{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:56:40Z","timestamp":1760237800334,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,26]],"date-time":"2022-07-26T00:00:00Z","timestamp":1658793600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Technology, Taiwan","award":["108-2221-E-262-003"],"award-info":[{"award-number":["108-2221-E-262-003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The grey Riccati model (GRM) is a generalization of the grey Verhulst model (GVM). Both the GRM and GVM generally perform well in simulating and forecasting the raw sequences with a bell-shaped or single peak feature. Although there are several methods to solve the Riccati differential equation, the existing time response functions of the GRM are generally complicated. In order to reduce the computational complexity of the time response function, this study attempts to transform the Riccati equation into a Bernoulli equation with the help of a known particular solution. Then, a unified time response function of the GRM is obtained by the proposed methodology. The simulation results demonstrate that the proposed unified grey Riccati model performs the same as the grey generalized Verhulst model (a kind of grey Riccati model) and is better than the traditional grey Verhulst model. The fact also reveals that the newly developed grey Riccati model is reasonable and effective.<\/jats:p>","DOI":"10.3390\/axioms11080364","type":"journal-article","created":{"date-parts":[[2022,7,26]],"date-time":"2022-07-26T22:03:42Z","timestamp":1658873022000},"page":"364","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Unified Grey Riccati Model"],"prefix":"10.3390","volume":"11","author":[{"given":"Ming-Feng","family":"Yeh","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Lunghwa University of Science and Technology, Taoyuan 33306, Taiwan"}]},{"given":"Ming-Hung","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Lunghwa University of Science and Technology, Taoyuan 33306, Taiwan"}]},{"given":"Ching-Chuan","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Lunghwa University of Science and Technology, Taoyuan 33306, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Liu, S.F., and Lin, Y. (2011). Grey Systems: Theory and Applications, Springer.","DOI":"10.1007\/978-3-642-16158-2"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"301032","DOI":"10.1155\/2014\/301032","article-title":"A grey NGM(1,1,k) self-memory coupling prediction model for energy consumption prediction","volume":"2014","author":"Guo","year":"2014","journal-title":"Sci. World J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"6234","DOI":"10.1016\/j.apm.2013.01.002","article-title":"Generalized GM (1,1) model and its application in forecasting of fuel production","volume":"37","author":"Zhou","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.compag.2012.03.007","article-title":"Forecasting agricultural output with an improved grey forecasting model based on the genetic algorithm","volume":"85","author":"Ou","year":"2012","journal-title":"Comput. Electron. Agric."},{"key":"ref_5","first-page":"256","article-title":"Using genetic algorithms grey theory to forecast high technology industrial output","volume":"195","author":"Wang","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.engappai.2015.12.011","article-title":"A self-adaptive intelligence grey predictive model with alterable structure and its application","volume":"50","author":"Zeng","year":"2016","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1016\/j.apm.2020.01.014","article-title":"A new-structure grey Verhulst model: Development and performance comparison","volume":"81","author":"Zeng","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7169130","DOI":"10.1155\/2018\/7169130","article-title":"The comparison of grey system and the Verhulst model for rainfall and water in dam prediction","volume":"2018","author":"Puripat","year":"2018","journal-title":"Adv. Meteorol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.neucom.2015.11.032","article-title":"An optimized nonlinear grey Bernoulli model and its applications","volume":"177","author":"Lu","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.chaos.2006.08.024","article-title":"Application of the novel nonlinear grey Bernoulli model for forecasting unemployment rate","volume":"37","author":"Chen","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","first-page":"13961","article-title":"Improvement of grey models by least squares","volume":"38","author":"Xu","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7557","DOI":"10.1016\/j.eswa.2010.04.088","article-title":"Forecasting Taiwan\u2019s major stock indices by the Nash nonlinear grey Bernoulli model","volume":"37","author":"Chen","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_13","first-page":"69","article-title":"The optimization of background value in GM(1,1) model","volume":"2","author":"Wang","year":"2007","journal-title":"J. Grey Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5640","DOI":"10.1016\/j.eswa.2010.02.048","article-title":"An approach to increase prediction precision of GM(1,1) model based on optimization of the initial condition","volume":"37","author":"Wang","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_15","first-page":"292","article-title":"Parameter optimization of nonlinear grey Bernoulli model using particle swarm optimization","volume":"207","author":"Zhou","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4318","DOI":"10.1016\/j.eswa.2009.11.068","article-title":"A genetic algorithm based nonlinear grey Bernoulli model for output forecasting in integrated circuit industry","volume":"37","author":"Hsu","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_17","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","author":"Cui","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_18","first-page":"242809","article-title":"Foundation settlement prediction based on a novel NGM model","volume":"2014","author":"Chen","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zeng, B., Zhou, M., and Zhang, J. (2017). Forecasting the energy consumption of China\u2019s manufacturing using a homologous grey prediction model. Sustainability, 11.","DOI":"10.3390\/su9111975"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4977","DOI":"10.1007\/s00500-019-04248-0","article-title":"The grey generalized Verhulst model and its application for forecasting Chinese pig price index","volume":"24","author":"Zhou","year":"2020","journal-title":"Soft Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"106555","DOI":"10.1016\/j.asoc.2020.106555","article-title":"Predicting China\u2019s energy consumption using a novel grey Riccati model","volume":"95","author":"Wu","year":"2020","journal-title":"Appl. Soft Comput. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"103863","DOI":"10.1016\/j.engappai.2020.103863","article-title":"A novel grey Riccati\u2013Bernoulli model and its application for the clean energy consumption prediction","volume":"95","author":"Xiao","year":"2020","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"118085","DOI":"10.1016\/j.energy.2020.118085","article-title":"A novel riccati equation grey model and its application in forecasting clean energy","volume":"205","author":"Luo","year":"2020","journal-title":"Energy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.neucom.2015.12.114","article-title":"Mean Absolute Percentage Error for regression models","volume":"5","author":"Myttenaere","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_25","unstructured":"Liu, S.F., and Lin, Y. (2006). Grey Information: Theory and Practical Applications, Springer."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/364\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:56:55Z","timestamp":1760140615000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/364"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,26]]},"references-count":25,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["axioms11080364"],"URL":"https:\/\/doi.org\/10.3390\/axioms11080364","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,7,26]]}}}