Editing Kalman filter (section)

== Fixed-lag smoother ==
{{More citations needed section|date=December 2010}}

The optimal fixed-lag smoother provides the optimal estimate of <math>\hat{\mathbf{x}}_{k-N \mid  k}</math> for a given fixed-lag <math>N</math> using the measurements from <math>\mathbf{z}_1</math> to <math>\mathbf{z}_k</math>.<ref>{{cite book|last1=Anderson|first1=Brian D. O.|last2=Moore|first2=John B.|title=Optimal Filtering|date=1979|publisher=Prentice Hall, Inc.|location=Englewood Cliffs, NJ|isbn=978-0-13-638122-8|pages=176–190}}</ref> It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following:
:<math>
 \begin{bmatrix}
  \hat{\mathbf{x}}_{t \mid t} \\
  \hat{\mathbf{x}}_{t-1 \mid t} \\
  \vdots \\
  \hat{\mathbf{x}}_{t-N+1 \mid t} \\
 \end{bmatrix}
 =
 \begin{bmatrix}
  \mathbf{I} \\
  0 \\
  \vdots \\
  0 \\
 \end{bmatrix}
 \hat{\mathbf{x}}_{t \mid t-1}
 +
 \begin{bmatrix}
  0          & \ldots & 0 \\
  \mathbf{I} & 0      & \vdots \\
  \vdots     & \ddots & \vdots \\
  0          & \ldots & \mathbf{I} \\
 \end{bmatrix}
 \begin{bmatrix}
  \hat{\mathbf{x}}_{t-1 \mid t-1} \\
  \hat{\mathbf{x}}_{t-2 \mid t-1} \\
  \vdots \\
  \hat{\mathbf{x}}_{t-N+1 \mid t-1} \\
 \end{bmatrix}
 +
 \begin{bmatrix}
  \mathbf{K}^{(0)} \\
  \mathbf{K}^{(1)} \\
  \vdots \\
  \mathbf{K}^{(N-1)} \\
 \end{bmatrix}
 \mathbf{y}_{t \mid t-1}
</math>

where:
* <math> \hat{\mathbf{x}}_{t \mid t-1} </math> is estimated via a standard Kalman filter;
* <math> \mathbf{y}_{t \mid t-1} = \mathbf{z}_t - \mathbf{H}\hat{\mathbf{x}}_{t \mid t-1} </math> is the innovation produced considering the estimate of the standard Kalman filter;
* the various <math> \hat{\mathbf{x}}_{t-i \mid t} </math> with <math> i = 1, \ldots, N-1 </math> are new variables; i.e., they do not appear in the standard Kalman filter;
* the gains are computed via the following scheme:
*:<math>
  \mathbf{K}^{(i+1)} =
  \mathbf{P}^{(i)} \mathbf{H}^\textsf{T}
  \left[
    \mathbf{H} \mathbf{P} \mathbf{H}^\textsf{T} + \mathbf{R}
  \right]^{-1}
</math>
:and
::<math>
    \mathbf{P}^{(i)} =
    \mathbf{P}
    \left[
      \left(
        \mathbf{F} - \mathbf{K} \mathbf{H}
      \right)^\textsf{T}
    \right]^i
  </math>
:where <math> \mathbf{P} </math> and <math> \mathbf{K} </math> are the prediction error covariance and the gains of the standard Kalman filter (i.e., <math> \mathbf{P}_{t \mid t-1} </math>).

If the estimation error covariance is defined so that
:<math>
\mathbf{P}_i :=
    E \left[
      \left(
        \mathbf{x}_{t-i} - \hat{\mathbf{x}}_{t-i \mid t}
      \right)^{*}
      \left(
        \mathbf{x}_{t-i} - \hat{\mathbf{x}}_{t-i \mid t}
       \right)
       \mid
       z_1 \ldots z_t
    \right],
</math>

then we have that the improvement on the estimation of <math> \mathbf{x}_{t-i} </math> is given by:
:<math>
  \mathbf{P} - \mathbf{P}_i =
  \sum_{j = 0}^i
  \left[
    \mathbf{P}^{(j)} \mathbf{H}^\textsf{T}
    \left(
      \mathbf{H} \mathbf{P} \mathbf{H}^\textsf{T} + \mathbf{R}
    \right)^{-1}
    \mathbf{H} \left( \mathbf{P}^{(i)} \right)^\textsf{T}
  \right]
</math>