1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Derivation of Kalman filter

  1. Apr 2, 2012 #1
    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 ),

    ξ_t ]

    Z_t=[1 1],

    Q_t=[σ_η^2 σ_ξη
    σ_ξη σ_ξ^2 ],

    T_t=[1 0
    0 0],

    R_t=[1 0
    0 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. jcsd
  3. Apr 2, 2012 #2

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
  4. Apr 3, 2012 #3
    Thank you for reply.

    And what about likelihood estimation of coefficients of system matrices if the transition matrix T depends on some exogenous variables (I suppose this is possible)?

    I have one more question about dynamic factor model. I consider transformation approach (yL_t=A_L y_t) and look for transformation matrix A=[A_L A_H]' such as:
    Ʃ_L=A_L H A´_L
    If I use Cholesky decomposition, the matrix C should be lower triangular. But how I choose it?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook