Kalman filtering & colored noises: the (autoregressive) moving-average case

The Kalman filter is a well-known and efficient recursive algorithm that estimates the state of a dynamic system from a series of indirect and noisy observations of this state. Its applications range from signal processing to machine learning, through speech processing or computer vision. The underlying model usually assumes white noises. Extensions to colored autoregressive (AR) noise model are classical. However, perhaps because of a lack of applications, moving-average (MA) or autoregressive moving-average (ARMA) noises seem not to have been considered before. Motivated by an application in reinforcement learning, the contribution of this paper is a generic approach to take MA and ARMA noises into account in the Kalman filtering paradigm.