{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T12:31:53Z","timestamp":1753360313363,"version":"3.37.3"},"reference-count":71,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T00:00:00Z","timestamp":1611014400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T00:00:00Z","timestamp":1611014400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC 2046","EXC 2046"],"award-info":[{"award-number":["EXC 2046","EXC 2046"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2021,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We investigate opinion dynamics based on an agent-based model and are interested in predicting the evolution of the percentages of the entire agent population that share an opinion. Since these opinion percentages can be seen as an aggregated observation of the full system state, the individual opinions of each agent, we view this in the framework of the Mori\u2013Zwanzig projection formalism. More specifically, we show how to estimate a nonlinear autoregressive model (NAR) with memory from data given by a time series of opinion percentages, and discuss its prediction capacities for various specific topologies of the agent interaction network. We demonstrate that the inclusion of memory terms significantly improves the prediction quality on examples with different network topologies.<\/jats:p>","DOI":"10.1007\/s00332-020-09673-2","type":"journal-article","created":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T13:02:56Z","timestamp":1611061376000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Memory-Based Reduced Modelling and Data-Based Estimation of Opinion Spreading"],"prefix":"10.1007","volume":"31","author":[{"given":"Niklas","family":"Wulkow","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P\u00e9ter","family":"Koltai","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christof","family":"Sch\u00fctte","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,1,19]]},"reference":[{"issue":"3","key":"9673_CR1","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1890\/13-1452.1","volume":"95","author":"K Aho","year":"2014","unstructured":"Aho, K., Derryberry, D., Peterson, T.: Model selection for ecologists: the worldviews of AIC and BIC. Ecology 95(3), 631\u2013636 (2014)","journal-title":"Ecology"},{"key":"9673_CR2","first-page":"943","volume":"6","author":"H An","year":"1996","unstructured":"An, H., Huang, F.: The geometrical ergodicity of nonlinear autoregressive models. Stat. Sin. 6, 943\u2013956 (1996)","journal-title":"Stat. Sin."},{"key":"9673_CR3","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/s11633-019-1169-8","volume":"16","author":"BDO Anderson","year":"2019","unstructured":"Anderson, B.D.O., Ye, M.: Recent advances in the modelling and analysis of opinion dynamics on influence networks. Int. J. Autom. Comput. 16, 129\u2013149 (2019)","journal-title":"Int. J. Autom. Comput."},{"issue":"16","key":"9673_CR4","doi-asserted-by":"publisher","first-page":"2096","DOI":"10.1137\/17M1125236","volume":"4","author":"H Arbabi","year":"2017","unstructured":"Arbabi, H., Mezic, I.: Ergodic theory, dynamic mode decomposition and computation of spectral properties of the koopman operator. SIAM J. Appl. Dyn. Syst. 4(16), 2096\u20132126 (2017)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"5","key":"9673_CR5","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1016\/0378-3758(81)90024-0","volume":"2","author":"J Baksalary","year":"1981","unstructured":"Baksalary, J., Kala, R.: Simple Least Squares estimation versus best linear unbiased prediction. J. Stat. Plan. Inference 2(5), 147\u2013151 (1981)","journal-title":"J. Stat. Plan. Inference"},{"key":"9673_CR6","doi-asserted-by":"publisher","first-page":"1450025","DOI":"10.1142\/S0219525914500258","volume":"12","author":"S Banisch","year":"2014","unstructured":"Banisch, S.: From microscopic heterogeneity to macroscopic complexity in the contrarian voter model. Adv. Complex Syst. 12, 1450025 (2014)","journal-title":"Adv. Complex Syst."