{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T01:13:16Z","timestamp":1781831596575,"version":"3.54.5"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,11,3]],"date-time":"2021-11-03T00:00:00Z","timestamp":1635897600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,11,3]],"date-time":"2021-11-03T00:00:00Z","timestamp":1635897600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput & Applic"],"published-print":{"date-parts":[[2022,3]]},"DOI":"10.1007\/s00521-021-06373-0","type":"journal-article","created":{"date-parts":[[2021,11,3]],"date-time":"2021-11-03T12:02:32Z","timestamp":1635940952000},"page":"4133-4150","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":77,"title":["An efficient multilayer RBF neural network and its application to regression problems"],"prefix":"10.1007","volume":"34","author":[{"given":"Qinghua","family":"Jiang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lailai","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0825-0883","authenticated-orcid":false,"given":"Chang","family":"Shu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vinothkumar","family":"Sekar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,11,3]]},"reference":[{"key":"6373_CR1","doi-asserted-by":"publisher","DOI":"10.1002\/9781118625590","volume-title":"Applied regression analysis","author":"NR Draper","year":"1998","unstructured":"Draper NR, Smith H (1998) Applied regression analysis. John Wiley & Sons, Hoboken"},{"key":"6373_CR2","doi-asserted-by":"publisher","DOI":"10.1002\/0471704091","volume-title":"Applied linear regression","author":"S Weisberg","year":"2005","unstructured":"Weisberg S (2005) Applied linear regression. John Wiley & Sons, Hoboken"},{"key":"6373_CR3","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316757","volume-title":"Nonlinear regression analysis and its applications","author":"DM Bates","year":"1988","unstructured":"Bates DM, Watts DG (1988) Nonlinear regression analysis and its applications. Wiley, New York"},{"key":"6373_CR4","volume-title":"Introduction to linear regression analysis","author":"DC Montgomery","year":"2012","unstructured":"Montgomery DC, Peck EA, Vining GG (2012) Introduction to linear regression analysis. John Wiley & Sons, Hoboken"},{"key":"6373_CR5","first-page":"345","volume-title":"A rank-invariant method of linear and polynomial regression analysis","author":"H Theil","year":"1992","unstructured":"Theil H (1992) A rank-invariant method of linear and polynomial regression analysis. Springer, Berlin, pp 345\u2013381"},{"key":"6373_CR6","doi-asserted-by":"publisher","DOI":"10.1002\/9781118548387","volume-title":"Applied logistic regression","author":"DW Hosmer Jr","year":"2013","unstructured":"Hosmer DW Jr, Lemeshow S, Sturdivant RX (2013) Applied logistic regression. John Wiley & Sons, Hoboken"},{"key":"6373_CR7","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1080\/00401706.1970.10488634","volume":"12","author":"AE Hoerl","year":"1970","unstructured":"Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12:55\u201367","journal-title":"Technometrics"},{"key":"6373_CR8","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B 58:267\u2013288","journal-title":"J R Stat Soc Ser B"},{"key":"6373_CR9","volume-title":"Bayesian inference in statistical analysis","author":"GEP Box","year":"2011","unstructured":"Box GEP, Tiao GC (2011) Bayesian inference in statistical analysis. John Wiley & Sons, Hoboken"},{"key":"6373_CR10","volume-title":"Deep learning","author":"I Goodfellow","year":"2016","unstructured":"Goodfellow I, Bengio Y, Courville A, Bengio Y (2016) Deep learning. MIT Press, Cambridge"},{"key":"6373_CR11","doi-asserted-by":"publisher","first-page":"683","DOI":"10.1109\/72.159058","volume":"3","author":"SK Pal","year":"1992","unstructured":"Pal SK, Mitra S (1992) Multilayer perceptron, fuzzy sets, and classification. IEEE Trans Neural Netw 3:683\u2013697","journal-title":"IEEE Trans Neural