Editing Kalman filter (section)

== Information filter ==
{{More citations needed section|date=April 2016}}

In cases where the dimension of the observation vector '''y''' is bigger than the dimension of the state space vector '''x''', the information filter can avoid the inversion of a bigger matrix in the Kalman gain calculation at the price of inverting a smaller matrix in the prediction step, thus saving computing time. Additionally, the information filter allows for system information initialization according to <math>{I_{1|0} = P^{-1}_{1|0} = 0}</math>, which would not be possible for the regular Kalman filter.<ref>{{Cite book |last=Gustafsson |first=Fredrik |title=Statistical sensor fusion |date=2018 |publisher=Studentlitteratur |isbn=978-91-44-12724-8 |edition=Third |location=Lund |pages=160–162}}</ref> In the information filter, or inverse covariance filter, the estimated covariance and estimated state are replaced by the [[Fisher information matrix|information matrix]] and [[Fisher information|information]] vector respectively. These are defined as:
:<math>\begin{align}
        \mathbf{Y}_{k \mid k} &= \mathbf{P}_{k \mid k}^{-1} \\
  \hat{\mathbf{y}}_{k \mid k} &= \mathbf{P}_{k \mid k}^{-1}\hat{\mathbf{x}}_{k \mid k}
\end{align}</math>

Similarly the predicted covariance and state have equivalent information forms, defined as:
:<math>\begin{align}
        \mathbf{Y}_{k \mid k-1} &= \mathbf{P}_{k \mid k-1}^{-1} \\
  \hat{\mathbf{y}}_{k \mid k-1} &= \mathbf{P}_{k \mid k-1}^{-1}\hat{\mathbf{x}}_{k \mid k-1}
\end{align}</math>

and the measurement covariance and measurement vector, which are defined as:
:<math>\begin{align}
  \mathbf{I}_k &= \mathbf{H}_k^\textsf{T} \mathbf{R}_k^{-1} \mathbf{H}_k \\
  \mathbf{i}_k &= \mathbf{H}_k^\textsf{T} \mathbf{R}_k^{-1} \mathbf{z}_k
\end{align}</math>

The information update now becomes a trivial sum.<ref name=terejanu>{{cite web |title=Discrete Kalman Filter Tutorial |author=Gabriel T. Terejanu |date=2012-08-04 |access-date=2016-04-13 |url=https://cse.sc.edu/~terejanu/files/tutorialKF.pdf |archive-date=2020-08-17 |archive-url=https://web.archive.org/web/20200817230005/https://cse.sc.edu/~terejanu/files/tutorialKF.pdf |url-status=dead }}</ref>
:<math>\begin{align}
        \mathbf{Y}_{k \mid k} &= \mathbf{Y}_{k \mid k-1} + \mathbf{I}_k \\
  \hat{\mathbf{y}}_{k \mid k} &= \hat{\mathbf{y}}_{k \mid k-1} + \mathbf{i}_k
\end{align}</math>

The main advantage of the information filter is that ''N'' measurements can be filtered at each time step simply by summing their information matrices and vectors.
:<math>\begin{align}
        \mathbf{Y}_{k \mid k} &= \mathbf{Y}_{k \mid k-1} + \sum_{j=1}^N \mathbf{I}_{k,j} \\
  \hat{\mathbf{y}}_{k \mid k} &= \hat{\mathbf{y}}_{k \mid k-1} + \sum_{j=1}^N \mathbf{i}_{k,j}
\end{align}</math>

To predict the information filter the information matrix and vector can be converted back to their state space equivalents, or alternatively the information space prediction can be used.<ref name=terejanu />
:<math>\begin{align}
                 \mathbf{M}_k &=
    \left[\mathbf{F}_k^{-1}\right]^\textsf{T} \mathbf{Y}_{k-1 \mid k-1} \mathbf{F}_k^{-1} \\
                 \mathbf{C}_k &=
    \mathbf{M}_k \left[\mathbf{M}_k + \mathbf{Q}_k^{-1}\right]^{-1} \\
                   \mathbf{L}_k &=
    \mathbf{I} - \mathbf{C}_k \\
        \mathbf{Y}_{k \mid k-1} &=
    \mathbf{L}_k \mathbf{M}_k +
    \mathbf{C}_k \mathbf{Q}_k^{-1} \mathbf{C}_k^\textsf{T} \\
  \hat{\mathbf{y}}_{k \mid k-1} &=
    \mathbf{L}_k \left[\mathbf{F}_k^{-1}\right]^\textsf{T} \hat{\mathbf{y}}_{k-1 \mid k-1}
\end{align}</math>