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Kalman smoother derivation subsitution

  1. Feb 22, 2013 #1
    Hi Im working through the derivation of Kalman smoothing and am stuck on probably a simple substitution problem for the conditional variance of the hidden state x_{t}|x{t+1}, conditioned on all previous observations

    Firstly the variance is given by:

    [itex]Var[x_{t}|x_{t+1},y_{o},..,y_{t}]=P_{t|t}-P_{t|t}A^{T}P^{-1}_{t+1|t}AP_{t|t}[/itex]

    with A being the system updat equation and P represent the respective covariance in the relevant states

    The substitution is of [itex]L{t}=P_{t|t}A^{T}P^{-1}_{t+1|t}[/itex]

    And the variance is thus supposed to equal.

    Thanks in advance

    [itex]P_{t|t}-L_{t}P_{t+1|t}L^T_{t}[/itex]

    But working through I obtain:

    [itex]P_{t|t}-P_{t|t}A^{T}P^{-1}_{t+1|t}P_{t+1|t}P_{t|t}A^{T}P^{-1}_{t+1|t}[/itex]

    which condenses to:

    [itex]P_{t|t}-P_{t|t}A^{T}P_{t|t}A^{T}P^{-1}_{t+1|t}[/itex]

    first three terms agree and can be cancelled so Im left with:

    [itex]P_{t|t}A^{T}P^{-1}_{t+1|t}[/itex]

    which should equal

    [itex]P^{-1}_{t+1|t}AP_{t|t}[/itex]

    Due to symmetry however

    [itex]P_{t|t}A^{T}=AP_{t|t}[/itex]

    So:

    [itex]AP_{t|t}P^{-1}_{t+1|t}=P^{-1}_{t+1|t}AP_{t|t}[/itex]

    The question is, does this final term hold, Im not sure if this relationship is commutative or not.
     
  2. jcsd
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