基于Groebner基估计由不可识别的状态空间模型和时间序列数据确定的隐藏结构的方法
Method for estimating hidden structures determined by unidentifiable state-space models and time-series data based on the Groebner basis
摘要：
在这项研究中，我们提出了一种方法，用于提取隐藏的代数结构的模型参数是唯一确定的观测时间序列数据和不可识别的状态空间模型，明确和详尽。状态空间模型通常基于域（例如，物理或生物）来构造。这样的模型包括被赋予与所考虑的系统相关的特定含义的参数，通过使用相应的数据估计参数来检查所考虑的系统。由于不可识别模型的参数不能从给定的数据中唯一确定，因此很难检查由此类模型描述的系统。为了克服这一困难，在现有的方法中，估计和分析多个可能的参数集;然而，一般来说，不能探索所有可能的参数;因此，对使用估计的参数的系统的考虑变得不足。在本研究中，我们针对由观测数据和模型唯一确定的某些结构，即使它们是不可识别的，我们引入了参数簇的概念。这是新定义和证明，形成代数簇，一般。一个计算代数方法，依赖于Groebner基础上推导出的品种的显式表示的支持理论。此外，它的应用程序中的一个模型，描述病毒动力学的分析。由此，发现了传统方法所忽略的动力学的新见解，证实了我们的想法和所提出的方法的适用性。
英文摘要：
In this study, we propose a method for extracting the hidden algebraic structures of model parameters that are uniquely determined by observed time-series data and unidentifiable state-space models, explicitly and exhaustively. State-space models are often constructed based on the domain, for example, physical or biological. Such models include parameters that are assigned specific meanings in relation to the system under consideration, which is examined by estimating the parameters using the corresponding data. As the parameters of unidentifiable models cannot be uniquely determined from the given data, it is difficult to examine the systems described by such models. To overcome this difficulty, multiple possible sets of parameters are estimated and analysed in the exiting approaches; however, in general, all the possible parameters cannot be explored; therefore, considerations on the system using the estimated parameters become insufficient. In this study, focusing on certain structures determined by the observed data and models uniquely, even if they are unidentifiable, we introduce the concept of parameter variety. This is newly defined and proven to form algebraic varieties, in general. A computational algebraic method that relies on the Groebner basis for deriving the explicit representation of the varieties is presented along with the supporting theory. Furthermore, its application in the analysis of a model that describes virus dynamics is presented. With this, new insight on the dynamics overlooked by the conventional approach are discovered, confirming the applicability of our idea and the proposed method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