},{"key":"9673_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-24877-6","volume-title":"Markov Chain Aggregation for Agent-Based Models","author":"S Banisch","year":"2016","unstructured":"Banisch, S.: Markov Chain Aggregation for Agent-Based Models, vol. 1. Springer, Berlin (2016)"},{"key":"9673_CR8","doi-asserted-by":"crossref","unstructured":"Banisch, S., Lima, R., Ara\u00fajo, T.: Agent based models and opinion dynamics as Markov chains. Soc. Netw. 34, 549\u2013561 (2011)","DOI":"10.1016\/j.socnet.2012.06.001"},{"issue":"5","key":"9673_CR9","doi-asserted-by":"publisher","first-page":"1530","DOI":"10.2307\/1940005","volume":"73","author":"AA Berryman","year":"1992","unstructured":"Berryman, A.A.: The orgins and evolution of predator-prey theory. Ecology 73(5), 1530\u20131535 (1992)","journal-title":"Ecology"},{"key":"9673_CR10","doi-asserted-by":"publisher","DOI":"10.1002\/9781118535561","volume-title":"Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains","author":"S Billings","year":"2013","unstructured":"Billings, S.: Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains, vol. 1. Wiley, Hoboken (2013)"},{"key":"9673_CR11","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s00332-017-9415-0","volume":"28","author":"A Bittracher","year":"2018","unstructured":"Bittracher, A., Koltai, P., Klus, S., Banisch, R., Dellnitz, M., Sch\u00fctte, C.: Transition manifolds of complex metastable systems. J. Nonlinear Sci. 28, 471\u2013512 (2018)","journal-title":"J. Nonlinear Sci."},{"key":"9673_CR12","doi-asserted-by":"publisher","first-page":"066117","DOI":"10.1103\/PhysRevE.85.066117","volume":"85","author":"GA B\u00f6hme","year":"2012","unstructured":"B\u00f6hme, G.A., Gross, T.: Fragmentation transitions in multistate voter models. Phys. Rev. E 85, 066117 (2012)","journal-title":"Phys. Rev. E"},{"key":"9673_CR13","first-page":"11","volume":"100","author":"P Bolzern","year":"2017","unstructured":"Bolzern, P., Colaneri, P., Nicolao, G.: Opinion influence and evolution in social networks: a markovian agents model. Automatica 100, 11 (2017)","journal-title":"Automatica"},{"key":"9673_CR14","unstructured":"Boschia, G., Cammarotaa, C., K\u00fchna, R.: Opinion dynamics with memory: how a society is shaped by its own past. arXiv:1909.12590, 09 (2019)"},{"key":"9673_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-007-7606-7","volume-title":"An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation","author":"GR Bowman","year":"2014","unstructured":"Bowman, G.R., Pande, V.S., No\u00e9, F.: An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation, vol. 1. Springer, Berlin (2014)"},{"key":"9673_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-0320-4","volume-title":"Time Series: Theory and Methods","author":"PJ Brockwell","year":"1991","unstructured":"Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods, vol. 2. Springer, Berlin (1991)"},{"key":"9673_CR17","doi-asserted-by":"crossref","unstructured":"Brunton, S., Brunton, B., Proctor, J., Kaiser, E., Kutz, J.: Chaos as an intermittently forced linear system. Nat. Commun. 8, 1\u20139 (2016)","DOI":"10.1038\/s41467-017-00030-8"},{"issue":"15","key":"9673_CR18","doi-asserted-by":"publisher","first-page":"3932","DOI":"10.1073\/pnas.1517384113","volume":"113","author":"SL Brunton","year":"2016","unstructured":"Brunton, S.L., Proctor, J.P.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. 113(15), 3932\u20133937 (2016)","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"49","key":"9673_CR19","doi-asserted-by":"publisher","first-page":"710","DOI":"10.1016\/j.ifacol.2016.10.249","volume":"18","author":"SL Brunton","year":"2016","unstructured":"Brunton, S.L., Proctor, J.P.L., Kutz, J.N.: Sparse Identification of Nonlinear Dynamics with Control (SINDYc). IFAC-PapersOnLine Issue 18(49), 710\u2013715 (2016)","journal-title":"IFAC-PapersOnLine Issue"},{"key":"9673_CR20","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1103\/RevModPhys.81.591","volume":"81","author":"C Castellano","year":"2009","unstructured":"Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591\u2013646 (2009)","journal-title":"Rev. Mod. Phys."