","article-title":"Stationary time autoregressive representation","volume":"60","author":"Storvik","year":"2002","journal-title":"Stat. Probab. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1111\/j.1467-9868.2005.00520.x","article-title":"Statistical methods for regular monitoring data","volume":"67","author":"Stein","year":"2005","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_13","first-page":"81","article-title":"Spatio-temporal models for some data sets in continuous space and discrete time","volume":"25","author":"Demel","year":"2015","journal-title":"Stat. Sin."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1198\/jasa.2010.tm09420","article-title":"Mat\u00e9rn cross-covariance functions for multivariate random fields","volume":"105","author":"Gneiting","year":"2010","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.jeconom.2006.09.010","article-title":"A spatial model for multivariate lattice data","volume":"140","author":"Sain","year":"2007","journal-title":"J. Econom."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1016\/j.jeconom.2018.11.018","article-title":"Multivariate spatial autoregressive model for large scale social networks","volume":"215","author":"Zhu","year":"2020","journal-title":"J. Econom."},{"key":"ref_17","first-page":"105199","article-title":"Covariance models for multivariate random fields resulting from pseudo cross-variograms","volume":"205","author":"Schlather","year":"2023","journal-title":"J. Multivar. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.aap.2018.05.003","article-title":"Predicting crash frequency for multi-vehicle collision types using multivariate Poisson-lognormal spatial model: A comparative analysis","volume":"118","author":"Hosseinpour","year":"2018","journal-title":"Accid. Anal. Prev."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Somayasa, W., Pasolon, Y.B., and Sutiari, D.K. (2020, January 24). Universal kriging of multivariate spatial data under multivariate isotropic power type variogram model. Proceedings of the 7th International Conference on Mathematics\u2014Pure, Applied and Computation (ICoMPAC 2020), Surabaya, Indonesia.","DOI":"10.1063\/5.0039429"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.jmva.2018.09.007","article-title":"A copula model for non-Gaussian multivariate spatial data","volume":"169","author":"Krupskii","year":"2019","journal-title":"J. Multivar. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.3150\/12-BEJSP06","article-title":"Strictly and non-strictly positive definite functions on spheres","volume":"19","author":"Gneiting","year":"2013","journal-title":"Bernoulli"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1007\/s11004-012-9411-8","article-title":"Stationary and isotropic vector random fields on spheres","volume":"44","author":"Ma","year":"2012","journal-title":"Math. Geosci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/s11004-013-9441-x","article-title":"Isotropic variogram matrix functions on spheres","volume":"45","author":"Du","year":"2013","journal-title":"Math. Geosci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1239\/aap\/1127483743","article-title":"Spatio-temporal variograms and covariance models","volume":"37","author":"Ma","year":"2005","journal-title":"Adv. Appl. Probab."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5921","DOI":"10.1109\/TSP.2011.2166391","article-title":"Spherically invariant vector random fields in space and time","volume":"59","author":"Du","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Cressie, N. (1993). Statistics for Spatial Data, Wiley. [rev. ed.].","DOI":"10.1002\/9781119115151"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1198\/016214502760047113","article-title":"Nonseparable, stationary covariance functions for space-time data","volume":"97","author":"Gneiting","year":"2002","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_28","first-page":"151","article-title":"Geostatistical space-time models, stationarity, separability, and full symmetry","volume":"107","author":"Gneiting","year":"2006","journal-title":"Monogr. Stat. Appl. Probab."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1080\/07362994.2011.532039","article-title":"Vector random fields with second-order moments or second-order increments","volume":"29","author":"Ma","year":"2011","journal-title":"Stoch. Anal. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Stein, M.L. (1999). Interpolation of Spatial Data: Some Theory for Kriging, Springer.","DOI":"10.1007\/978-1-4612-1494-6"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Wackernagel, H. (2003). Multivariate Geostatistics, Springer. [3rd ed.].","DOI":"10.1007\/978-3-662-05294-5"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/8\/837\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:24:41Z","timestamp":1760034281000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/8\/837"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,6]]},"references-count":31,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["e27080837"],"URL":"https:\/\/doi.org\/10.3390\/e27080837","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,8,6]]}}}