Fast CKMS filter (or fast Chandrasekhar recursions)
When the model is time invariant (which also implies that we don’t have missing values), it is usually much more efficient to use the Chandrasekhar recursions, which propagate the differences of the covariance matrix:
\[\Delta P_t = - L_t M_t^{-1} L_t'\]If we write:
\[K_t = T P_t Z'\] \[F_t = Z P_t Z' + H\]the CKMS recursions are:
\[\Delta K_t = -T L_t M_t^{-1} L_t' Z'\] \[\Delta F_t = -Z L_t M_t^{-1} L_t' Z'\] \[L_{t+1} = \left( T - K_t F_t^{-1}Z \right) L_t\] \[\Delta M_t = - Z L_t F_t^{-1} L_t'Z'\]It is clear that if we can take $M_0 = F_0$ (see below for stationary models with their unconditional initial distribution), the last equation is redundant ( $M_t=F_t$ ). Then, the recursions are
\[\Delta K_t = -T L_t F_t^{-1} L_t' Z'\] \[\Delta F_t = -Z L_t F_t^{-1} L_t' Z'\] \[L_{t+1} = \left( T - K_t F_t^{-1}Z \right) L_t\]In the case of stationary models, the usual (unconditional) distribution of the state is defined by the relationship:
\[P_* = T P_* T' + V\]So, we have:
\[\Delta P_0 = P_0 - P_* = T P_* T' - K_0 F_0^{-1} K_0' + V - P_* = - K_0 F_0^{-1} K_0'\]and we can take:
\[L_0 = K_0, \: M_0 = F_0\]It can be shown that the CKMS algorithm is also valid for non-stationary models, after their initialization period.