Method for estimating hidden structures determined by unidentifiable state-space models and time-series data based on the Groebner basis

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.