Kalmansmoothing

Kalman smoothing refers to a family of algorithms for estimating the complete sequence of a dynamic system’s hidden states from noisy observations, by using all available measurements. It is typically used in offline analysis, where data has already been collected, in contrast to the online Kalman filter which updates estimates in real time.

In the standard linear Gaussian setting, the system is modeled with a state equation x_{k+1} = F_k x_k + w_k and an observation equation z_k = H_k x_k + v_k, where w_k and v_k are uncorrelated zero-mean Gaussian noises with covariances Q_k and R_k, respectively. The Kalman smoother aims to compute the smoothed state estimates x_{k|T} and their covariances P_{k|T}, for k = 1,...,T, conditioning on the entire observation sequence z_1,...,z_T.

A common approach is the Rauch–Tung–Striebel (RTS) smoother. It starts by running the forward Kalman filter to obtain the filtered means x_{k|k} and covariances P_{k|k}. It then performs a backward pass, computing the smoothing gain G_k = P_{k|k} F_k^T P_{k+1|k}^{-1} and updating the smoothed estimates via x_{k|T} = x_{k|k} + G_k (x_{k+1|T} - x_{k+1|k}), with a corresponding update to P_{k|T}. This two-pass procedure yields improved state estimates by incorporating future information.

Variants exist for nonlinear models, including extended Kalman smoothers, unscented smoothers, and particle smoothers. Kalman smoothing has wide use in navigation, signal processing, robotics, and econometrics, and forms a foundational tool in data assimilation and target tracking. The method originated with the Kalman filter (1960) and was extended to smoothing by Rauch, Tung, and Striebel in 1965.