# Relation between Kalman filter and Sequential linear MMSE estimation

Are the results of applying Kalman filter and recursive linear MMSE estimation process the same ?

## 1 Answer

The system model for the Kalman filter is:

$x_k = F_k x_{k-1} + B_k u_{k-1} + w_k \\ z_k = H_k x_k + v_k$

The system model for sequential linear MMSE estimation might be:

$x_k = x_{k-1} \\ z_k = H_k x_k + v_k$

Those are equivalent if $F_k$ is the identity, $B_k=0$ and $w_k=0$ and I believe the estimators turn out to be the same in that case. If $F_k$ is not the identity, or if either $B_k$ or $w_k$ are non-zero, then I see no immediate way to match sequential linear MMSE estimation to the Kalman filter system model.

On the other hand, let's consider the system equations for $k=1$, $k=2$ jointly:

$z_1 = H_1 F_1 x_0 + F_1 B_1 u_0 + F_1 w_0 + v_1 \\ z_2 = H_2 F_2 (F_1 x_0 + B_1 u_0 + w_0) + F_2 B_2 u_1 + F_2 w_1 + v_2$

This can be stacked into one equation:

$z = A x_0 + \tilde{v} \,,$

where $z$ is a column vector stacking $z_1$ and $z_2$. $A$ contains $H_1, F_1, H_2, F_2$, and $\tilde{v}$ contains $H_2, F_2, B_1, u_0, u_1, w_0, w_1, v_1, v_2$. Both the Kalman filter and the linear MMSE estimator can be applied to this linear model, and I would expect them to produce the same result. The Wikipedia article assumes that the error covariance for $\tilde{v}$ is zero, but we can trivially fix that by subtracting the error mean (which arises from the known inputs $u_0, u_1$).

So if we model the system with joint equations, rather than sequential, iterative equations, then linear MMSE estimation applies directly to the general model, and Kalman filtering gives no improvement for this more general problem.

• I should have noted that the MMSE estimation applied to $z = A x_0 + \tilde{v}$ will only give an estimate for $x_0$, and that we would need similar equations for $x_1$, $x_2$. We can replace $F_k x_{k-1}$ with $x_k - B_k u_{k-1} - w_k$, recursively, to get to equations for $x_1$ and $x_2$. – drizzd Nov 3 '13 at 20:40