},{"key":"9673_CR21","first-page":"1","volume":"04","author":"G Chen","year":"2018","unstructured":"Chen, G., Duan, X., Friedkin, N., Bullo, F.: Social power dynamics over switching and stochastic influence networks. IEEE Trans. Autom. Control 04, 1 (2018)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"7","key":"9673_CR22","doi-asserted-by":"publisher","first-page":"2968","DOI":"10.1073\/pnas.97.7.2968","volume":"97","author":"AJ Chorin","year":"2000","unstructured":"Chorin, A.J., Hald, O.H., Kupferman, R.: Optimal prediction and the Mori\u2013Zwanzig representation of irreversible processes. Proc. Natl. Acad. Sci. 97(7), 2968\u20132973 (2000)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"9673_CR23","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/S0167-2789(02)00446-3","volume":"166","author":"AJ Chorin","year":"2002","unstructured":"Chorin, A.J., Hald, O.H., Kupferman, R.: Optimal prediction with memory. Phys. D 166, 239\u2013257 (2002)","journal-title":"Phys. D"},{"key":"9673_CR24","doi-asserted-by":"crossref","unstructured":"Davis, R., Zang, P., Zheng, T.: Sparse vector autoregressive modeling. J. Comput. Graph. Stat. 30, 1077\u20131096 (2012)","DOI":"10.1080\/10618600.2015.1092978"},{"key":"9673_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3343483","volume":"13","author":"A De","year":"2019","unstructured":"De, A., Bhattacharya, S., Bhattacharya, P., Ganguly, N., Chakrabarti, S.: Learning linear influence models in social networks from transient opinion dynamics. ACM Trans. Web 13, 1\u201333 (2019)","journal-title":"ACM Trans. Web"},{"key":"9673_CR26","unstructured":"Devroye, L., Gy\u00f6rfi, L., Lugosi, G.: A probabilistic theory of pattern recognition, vol. 31. Springer, Berlin (2013)"},{"key":"9673_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-62010-2","volume-title":"Geometric Measure Theory","author":"H Federer","year":"1996","unstructured":"Federer, H.: Geometric Measure Theory, vol. 1. Springer, Berlin (1996)"},{"key":"9673_CR28","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1186\/1752-0509-1-39","volume":"1","author":"A Fujita","year":"2007","unstructured":"Fujita, A., Sato, J., Garay, M., Yamaguchi, R., Miyano, S., Sogayar, M., Ferreira, C.: Modeling gene expression regulatory networks with the sparse vector autoregressive model. BMC Syst. Biol. 1, 39 (2007)","journal-title":"BMC Syst. Biol."},{"key":"9673_CR29","doi-asserted-by":"crossref","unstructured":"Gilani, F., Giannakis, D., Harlim, J.: Kernel-based prediction of non-markovian time series, 07 (2020)","DOI":"10.1016\/j.physd.2020.132829"},{"key":"9673_CR30","doi-asserted-by":"crossref","unstructured":"Hansen, P., D. O\u2019leary, : The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14, 1487\u20131503 (1993)","DOI":"10.1137\/0914086"},{"issue":"1","key":"9673_CR31","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/BF01608556","volume":"50","author":"M H\u00e9non","year":"1976","unstructured":"H\u00e9non, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50(1), 69\u201377 (1976)","journal-title":"Commun. Math. Phys."},{"key":"9673_CR32","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1039\/B902479B","volume":"144","author":"C Hij\u00f3n","year":"2010","unstructured":"Hij\u00f3n, C., Espa\u00f1ol, P., Vanden-Eijnden, E., Delgado-Buscalioni, R.: Mori\u2013Zwanzig formalism as a practical computational tool. Faraday Discuss. 144, 301\u201322 (2010). discussion 323","journal-title":"Faraday Discuss."},{"key":"9673_CR33","doi-asserted-by":"publisher","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","volume":"9","author":"S Hochreiter","year":"1997","unstructured":"Hochreiter, S., Schmidhuber, J.: Long short-term memory. Neural Comput. 9, 1735\u201380 (1997)","journal-title":"Neural Comput."