Netw"},{"key":"6373_CR12","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1038\/nature14539","volume":"521","author":"Y LeCun","year":"2015","unstructured":"LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521:436\u2013444","journal-title":"Nature"},{"key":"6373_CR13","doi-asserted-by":"publisher","first-page":"2627","DOI":"10.1016\/S1352-2310(97)00447-0","volume":"32","author":"MW Gardner","year":"1998","unstructured":"Gardner MW, Dorling SR (1998) Artificial neural networks (the multilayer perceptron)\u2014a review of applications in the atmospheric sciences. Atmos Environ 32:2627\u20132636","journal-title":"Atmos Environ"},{"key":"6373_CR14","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1016\/0893-6080(92)90012-8","volume":"5","author":"V K\u016frkov\u00e1","year":"1992","unstructured":"K\u016frkov\u00e1 V (1992) Kolmogorov\u2019s theorem and multilayer neural networks. Neural Netw 5:501\u2013506. https:\/\/doi.org\/10.1016\/0893-6080(92)90012-8","journal-title":"Neural Networks"},{"key":"6373_CR15","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1109\/72.392253","volume":"6","author":"T Chen","year":"1995","unstructured":"Chen T, Chen H (1995) Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans Neural Netw 6:911\u2013917","journal-title":"IEEE Trans Neural Netw"},{"key":"6373_CR16","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/S0925-2312(00)00305-2","volume":"34","author":"BB Chaudhuri","year":"2000","unstructured":"Chaudhuri BB, Bhattacharya U (2000) Efficient training and improved performance of multilayer perceptron in pattern classification. Neurocomputing 34:11\u201327","journal-title":"Neurocomputing"},{"key":"6373_CR17","doi-asserted-by":"publisher","first-page":"1082","DOI":"10.1109\/14.249382","volume":"28","author":"AA Mazroua","year":"1993","unstructured":"Mazroua AA, Salama MMA, Bartnikas R (1993) PD pattern recognition with neural networks using the multilayer perceptron technique. IEEE Trans Electr Insul 28:1082\u20131089","journal-title":"IEEE Trans Electr Insul"},{"key":"6373_CR18","doi-asserted-by":"publisher","first-page":"594","DOI":"10.1080\/07474938.2010.481556","volume":"29","author":"NK Ahmed","year":"2010","unstructured":"Ahmed NK, Atiya AF, El GN, El-Shishiny H (2010) An empirical comparison of machine learning models for time series forecasting. Econom Rev 29:594\u2013621","journal-title":"Econom Rev"},{"key":"6373_CR19","unstructured":"Koskela T, Lehtokangas M, Saarinen J, Kaski K (1996) Time series prediction with multilayer perceptron, FIR and Elman neural networks. In: Proceedings of the world congress on neural networks. Citeseer, pp 491\u2013496"},{"key":"6373_CR20","doi-asserted-by":"crossref","unstructured":"Shedeed HA, Issa MF, El-Sayed SM (2013) Brain EEG signal processing for controlling a robotic arm. In: Proceedings\u20142013 8th international conference on computer engineering and systems, ICCES 2013. IEEE, pp 152\u2013157","DOI":"10.1109\/ICCES.2013.6707191"},{"key":"6373_CR21","doi-asserted-by":"crossref","unstructured":"Nielsen JLG, Holmgaard S, Jiang N et al (2009) Enhanced EMG signal processing for simultaneous and proportional myoelectric control. In: Proceedings of the 31st annual international conference of the IEEE engineering in medicine and biology society: engineering the future of biomedicine, EMBC 2009. IEEE, pp 4335\u20134338","DOI":"10.1109\/IEMBS.2009.5332745"},{"key":"6373_CR22","doi-asserted-by":"crossref","unstructured":"Sezer OB, Ozbayoglu AM, Dogdu E (2017) An artificial neural network-based stock trading system using technical analysis and big data framework. In: arXiv, pp 223\u2013226","DOI":"10.1145\/3077286.3077294"},{"key":"6373_CR23","doi-asserted-by":"publisher","first-page":"105023","DOI":"10.1016\/j.compag.2019.105023","volume":"166","author":"T Moon","year":"2019","unstructured":"Moon T, Hong S, Choi HY et al (2019) Interpolation of greenhouse environment data using multilayer perceptron. Comput Electron Agric 166:105023","journal-title":"Comput