Editing Kalman filter (section)

== Fixed-interval smoothers ==
The optimal fixed-interval smoother provides the optimal estimate of <math>\hat{\mathbf{x}}_{k \mid  n}</math> (<math>k < n</math>) using the measurements from a fixed interval <math>\mathbf{z}_1</math> to <math>\mathbf{z}_n</math>. This is also called "Kalman Smoothing". There are several smoothing algorithms in common use.

=== 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.

=== Modified Bryson–Frazier smoother ===
An alternative to the RTS algorithm is the modified Bryson–Frazier (MBF) fixed interval smoother developed by Bierman.<ref name=bierman>{{cite journal | last = Bierman | first =  G.J. | title = Factorization Methods for Discrete Sequential Estimation | year = 1977 | bibcode = 1977fmds.book.....B | journal = Factorization Methods for Discrete Sequential Estimation }}</ref> This also uses a backward pass that processes data saved from the Kalman filter forward pass. The equations for the backward pass involve the recursive
computation of data which are used at each observation time to compute the smoothed state and covariance.

The recursive equations are
:<math>\begin{align}
    \tilde{\Lambda}_k &= \mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1} \mathbf{H}_k + \hat{\mathbf{C}}_k^\textsf{T} \hat{\Lambda}_k \hat{\mathbf{C}}_k \\
  \hat{\Lambda}_{k-1} &= \mathbf{F}_k^\textsf{T}\tilde{\Lambda}_k\mathbf{F}_k \\
      \hat{\Lambda}_n &= 0 \\
    \tilde{\lambda}_k &=  -\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1} \mathbf{y}_k + \hat{\mathbf{C}}_k^\textsf{T} \hat{\lambda}_k \\
  \hat{\lambda}_{k-1} &= \mathbf{F}_k^\textsf{T}\tilde{\lambda}_k \\
      \hat{\lambda}_n &= 0
\end{align}</math>

where <math>\mathbf{S}_k</math> is the residual covariance and <math>\hat{\mathbf{C}}_k = \mathbf{I} - \mathbf{K}_k \mathbf{H}_k</math>. The smoothed state and covariance can then be found by substitution in the equations
:<math>\begin{align}
  \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k} - \mathbf{P}_{k \mid k}\hat{\Lambda}_k\mathbf{P}_{k \mid k} \\
  \mathbf{x}_{k \mid n} &= \mathbf{x}_{k \mid k} - \mathbf{P}_{k \mid k}\hat{\lambda}_k
\end{align}</math>

or
:<math>\begin{align}
  \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k-1} - \mathbf{P}_{k \mid k-1}\tilde{\Lambda}_k\mathbf{P}_{k \mid k-1} \\
  \mathbf{x}_{k \mid n} &= \mathbf{x}_{k \mid k-1} - \mathbf{P}_{k \mid k-1}\tilde{\lambda}_k.
\end{align}</math>

An important advantage of the MBF is that it does not require finding the inverse of the covariance matrix.  Bierman's derivation is based on the RTS smoother, which assumes that the underlying distributions are Gaussian. However, a derivation of the MBF based on the concept of the fixed point smoother, which does not require the Gaussian assumption, is given by Gibbs.<ref name=gibbs1>{{cite journal |last1=Gibbs |first1=Richard G. |date=February 2011 |title=Square Root Modified Bryson–Frazier Smoother |url=https://ieeexplore.ieee.org/document/5609189 |journal=IEEE Transactions on Automatic Control |volume=56 |issue=2 |pages=452–456 |doi=10.1109/TAC.2010.2089753 |access-date=}}</ref>

The MBF can also be used to perform consistency checks on the filter residuals and the difference between the value of a filter state after an update and the smoothed value of the state, that is <math>\mathbf{x}_{k \mid k} - \mathbf{x}_{k \mid n}</math>.<ref name=gibbs2>{{cite journal |last1=Gibbs |first1=Richard G. |date=2013 |title=New Kalman filter and smoother consistency tests |url=https://www.sciencedirect.com/science/article/pii/S0005109813003610 |journal=Automatica |volume=49 |issue=10 |pages=3141–3144 |doi=10.1016/j.automatica.2013.07.013 }}</ref>