},{"key":"9673_CR34","doi-asserted-by":"publisher","first-page":"1700","DOI":"10.1137\/100790549","volume":"9","author":"I Horenko","year":"2011","unstructured":"Horenko, I.: On analysis of nonstationary categorical data time series: dynamical dimension reduction, model selection, and applications to computational sociology. Multiscale Model. Simul. 9, 1700\u20131726 (2011)","journal-title":"Multiscale Model. Simul."},{"key":"9673_CR35","doi-asserted-by":"publisher","first-page":"016706","DOI":"10.1103\/PhysRevE.76.016706","volume":"76","author":"I Horenko","year":"2007","unstructured":"Horenko, I., Hartmann, C., Sch\u00fctte, C., Noe, F.: Data-based parameter estimation of generalized multidimensional Langevin processes. Phys. Rev. E 76, 016706 (2007)","journal-title":"Phys. Rev. E"},{"key":"9673_CR36","first-page":"03","volume":"505","author":"A Jedrzejewski","year":"2018","unstructured":"Jedrzejewski, A., Sznajd-Weron, K.: Impact of memory on opinion dynamics. Phy. A Stat. Mech. Appl. 505, 03 (2018)","journal-title":"Phy. A Stat. Mech. Appl."},{"key":"9673_CR37","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1023\/A:1010090405266","volume":"1","author":"N Jennings","year":"1998","unstructured":"Jennings, N., Sycara, K., Wooldridge, M.: A roadmap of agent research and development. Auton. Agents Multi-Agent Syst. 1, 7\u201338 (1998)","journal-title":"Auton. Agents Multi-Agent Syst."},{"key":"9673_CR38","doi-asserted-by":"crossref","unstructured":"Jovanovic, M., Schmid, P., Nichols, J.: Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26, 1\u201322 (2013)","DOI":"10.1063\/1.4863670"},{"key":"9673_CR39","doi-asserted-by":"crossref","unstructured":"Kaiser, E., Kutz, J.N., Brunton, S.L.: Sparse identification of nonlinear dynamics for model predictive control in the low-data limit. Proc. R. Soc. A 474, 20180335 (2018)","DOI":"10.1098\/rspa.2018.0335"},{"key":"9673_CR40","doi-asserted-by":"crossref","unstructured":"Klimek, P., Lambiotte, R., Thurner, S.: Opinion formation in laggard societies. Europhys. Lett. EPL 82, 1\u20135 (2007)","DOI":"10.1209\/0295-5075\/82\/18007"},{"key":"9673_CR41","doi-asserted-by":"publisher","first-page":"132416","DOI":"10.1016\/j.physd.2020.132416","volume":"406","author":"S Klus","year":"2020","unstructured":"Klus, S., N\u00fcske, F., Peitz, S., Niemann, J.-H., Clementi, C., Schuette, C.: Data-driven approximation of the Koopman generator: model reduction, system identification, and control. Phys. D 406, 132416 (2020)","journal-title":"Phys. D"},{"key":"9673_CR42","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/j.physd.2014.12.005","volume":"297","author":"D Kondrashov","year":"2015","unstructured":"Kondrashov, D., Chekroun, M.D., Ghil, M.: Data-driven non-Markovian closure models. Phys. D Nonlinear Phenom. 297, 33\u201355 (2015)","journal-title":"Phys. D Nonlinear Phenom."},{"key":"9673_CR43","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-71887-3","volume-title":"Information Criteria and Statistical Modeling","author":"S Konishi","year":"2008","unstructured":"Konishi, S., Kitagawa, G.: Information Criteria and Statistical Modeling, vol. 1. Springer, Berlin (2008)"},{"issue":"5","key":"9673_CR44","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1073\/pnas.17.5.315","volume":"17","author":"BO Koopman","year":"1931","unstructured":"Koopman, B.O.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. 17(5), 315\u2013318 (1931)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"9673_CR45","doi-asserted-by":"crossref","unstructured":"Laubenbacher, R., Jarrah, A.\u00a0S., Mortveit, H.\u00a0S., Ravi, S.: Agent Based Modeling, Mathematical Formalism for, pp. 160\u2013176. Springer New York, New York, NY, (2009)","DOI":"10.1007\/978-0-387-30440-3_10"},{"issue":"50","key":"9673_CR46","doi-asserted-by":"publisher","first-page":"14183","DOI":"10.1073\/pnas.1609587113","volume":"113","author":"H Lei","year":"2016","unstructured":"Lei, H., Baker, N.A., Li, X.: Data-driven parameterization of the generalized Langevin equation. Proc. Natl. Acad. Sci. 113(50), 14183\u201314188 (2016)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"9673_CR47","first-page":"539","volume":"17","author":"X Li","year":"2017","unstructured":"Li, X., Chu, W.: The Mori-Zwanzig formalism for the derivation of a fluctuating heat conduction model from molecular dynamics. Commun. Math. Sci. 17, 539\u2013563 (2017)","journal-title":"Commun. Math. Sci."