On the Observability of General Nonlinear Gaussian State Space Models Using Discrete Distributional Approximations

We consider arbitrary nonlinear stochastic discrete-time state space models (SSMs) with time-invariant parameters and nonadditive Gaussian disturbances. Given an observation trajectory, the goal is to obtain an estimate of the augmented state (the underlying state trajectory and the time-invariant parameters of the model). A numerical approach to checking this type of observability is given, and a quantitative assessment of the degree of observability is provided. In general, no global statements for all observation trajectories can be made on the observability of nonlinear SSMs. However, we can find regions of the state-observation space (consisting of all possible observation trajectories and corresponding state trajectories) in which an estimate of the augmented state can be obtained. This is achieved by approximating the continuous distribution of observation trajectories and state trajectories using an optimal discrete distribution. The associated locations of the point masses are called design values. For these design values, we can then check whether the augmented state can be recovered. We could also use random realizations of the observation trajectory. However, when using design values, a smaller number of considered observation trajectories is required to achieve a good coverage of the space. We illustrate our approach to checking observability for different specifications of discrete-time SSMs in univariate and multivariate settings.