Electron Agric"},{"key":"6373_CR24","first-page":"1","volume-title":"Introduction to radial basis function networks","author":"MJL Orr","year":"1996","unstructured":"Orr MJL (1996) Introduction to radial basis function networks. University of Edinburgh, Edinburgh, pp 1\u201367"},{"key":"6373_CR25","unstructured":"Broomhead DS, Lowe D (1988) Radial basis functions, multi-variable functional interpolation and adaptive networks"},{"key":"6373_CR26","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/0003-2670(96)00202-4","volume":"331","author":"B Walczak","year":"1996","unstructured":"Walczak B, Massart DL (1996) The radial basis functions\u2014partial least squares approach as a flexible non-linear regression technique. Anal Chim Acta 331:177\u2013185","journal-title":"Anal Chim Acta"},{"key":"6373_CR27","doi-asserted-by":"publisher","first-page":"941","DOI":"10.1016\/S0045-7825(02)00618-7","volume":"192","author":"C Shu","year":"2003","unstructured":"Shu C, Ding H, Yeo KS (2003) Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier\u2013Stokes equations. Comput Methods Appl Mech Eng 192:941\u2013954","journal-title":"Comput Methods Appl Mech Eng"},{"key":"6373_CR28","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1016\/j.enganabound.2003.02.001","volume":"28","author":"C Shu","year":"2004","unstructured":"Shu C, Ding H, Yeo KS (2004) Solution of partial differential equations by a global radial basis function-based differential quadrature method. Eng Anal Bound Elem 28:1217\u20131226","journal-title":"Eng Anal Bound Elem"},{"key":"6373_CR29","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1080\/10407780590889545","volume":"47","author":"H Ding","year":"2005","unstructured":"Ding H, Shu C, Yeo KS, Lu ZL (2005) Simulation of natural convection in eccentric annuli between a square outer cylinder and a circular inner cylinder using local MQ-DQ method. Numer Heat Transf Part A Appl 47:291\u2013313","journal-title":"Numer Heat Transf Part A Appl"},{"key":"6373_CR30","doi-asserted-by":"publisher","first-page":"2161","DOI":"10.1016\/j.neucom.2005.07.010","volume":"69","author":"C Harpham","year":"2006","unstructured":"Harpham C, Dawson CW (2006) The effect of different basis functions on a radial basis function network for time series prediction: a comparative study. Neurocomputing 69:2161\u20132170","journal-title":"Neurocomputing"},{"key":"6373_CR31","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/S0893-6080(01)00027-2","volume":"14","author":"F Schwenker","year":"2001","unstructured":"Schwenker F, Kestler HA, Palm G (2001) Three learning phases for radial-basis-function networks. Neural Netw 14:439\u2013458. https:\/\/doi.org\/10.1016\/S0893-6080(01)00027-2","journal-title":"Neural Netw"},{"key":"6373_CR32","doi-asserted-by":"crossref","unstructured":"Na S, Xumin L, Yong G (2010) Research on k-means clustering algorithm: an improved k-means clustering algorithm. In: IEEE, pp 63\u201367","DOI":"10.1109\/IITSI.2010.74"},{"key":"6373_CR33","unstructured":"Broomhead DS, Lowe D (1988) Multivariable functional interpolation and adaptive networks, complex systems, vol 2"},{"key":"6373_CR34","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1109\/37.55123","volume":"10","author":"F-C Chen","year":"1990","unstructured":"Chen F-C (1990) Back-propagation neural networks for nonlinear self-tuning adaptive control. IEEE Control Syst Mag 10:44\u201348","journal-title":"IEEE Control Syst Mag"},{"key":"6373_CR35","first-page":"65","volume-title":"Theory of the backpropagation neural network","author":"R Hecht-Nielsen","year":"1992","unstructured":"Hecht-Nielsen R (1992) Theory of the backpropagation neural network. Elsevier, New York, pp 65\u201393"},{"key":"6373_CR36","first-page":"1264","volume":"47","author":"P Sibi","year":"2013","unstructured":"Sibi P, Jones SA, Siddarth P (2013) Analysis of different activation functions using back propagation neural networks. J Theor Appl Inf Technol 