Array filter
We shall use the following notations:
\[P_t = L_t L_T'\] \[F_t = Z_t P_t Z_t' + H_t = Z_t P_t Z_t' + G_t G_t' = R_t R_t'\] \[K_t = T_t P_t Z_t' {R_t^{-1}}'\]where $L_t$ and $R_t$ are lower triangular matrices.
The array form of the Kalman filter is described by the following sequence:
\[\begin{pmatrix}R_{t-1} & 0 & 0 \\ K_{t-1} & L_t & 0\end{pmatrix} \rightarrow \begin{pmatrix}G_t & Z_t L_t & 0 \\ 0 & T_t L_t & S_t\end{pmatrix} \rightarrow \begin{pmatrix}R_t & 0 & 0 \\ K_t & L_{t+1} & 0\end{pmatrix}\]Given $L_t$, we can easily form the “pre-array” matrix defined in the second step. That matrix is then triangularized by means of usual orthogonal transformations. JD+ uses Givens rotations when the pre-array form is quasi-triangular and Householder reflections in the other cases.
The result of the triangularization corresponds necessarily to the matrices of the third step.
Indeed, we have that
\[\begin{pmatrix}G_t & Z_t L_t & 0 \\ 0 & T_t L_t & S_t\end{pmatrix}\begin{pmatrix}G_t & Z_t L_t & 0 \\ 0 & T_t L_t & S_t\end{pmatrix}'=\begin{pmatrix}F_t & R_t K_t' \\ K_t R_t' & T_t P_t T_t' +V_t\end{pmatrix}\] \[=\begin{pmatrix}A & 0 & 0\\ B & C & 0\end{pmatrix}\begin{pmatrix}A & 0 & 0\\ B & C & 0\end{pmatrix}'=\begin{pmatrix}AA' & AB' \\ BA' & BB' + CC'\end{pmatrix}\Leftrightarrow \begin{cases}A=R_t \\B=K_t \\ C = L_{t+1} \end{cases}\]The last equality comes from: $P_{t+1}=T_t P_t T_t’ -K_t K_t’ + V_t \Leftrightarrow T_t P_t T_t’ + V_t = K_t K_t’ + P_{t+1}$
The other quantities are computed as usual:
\[u_t = R_t^{-1} e_t\] \[a_{t+1} = T_t a_{t} + K_t u_t\]Implementation
The array filter is implemented in the classes jdplus.ssf.array.ArrayFilter and jdplus.ssf.array.MultivariateArrayFilter.
Those filters will use a modified representation of the state, which contains the Cholesky factor of its covariance matrix (see jdplus.ssf.array.LState