},{"key":"9673_CR48","first-page":"10","volume":"151","author":"Q Li","year":"2012","unstructured":"Li, Q., Braunstein, L., Wang, H., Shao, J., Stanley, H., Havline, S.: Non-consensus opinion models on complex networks. J. Stat. Phys. 151, 10 (2012)","journal-title":"J. Stat. Phys."},{"key":"9673_CR49","unstructured":"Lin, K.K., Lu, F.: Data-driven model reduction, Wiener projections, and the Mori\u2013Zwanzig formalism. arXiv:1908.07725v1, (2019)"},{"key":"9673_CR50","doi-asserted-by":"publisher","first-page":"06","DOI":"10.1073\/pnas.1822012116","volume":"116","author":"F Lu","year":"2019","unstructured":"Lu, F., Maggioni, M., Tang, S., Zhong, M.: Nonparametric inference of interaction laws in systems of agents from trajectory data. Proc. Natl. Acad. Sci. 116, 06 (2019)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"9673_CR51","first-page":"343","volume":"17","author":"AK Misra","year":"2012","unstructured":"Misra, A.K.: A simple mathematical model for the spread of two political parties. Nonlinear Anal. Model. Control, 2012, No. 3 17, 343\u2013354 (2012)","journal-title":"Nonlinear Anal. Model. Control, 2012, No. 3"},{"key":"9673_CR52","doi-asserted-by":"publisher","first-page":"e78433","DOI":"10.1371\/journal.pone.0078433","volume":"8","author":"M Moussa\u00efd","year":"2013","unstructured":"Moussa\u00efd, M., K\u00e4mmer, J., Analytis, P., Neth, H.: Social influence and the collective dynamics of opinion formation. PloS One 8, e78433 (2013)","journal-title":"PloS One"},{"key":"9673_CR53","doi-asserted-by":"publisher","first-page":"158701","DOI":"10.1103\/PhysRevLett.100.158701","volume":"100","author":"C Nardini","year":"2008","unstructured":"Nardini, C., Kozma, B., Barrat, A.: Who\u2019s talking first? Consensus or lack thereof in coevolving opinion formation models. Phys. Rev. Lett. 100, 158701 (2008)","journal-title":"Phys. Rev. Lett."},{"key":"9673_CR54","first-page":"2018","volume":"1\u201326","author":"S Pan","year":"2018","unstructured":"Pan, S., Duraisamy, K.: Long-time predictive modeling of nonlinear dynamical systems using neural networks. Complexity 1\u201326, 2018 (2018)","journal-title":"Complexity"},{"key":"9673_CR55","doi-asserted-by":"publisher","first-page":"458","DOI":"10.1093\/biomet\/36.3-4.458","volume":"36","author":"R Plackett","year":"1949","unstructured":"Plackett, R.: A Historical Note on the Method of Least Squares. Biometrika, No. 3\/4 36, 458\u2013460 (1949)","journal-title":"Biometrika, No. 3\/4"},{"key":"9673_CR56","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1111\/j.2517-6161.1985.tb01383.x","volume":"47","author":"AE Raftery","year":"1985","unstructured":"Raftery, A.E.: A model for high-order Markov chains. J. R. Stat. Soc. Ser. B (Methodological) No. 3 47, 528\u2013539 (1985)","journal-title":"J. R. Stat. Soc. Ser. B (Methodological) No. 3"},{"key":"9673_CR57","doi-asserted-by":"crossref","unstructured":"Ravazzi, C., Hojjatinia, S., Lagoa, C., Dabbene, F.: Randomized opinion dynamics over networks: Influence estimation from partial observations. In: Proceedings of the IEEE Conference on Decision and Control, pp. 2452\u20132457. Institute of Electrical and Electronics Engineers Inc., January (2019)","DOI":"10.1109\/CDC.2018.8619770"},{"key":"9673_CR58","first-page":"11","volume":"656","author":"P Schmid","year":"2008","unstructured":"Schmid, P., Sesterhenn, J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 11 (2008)","journal-title":"J. Fluid Mech."},{"key":"9673_CR59","doi-asserted-by":"crossref","unstructured":"S\u00eerbu, A., Loreto, V., Servedio, V., Tria, F.: Opinion Dynamics: Models, Extensions and External Effects, pp. 363\u2013401. 