47:1264\u20131268","journal-title":"J Theor Appl Inf Technol"},{"key":"6373_CR37","unstructured":"Kingma DP, Ba J (2014) Adam: a method for stochastic optimization. arXiv14126980"},{"key":"6373_CR38","doi-asserted-by":"crossref","unstructured":"Chao J, Hoshino M, Kitamura T, Masuda T (2001) A multilayer RBF network and its supervised learning. In: Proceedings of the international joint conference on neural networks. IEEE, pp 1995\u20132000","DOI":"10.1109\/IJCNN.2001.938470"},{"key":"6373_CR39","doi-asserted-by":"crossref","unstructured":"Mhaskar H, Liao Q, Poggio T (2017) When and why are deep networks better than shallow ones? In: Proceedings of the AAAI conference on artificial intelligence","DOI":"10.1609\/aaai.v31i1.10913"},{"key":"6373_CR40","doi-asserted-by":"crossref","unstructured":"Bottou L (2010) Large-scale machine learning with stochastic gradient descent. In: Proceedings of COMPSTAT\u20192010. Springer, pp 177\u2013186","DOI":"10.1007\/978-3-7908-2604-3_16"},{"key":"6373_CR41","unstructured":"Glorot X, Bengio Y (2010) Understanding the difficulty of training deep feedforward neural networks. In: Journal of Machine Learning Research, pp 249\u2013256"},{"key":"6373_CR42","unstructured":"Chollet F (2018) Deep Learning mit Python und Keras: Das Praxis-Handbuch vom Entwickler der Keras-Bibliothek. MITP-Verlags GmbH & Co. KG"},{"key":"6373_CR43","unstructured":"Abadi M, Agarwal A, Barham P et al (2016) TensorFlow: large-scale machine learning on heterogeneous distributed systems"},{"key":"6373_CR44","doi-asserted-by":"publisher","first-page":"109136","DOI":"10.1016\/j.jcp.2019.109136","volume":"404","author":"AD Jagtap","year":"2020","unstructured":"Jagtap AD, Kawaguchi K, Karniadakis GE (2020) Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J Comput Phys 404:109136","journal-title":"J Comput Phys"},{"key":"6373_CR45","doi-asserted-by":"publisher","first-page":"665","DOI":"10.1109\/21.256541","volume":"23","author":"J-S Jang","year":"1993","unstructured":"Jang J-S (1993) ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans Syst Man Cybern 23:665\u2013685","journal-title":"IEEE Trans Syst Man Cybern"},{"key":"6373_CR46","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1016\/j.neucom.2005.02.006","volume":"69","author":"Y Chen","year":"2006","unstructured":"Chen Y, Yang B, Dong J (2006) Time-series prediction using a local linear wavelet neural network. Neurocomputing 69:449\u2013465","journal-title":"Neurocomputing"},{"key":"6373_CR47","doi-asserted-by":"publisher","first-page":"750","DOI":"10.1016\/j.snb.2007.09.060","volume":"129","author":"S De Vito","year":"2008","unstructured":"De Vito S, Massera E, Piga M et al (2008) On field calibration of an electronic nose for benzene estimation in an urban pollution monitoring scenario. Sens Actuators B Chem 129:750\u2013757","journal-title":"Sens Actuators B Chem"},{"key":"6373_CR48","unstructured":"http:\/\/archive.ics.uci.edu\/ml\/datasets\/Air+Quality"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-021-06373-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00521-021-06373-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-021-06373-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,11]],"date-time":"2024-09-11T11:02:26Z","timestamp":1726052546000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00521-021-06373-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,3]]},"references-count":48,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,3]]}},"alternative-id":["6373"],"URL":"https:\/\/doi.org\/10.1007\/s00521-021-06373-0","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,3]]},"assertion":[{"value":"28 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 July 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 November 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declaration"}},{"value":"The authors declare that they have no conflict of interest to this work.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}