Convergence Analysis of the Factorial Kalman Filter

The linear dynamical system is a common method for modeling time series whose hidden states evolve linearly. However, it assumes the existence of only a single hidden state at each time step, which constrains its capability to represent a combination of multiple time series. Therefore, we previously proposed the factorial linear dynamical systems and an iterative factorial Kalman filter algorithm for state estimations. Based on previous work, this letter analyzes the convergence property of the factorial Kalman filter, which ensures that the Kalman gain converges when $t\to \infty$. Specifically, a sufficient convergence condition of the Kalman gain is provided without observation biases. Furthermore, a sufficient convergence condition of the average innovation is derived with observation biases. Numerical simulations are provided to corroborate the goodness and effectiveness of the derived results.