# Derivation of Kalman filter

1. Apr 2, 2012

### spirall

Hi all,

I have a standard local level model, but the disturbances are not independent:
y_t=μ_t+ε_t, μ_t+1=μ_t+η_t, E(ε_t η_t) =/= 0

In order to derive the Kalman filter, I rewrite this model in state space form
y_t=Z_t α_t+ε_t, ε_t~NID(0,H_t ),
α_(t+1)=T_t α_t+R_t η_t, η_t~NID(0,Q_t ),
α_1~N(a_1,P_1 ),

α_t=[μ_t
ξ_t ]

Z_t=[1 1],
H_t=0,

Q_t=[σ_η^2 σ_ξη
σ_ξη σ_ξ^2 ],

T_t=[1 0
0 0],

R_t=[1 0
0 1],

η_t=[η_t
ξ_(t+1)].

I wonder whether there is any difference in the derivation of the Kalman filter, since the matrix Q in not diagonal.

Thank you

2. Apr 2, 2012

### D H

Staff Emeritus
There's nothing in the formulation or derivation of the Kalman filter that requires the noise matrix Q to be diagonal. It just needs to qualify as a covariance matrix. In other words, it needs to be a symmetric positive semidefinite matrix.

3. Apr 3, 2012