Editing Kalman filter (section)

=== 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>