{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T14:03:00Z","timestamp":1773237780133,"version":"3.50.1"},"reference-count":42,"publisher":"Emerald","issue":"2","license":[{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.emerald.com\/insight\/site-policies"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["GS"],"published-print":{"date-parts":[[2019,4,1]]},"abstract":"<jats:sec><jats:title content-type=\"abstract-subheading\">Purpose<\/jats:title><jats:p>The purpose of this paper is to study a fractional grey model FAGM(1,1,<jats:italic>t<\/jats:italic><jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>) based on the GM(1,1,<jats:italic>t<\/jats:italic><jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>) model and the fractional accumulated generating operation, and then predict the national health expenditure, the government health expenditure and the out-of-pocket health expenditure of China.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-subheading\">Design\/methodology\/approach<\/jats:title><jats:p>The presented univariate grey model is systematically studied by using the grey modelling technique, the fractional accumulated generating operation and the trapezoid approximation formula of definite integral. The optimal system parameters<jats:italic>r<\/jats:italic>and<jats:italic>\u03b1<\/jats:italic>are evaluated by the particle swarm optimisation algorithm.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-subheading\">Findings<\/jats:title><jats:p>The expressions of the time response function and the restored values of this model are derived. The GM(1,1), NGM(1,1,<jats:italic>k<\/jats:italic>,<jats:italic>c<\/jats:italic>) and GM(1,1,<jats:italic>t<\/jats:italic><jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>) models are particular cases of the FAGM(1,1,<jats:italic>t<\/jats:italic><jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>) model with deterministic<jats:italic>r<\/jats:italic>and<jats:italic>\u03b1<\/jats:italic>. Compared with other forecasting models, the results of the FAGM(1,1,<jats:italic>t<\/jats:italic><jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>) model have higher precision.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-subheading\">Practical implications<\/jats:title><jats:p>The superiority of the new model has high potential to be used in the medicine and health fields and others. Results can provide a guideline for government decision making.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-subheading\">Originality\/value<\/jats:title><jats:p>The univariate fractional grey model FAGM (1,1,<jats:italic>t<\/jats:italic><jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>) successfully studies the China\u2019s health expenditure.<\/jats:p><\/jats:sec>","DOI":"10.1108\/gs-11-2018-0053","type":"journal-article","created":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T05:38:42Z","timestamp":1557898722000},"page":"232-250","source":"Crossref","is-referenced-by-count":43,"title":["Analysis of novel FAGM(1,1,<i>t<\/i><sup><i>\u03b1<\/i><\/sup>) model to forecast health expenditure of China"],"prefix":"10.1108","volume":"9","author":[{"given":"Wenqing","family":"Wu","sequence":"first","affiliation":[]},{"given":"Xin","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Yuanyuan","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Yong","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Xinxing","family":"Wu","sequence":"additional","affiliation":[]}],"member":"140","reference":[{"issue":"6","key":"key2020092404231335100_ref001","doi-asserted-by":"crossref","first-page":"1194","DOI":"10.1016\/j.cnsns.2006.08.008","article-title":"Forecasting of foreign exchange rates of Taiwan\u2019s major trading partners by novel nonlinear grey Bernoulli model NGBM(1, 1)","volume":"13","year":"2008","journal-title":"Communications in Nonlinear Science and Numerical Simulation"},{"issue":"1","key":"key2020092404231335100_ref002","first-page":"1","article-title":"Foundation settlement prediction based on a novel NGM model","volume":"2014","year":"2014","journal-title":"Mathematical Problems in Engineering"},{"issue":"6","key":"key2020092404231335100_ref003","doi-asserted-by":"crossref","first-page":"4399","DOI":"10.1016\/j.apm.2012.09.052","article-title":"A novel grey forecasting model and its optimization","volume":"37","year":"2013","journal-title":"Applied Mathematical Modelling"},{"issue":"5","key":"key2020092404231335100_ref004","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1016\/S0167-6911(82)80025-X","article-title":"Control problems of grey systems","volume":"1","year":"1982","journal-title":"Systems & Control Letters"},{"issue":"15","key":"key2020092404231335100_ref005","first-page":"314","article-title":"Forecasting China\u2019s electricity consumption using a new grey prediction model","volume":"149","year":"2018","journal-title":"Energy"},{"issue":"4","key":"key2020092404231335100_ref006","first-page":"114","article-title":"Estimation of Chinese CO2 emission based on a discrete fractional accumulation grey model","volume":"27","year":"2015","journal-title":"The Journal of Grey System"},{"issue":"2","key":"key2020092404231335100_ref007","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1108\/GS-01-2018-0005","article-title":"Comparison study on the nonlinear parameter optimization of nonlinear grey Bernoulli model (NGBM(1,1)) between intelligent optimizers","volume":"8","year":"2018","journal-title":"Grey Systems: Theory and Application"},{"issue":"6-7","key":"key2020092404231335100_ref008","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1080\/09720502.2017.1382150","article-title":"Non-equidistance and nonhomogeneous grey model NNFGM(1,1) with the fractional order accumulation and its application","volume":"20","year":"2017","journal-title":"Journal of Interdisciplinary Mathematics"},{"issue":"3","key":"key2020092404231335100_ref009","first-page":"58","article-title":"Grey forecasting model with polynomial term and its optimization","volume":"29","year":"2017","journal-title":"Journal of Grey Systems"},{"issue":"6","key":"key2020092404231335100_ref010","first-page":"822","article-title":"Do commercial building sector-derived carbon emissions decouple from the economic growth in tertiary industry? 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