{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:23:11Z","timestamp":1787329391229,"version":"3.56.0"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>This paper concerns kernel-based interpolation methods for vector data with correlated components. It gives conditions for a matrix kernel to be conditionally positive definite in an appropriate sense. The conditions allow construction of matrix kernels from nonsymmetric mixtures and scalings of scalar kernels. In particular the kernel used to model the influence of component i on component j can be different from that used to model the influence of component j on component i. The vector modeling techniques considered are particularly appropriate when there are relatively few measurements of one quantity and relatively many of another \u201ccorrelated\u201d quantity. The paper concludes with some numerical tests on model problems.<\/jats:p>","DOI":"10.1137\/090758076","type":"journal-article","created":{"date-parts":[[2011,8,16]],"date-time":"2011-08-16T18:13:23Z","timestamp":1313518403000},"page":"1975-1995","source":"Crossref","is-referenced-by-count":3,"title":["Kernel-Based Methods for Vector-Valued Data with Correlated Components"],"prefix":"10.1137","volume":"33","author":[{"given":"R. K.","family":"Beatson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"W. zu","family":"Castell","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S. J.","family":"Schr\u00f6dl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,8,16]]},"reference":[{"key":"R1","unstructured":"M. Abramowitz and I. A. Stegun,\n                      Handbook of Mathematical Functions\n                      , Dover Publications, New York, 1965."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9045(91)90025-6"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asp078"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"R5","unstructured":"M. D. Buhmann,\n                      Radial Basis Functions\n                      , Cambridge Monogr. Appl. Comput. Math., Cambridge University Press, Cambridge, UK, 2003."},{"key":"R6","first-page":"1615","volume":"9","author":"Caponnetto A.","year":"2008","journal-title":"J. Mach. Learning Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"J. P. Chil\u00e9s and P. Delfiner,\n                      Geostatistics, Modeling Spatial Uncertainty\n                      , John Wiley and Sons, New York, 1999.","DOI":"10.1002\/9780470316993"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-07-02096-0"},{"key":"R9","unstructured":"A. Graham,\n                      Kronecker Products and Matrix Calculus with Applications\n                      , Halsted Press, John Wiley and Sons, New York, 1981."},{"key":"R10","unstructured":"G. Matheron,\n                      Les variables R\u00e9gionalis\u00e9es et Leur Estimation\n                      , Masson, Paris, 1965 (in French)."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01893414"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1162\/0899766052530802"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1994-1254147-6"},{"key":"R14","unstructured":"G. Pedrick,\n                      Theory of Reproducing Kernel Hilbert Spaces of Vector Valued Functions\n                      , Technical Report 19, University of Kansas, Lawrence, KS, 1957."},{"key":"R15","doi-asserted-by":"crossref","unstructured":"B. K. Sch\u00f6lkopf and A. Smola,\n                      Learning with Kernels\n                      , MIT Press, Cambridge, MA, 2002.","DOI":"10.7551\/mitpress\/4175.001.0001"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/BF00893273"},{"key":"R17","unstructured":"H. Wendland,\n                      Scattered Data Approximation\n                      , Cambridge Monogr. Appl. Comput. 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