Editing Kalman filter (section)

=== Rauch–Tung–Striebel ===
The Rauch–Tung–Striebel (RTS) smoother is an efficient two-pass algorithm for fixed interval smoothing.<ref>{{cite journal | last1 = Rauch | first1 =  H.E. | last2 = Tung | first2 = F. | last3 = Striebel | first3 = C. T. | title = Maximum likelihood estimates of linear dynamic systems | journal = AIAA Journal| volume = 3 | issue = 8 | pages = 1445–1450 | date=August 1965 | doi = 10.2514/3.3166 | bibcode = 1965AIAAJ...3.1445R }}</ref>

The forward pass is the same as the regular Kalman filter algorithm. These ''filtered'' a-priori and a-posteriori state estimates <math>\hat{\mathbf{x}}_{k \mid k-1}</math>, <math>\hat{\mathbf{x}}_{k \mid k}</math> and covariances <math>\mathbf{P}_{k \mid k-1}</math>, <math>\mathbf{P}_{k \mid k}</math> are saved for use in the backward pass (for [[retrodiction]]).

In the backward pass, we compute the ''smoothed'' state estimates <math>\hat{\mathbf{x}}_{k \mid n}</math> and covariances <math>\mathbf{P}_{k \mid n}</math>. We start at the last time step and proceed backward in time using the following recursive equations:
:<math>\begin{align}
  \hat{\mathbf{x}}_{k \mid n} &= \hat{\mathbf{x}}_{k \mid k} + \mathbf{C}_k \left(\hat{\mathbf{x}}_{k+1 \mid n} - \hat{\mathbf{x}}_{k+1 \mid k}\right) \\
        \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k} + \mathbf{C}_k \left(\mathbf{P}_{k+1 \mid n} - \mathbf{P}_{k+1 \mid k}\right) \mathbf{C}_k^\textsf{T}
\end{align}</math>

where
:<math>\mathbf{C}_k = \mathbf{P}_{k \mid k} \mathbf{F}_{k+1}^\textsf{T} \mathbf{P}_{k+1 \mid k}^{-1}.</math>

<math> \mathbf{x}_{k \mid k}</math> is the a-posteriori state estimate of timestep <math>k</math> and <math>\mathbf{x}_{k+1 \mid k}</math> is the a-priori state estimate of timestep <math>k + 1</math>. The same notation applies to the covariance.