  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.
