Editing Kalman filter (section)

=== 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).