=== Minimum-variance smoother ===
The minimum-variance smoother can attain the best-possible error performance, provided that the models are linear, their parameters and the noise statistics are known precisely.<ref>{{cite journal | last = Einicke | first =  G.A. | title = Optimal and Robust Noncausal Filter Formulations | journal =  IEEE Transactions on Signal Processing| volume = 54 | issue = 3 | pages = 1069–1077 | date=March 2006 | bibcode = 2006ITSP...54.1069E | doi = 10.1109/TSP.2005.863042 | s2cid = 15376718 }}</ref> This smoother is a time-varying state-space generalization of the optimal non-causal [[Wiener filter]].

The smoother calculations are done in two passes. The forward calculations involve a one-step-ahead predictor and are given by
:<math>\begin{align}
  \hat{\mathbf{x}}_{k+1 \mid k} &= (\mathbf{F}_k - \mathbf{K}_k\mathbf{H}_k)\hat{\mathbf{x}}_{k \mid k-1} + \mathbf{K}_k\mathbf{z}_k \\
                       \alpha_k &= -\mathbf{S}_k^{-\frac{1}{2}}\mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1} + \mathbf{S}_k^{-\frac{1}{2}}\mathbf{z}_k
\end{align}</math>

The above system is known as the inverse Wiener-Hopf factor. The backward recursion is the adjoint of the above forward system. The result of the backward pass <math>\beta_k</math>  may be calculated by operating the forward equations on the time-reversed <math>\alpha_k</math>  and time reversing the result. In the case of output estimation, the smoothed estimate is given by
:<math>\hat{\mathbf{y}}_{k \mid N} = \mathbf{z}_k - \mathbf{R}_k\beta_k</math>

Taking the causal part of this minimum-variance smoother yields
:<math>\hat{\mathbf{y}}_{k \mid k} = \mathbf{z}_k - \mathbf{R}_k \mathbf{S}_k^{-\frac{1}{2}} \alpha_k</math>

which is identical to the minimum-variance Kalman filter. The above solutions minimize the variance of the output estimation error. Note that the Rauch–Tung–Striebel smoother derivation assumes that the underlying distributions are Gaussian, whereas the minimum-variance solutions do not. Optimal smoothers for state estimation and input estimation can be constructed similarly.

A continuous-time version of the above smoother is described in.<ref>{{cite journal | last = Einicke | first =  G.A. | title = Asymptotic Optimality of the Minimum-Variance Fixed-Interval Smoother | journal =  IEEE Transactions on Signal Processing| volume = 55 | issue = 4 | pages = 1543–1547 | date=April 2007 | bibcode = 2007ITSP...55.1543E | doi = 10.1109/TSP.2006.889402 | s2cid = 16218530 }}</ref><ref>{{cite journal | last1 = Einicke | first1 =  G.A. | last2 = Ralston | first2 = J.C. | last3 = Hargrave | first3 = C.O. | last4 = Reid | first4 = D.C. | last5 = Hainsworth | first5 = D.W. | title = Longwall Mining Automation. An Application of Minimum-Variance Smoothing | journal = IEEE Control Systems Magazine | volume = 28 | issue = 6 |pages=28–37 |doi=10.1109/MCS.2008.929281 | date=December 2008 | s2cid = 36072082 }}</ref>

[[Expectation–maximization algorithm]]s may be employed to calculate approximate [[maximum likelihood]] estimates of unknown state-space parameters within minimum-variance filters and smoothers. Often uncertainties remain within problem assumptions. A smoother that accommodates uncertainties can be designed by adding a positive definite term to the Riccati equation.<ref>{{cite journal | last = Einicke | first =  G.A. | title = Asymptotic Optimality of the Minimum-Variance Fixed-Interval Smoother | journal = IEEE Transactions on Automatic Control | volume = 54 | issue = 12 | pages = 2904–2908 | date=December 2009 | bibcode = 2007ITSP...55.1543E | doi = 10.1109/TSP.2006.889402 | s2cid = 16218530 }}</ref>

In cases where the models are nonlinear, step-wise linearizations may be within the minimum-variance filter and smoother recursions ([[extended Kalman filter]]ing).