{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T16:14:11Z","timestamp":1791303251304,"version":"4.1.0"},"reference-count":0,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2000,7,1]],"date-time":"2000-07-01T00:00:00Z","timestamp":962409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2000,7,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>Penalized likelihood methods provide a range of practical modelling tools, including spline smoothing, generalized additive models and variants of ridge regression. Selecting the correct weights for penalties is a critical part of using these methods and in the single-penalty case the analyst has several well-founded techniques to choose from. However, many modelling problems suggest a formulation employing multiple penalties, and here general methodology is lacking. A wide family of models with multiple penalties can be fitted to data by iterative solution of the generalized ridge regression problem minimize ||W \u00a01\/2 (Xp\u2212y) ||2\u03c1+\u03a3i=1 \u00a0m\u2003\u03b8i \u00a0p\u2032S \u00a0i \u00a0p (p is a parameter vector, X a design matrix, S \u00a0i a non-negative definite coefficient matrix defining the ith penalty with associated smoothing parameter \u03b8i, W a diagonal weight matrix, y a vector of data or pseudodata and \u03c1 an \u2018overall\u2019 smoothing parameter included for computational efficiency). This paper shows how smoothing parameter selection can be performed efficiently by applying generalized cross-validation to this problem and how this allows non-linear, generalized linear and linear models to be fitted using multiple penalties, substantially increasing the scope of penalized modelling methods. Examples of non-linear modelling, generalized additive modelling and anisotropic smoothing are given.<\/jats:p>","DOI":"10.1111\/1467-9868.00240","type":"journal-article","created":{"date-parts":[[2003,3,12]],"date-time":"2003-03-12T19:36:21Z","timestamp":1047497781000},"page":"413-428","source":"Crossref","is-referenced-by-count":546,"title":["Modelling and Smoothing Parameter Estimation With Multiple Quadratic Penalties"],"prefix":"10.1093","volume":"62","author":[{"given":"S. N.","family":"Wood","sequence":"first","affiliation":[{"name":"University of St Andrews , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2002,1,6]]},"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2F1467-9868.00240","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/62\/2\/413\/49589723\/jrsssb_62_2_413.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/62\/2\/413\/49589723\/jrsssb_62_2_413.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T06:59:06Z","timestamp":1679468346000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/62\/2\/413\/7083238"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,7,1]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2002,1,6]]},"published-print":{"date-parts":[[2000,7,1]]}},"URL":"https:\/\/doi.org\/10.1111\/1467-9868.00240","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2000,7]]},"published":{"date-parts":[[2000,7,1]]}}}