Editing Kalman filter (section)

=== Kalman gain derivation ===
The Kalman filter is a [[minimum mean-square error| minimum mean-square error (MMSE)]] estimator. The error in the ''a posteriori'' state estimation is
:<math>\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}</math>

We seek to minimize the expected value of the square of the magnitude of this vector, <math>\operatorname{E}\left[\left\|\mathbf{x}_{k} - \hat{\mathbf{x}}_{k|k}\right\|^2\right]</math>. This is equivalent to minimizing the [[trace (matrix)|trace]] of the ''a posteriori'' estimate [[covariance matrix]] <math> \mathbf{P}_{k|k} </math>. By expanding out the terms in the equation above and collecting, we get:
:<math>\begin{align}
  \mathbf{P}_{k\mid k} & = \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1} - \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T} + \mathbf{K}_k \left(\mathbf{H}_k \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k\right) \mathbf{K}_k^\textsf{T} \\[6pt]
                       &= \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1} - \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T} + \mathbf{K}_k \mathbf{S}_k\mathbf{K}_k^\textsf{T}
\end{align}</math>

The trace is minimized when its [[matrix calculus|matrix derivative]] with respect to the gain matrix is zero. Using the [[matrix calculus#Identities|gradient matrix rules]] and the symmetry of the matrices involved we find that
:<math>\frac{\partial \; \operatorname{tr}(\mathbf{P}_{k\mid k})}{\partial \;\mathbf{K}_k} = -2 \left(\mathbf{H}_k \mathbf{P}_{k\mid k-1}\right)^\textsf{T} + 2 \mathbf{K}_k \mathbf{S}_k = 0.</math>

Solving this for '''K'''<sub>''k''</sub> yields the Kalman gain:
:<math>\begin{align}
  \mathbf{K}_k \mathbf{S}_k &= \left(\mathbf{H}_k \mathbf{P}_{k\mid k-1}\right)^\textsf{T} = \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \\
   \Rightarrow \mathbf{K}_k &= \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1}
\end{align}</math>

This gain, which is known as the ''optimal Kalman gain'', is the one that yields MMSE estimates when used.