Identifiability of the stochastic state-space models

State-Space Models (SSMs) are deterministic or stochastic dynamical systems defined by two processes. The state process, which is not observed directly, models the transformation of the states over time. On another hand, the observation process produces the observables on which model fitting and prediction are based. Ecology frequently uses stochastic SSMs to represent the imperfectly observed dynamics of population sizes or animal movement. However, several simulation-based evaluations of model performance suggest broad identifiability issues in ecological SSMs. Formal SSM identifiability is typically investigated using exhaustive summaries, which are simplified representations of the model. The theory on exhaustive summaries is largely based on continuous-time deterministic modelling and those for ecological SSMs have developed by analogy. While the discreteness of time does not constitute a challenge, finding a good exhaustive summary for a stochastic SSM is more difficult. The strategy adopted so far has been to create exhaustive summaries based on a transfer function of the expectations of the stochastic process. However, this evaluation of identifiability does not allow to take into account the possible dependency between the variance parameters and the process parameters. We show that the output spectral density plays a key role in stochastic SSM identifiability assessment. This allows us to define a new suitable exhaustive summary. Using several ecological examples, we show that usual ecological models are often theoretically identifiable, suggesting that most SSM estimation problems are due to practical rather than theoretical identifiability issues.