05 (2017)","DOI":"10.1007\/978-3-319-25658-0_17"},{"key":"9673_CR60","doi-asserted-by":"publisher","DOI":"10.4324\/9781315677163","volume-title":"Social Groups in Action and Interaction","author":"C Stangor","year":"2015","unstructured":"Stangor, C.: Social Groups in Action and Interaction, vol. 2. Routledge, London (2015)"},{"key":"9673_CR61","doi-asserted-by":"publisher","first-page":"734","DOI":"10.1038\/344734a0","volume":"344","author":"G Sugihara","year":"1990","unstructured":"Sugihara, G., May, R.M.: Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature 344, 734\u2013741 (1990)","journal-title":"Nature"},{"key":"9673_CR62","doi-asserted-by":"crossref","unstructured":"Takens, F.: Detecting strange attractors in turbulence. 898, 366\u2013381 (1981)","DOI":"10.1007\/BFb0091924"},{"key":"9673_CR63","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.: Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. Ser. B (Methodological), no. 1 58, 267\u2013288 (1996)","journal-title":"J. R. Stat. Soc. Ser. B (Methodological), no. 1"},{"key":"9673_CR64","doi-asserted-by":"publisher","first-page":"391","DOI":"10.3934\/jcd.2014.1.391","volume":"1","author":"JH Tu","year":"2014","unstructured":"Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1, 391\u2013421 (2014)","journal-title":"J. Comput. Dyn."},{"key":"9673_CR65","first-page":"1","volume":"57","author":"L Tuyen","year":"2018","unstructured":"Tuyen, L.: A higher order Markov model for time series forecasting. Int. J. Appl. Math. Stat. 57, 1\u201318 (2018)","journal-title":"Int. J. Appl. Math. Stat."},{"key":"9673_CR66","doi-asserted-by":"publisher","first-page":"892","DOI":"10.1007\/s10955-017-1761-7","volume":"167","author":"SC Venkataramani","year":"2017","unstructured":"Venkataramani, S.C., Venkataramani, R.C., Restrepo, J.M.: Dimension reduction for systems with slow relaxation. J. Stat. Phys. 167, 892\u2013933 (2017)","journal-title":"J. Stat. Phys."},{"key":"9673_CR67","doi-asserted-by":"crossref","unstructured":"Williams, M., Kevrekidis, I., Rowley, C.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25, 1307\u20131346 (2014)","DOI":"10.1007\/s00332-015-9258-5"},{"key":"9673_CR68","first-page":"1","volume":"04","author":"X Wu","year":"2018","unstructured":"Wu, X., Wai, H.-T., Scaglione, A.: Estimating social opinion dynamics models from voting records. IEEE Trans. Signal Process. 04, 1 (2018)","journal-title":"IEEE Trans. Signal Process."},{"key":"9673_CR69","first-page":"72","volume":"2","author":"H Xia","year":"2011","unstructured":"Xia, H., Wang, H., Xuan, Z.: Opinion dynamics: a multidisciplinary review and perspective on future research. IJKSS 2, 72\u201391 (2011)","journal-title":"IJKSS"},{"key":"9673_CR70","doi-asserted-by":"publisher","first-page":"103501","DOI":"10.1063\/1.5003467","volume":"59","author":"Y Zhu","year":"2018","unstructured":"Zhu, Y., Dominy, J., Venturi, D.: On the estimation of the Mori\u2013Zwanzig memory integral. J. Math. Phys. 59, 103501 (2018)","journal-title":"J. Math. Phys."},{"key":"9673_CR71","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195140187.001.0001","volume-title":"Nonequilibrium Statistical Mechanics","author":"R Zwanzig","year":"2001","unstructured":"Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, Oxford (2001)"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-020-09673-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-020-09673-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-020-09673-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T17:17:38Z","timestamp":1724347058000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-020-09673-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,19]]},"references-count":71,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["9673"],"URL":"https:\/\/doi.org\/10.1007\/s00332-020-09673-2","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"type":"print","value":"0938-8974"},{"type":"electronic","value":"1432-1467"}],"subject":[],"published":{"date-parts":[[2021,1,19]]},"assertion":[{"value":"23 June 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 December 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"19"}}