{"title":"从单变量时间序列估计卡勒曼算子。","authors":"Sherehe Semba, Huijie Yang, Xiaolu Chen, Huiyun Wan, Changgui Gu","doi":"10.1063/5.0209612","DOIUrl":null,"url":null,"abstract":"<p><p>Reconstructing a nonlinear dynamical system from empirical time series is a fundamental task in data-driven analysis. One of the main challenges is the existence of hidden variables; we only have records for some variables, and those for hidden variables are unavailable. In this work, the techniques for Carleman linearization, phase-space embedding, and dynamic mode decomposition are integrated to rebuild an optimal dynamical system from time series for one specific variable. Using the Takens theorem, the embedding dimension is determined, which is adopted as the dynamical system's dimension. The Carleman linearization is then used to transform this finite nonlinear system into an infinite linear system, which is further truncated into a finite linear system using the dynamic mode decomposition technique. We illustrate the performance of this integrated technique using data generated by the well-known Lorenz model, the Duffing oscillator, and empirical records of electrocardiogram, electroencephalogram, and measles outbreaks. The results show that this solution accurately estimates the operators of the nonlinear dynamical systems. This work provides a new data-driven method to estimate the Carleman operator of nonlinear dynamical systems.</p>","PeriodicalId":9974,"journal":{"name":"Chaos","volume":null,"pages":null},"PeriodicalIF":2.7000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Estimation of Carleman operator from a univariate time series.\",\"authors\":\"Sherehe Semba, Huijie Yang, Xiaolu Chen, Huiyun Wan, Changgui Gu\",\"doi\":\"10.1063/5.0209612\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>Reconstructing a nonlinear dynamical system from empirical time series is a fundamental task in data-driven analysis. One of the main challenges is the existence of hidden variables; we only have records for some variables, and those for hidden variables are unavailable. In this work, the techniques for Carleman linearization, phase-space embedding, and dynamic mode decomposition are integrated to rebuild an optimal dynamical system from time series for one specific variable. Using the Takens theorem, the embedding dimension is determined, which is adopted as the dynamical system's dimension. The Carleman linearization is then used to transform this finite nonlinear system into an infinite linear system, which is further truncated into a finite linear system using the dynamic mode decomposition technique. We illustrate the performance of this integrated technique using data generated by the well-known Lorenz model, the Duffing oscillator, and empirical records of electrocardiogram, electroencephalogram, and measles outbreaks. The results show that this solution accurately estimates the operators of the nonlinear dynamical systems. This work provides a new data-driven method to estimate the Carleman operator of nonlinear dynamical systems.</p>\",\"PeriodicalId\":9974,\"journal\":{\"name\":\"Chaos\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.7000,\"publicationDate\":\"2024-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Chaos\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1063/5.0209612\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Chaos","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1063/5.0209612","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Estimation of Carleman operator from a univariate time series.
Reconstructing a nonlinear dynamical system from empirical time series is a fundamental task in data-driven analysis. One of the main challenges is the existence of hidden variables; we only have records for some variables, and those for hidden variables are unavailable. In this work, the techniques for Carleman linearization, phase-space embedding, and dynamic mode decomposition are integrated to rebuild an optimal dynamical system from time series for one specific variable. Using the Takens theorem, the embedding dimension is determined, which is adopted as the dynamical system's dimension. The Carleman linearization is then used to transform this finite nonlinear system into an infinite linear system, which is further truncated into a finite linear system using the dynamic mode decomposition technique. We illustrate the performance of this integrated technique using data generated by the well-known Lorenz model, the Duffing oscillator, and empirical records of electrocardiogram, electroencephalogram, and measles outbreaks. The results show that this solution accurately estimates the operators of the nonlinear dynamical systems. This work provides a new data-driven method to estimate the Carleman operator of nonlinear dynamical systems.
期刊介绍:
Chaos: An Interdisciplinary Journal of Nonlinear Science is a peer-reviewed journal devoted to increasing the understanding of nonlinear phenomena and describing the manifestations in a manner comprehensible to researchers from a broad spectrum of disciplines.