## 5 Weighted Least-Squares Approach to State Estimation
### Future Form  
Now, it shall be assumed that the state space model is formulated in the future form; see Equations (1) and (3). The corresponding temporal shift in the dynamical noise term, as compared to the state space model in the contemporaneous form, has the effect that, within the matrix \(\mathcal{R}\), the cross-covariance matrix \(\mathsf{S}_{t}\) moves to different block positions:  
\[\mathcal{R}= \begin{pmatrix}\mathsf{Q}_{0}&0&0&\ldots&0&0\\ 0&\mathsf{R}_{1}&\mathsf{S}_{1}^{\mathsf{T}}&\ldots&0&0&0\\ 0&\mathsf{S}_{1}&\mathsf{Q}_{1}&\ldots&0&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&0&\ldots&\mathsf{R}_{N-1}&\mathsf{S}_{N-1}^{\mathsf{T}}&0\\ 0&0&0&\ldots&\mathsf{S}_{N-1}&\mathsf{Q}_{N-1}&0\\ 0&0&0&\ldots&0&0&\mathsf{R}_{N}\end{pmatrix}  \tag{75}\]  
The inverse of \(\mathcal{R}\) is given by  
\[\mathcal{R}^{-1}= \begin{pmatrix}\mathbf{\Theta}_{0}&0&0&\ldots&0&0&0\\ 0&\mathbf{\Psi}_{1}&\mathbf{\Sigma}_{1}^{\mathsf{T}}&\ldots&0&0&0\\ 0&\mathbf{\Sigma}_{1}&\mathbf{\Omega}_{1}&\ldots&0&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&0&\ldots&\mathbf{\Psi}_{N-1}&\mathbf{\Sigma}_{N-1}^{\mathsf{T}}&0\\ 0&0&0&\ldots&\mathbf{\Sigma}_{N-1}&\mathbf{\Omega}_{N-1}&0\\ 0&0&0&\ldots&0&0&\mathbf{\Phi}_{N}\end{pmatrix}  \tag{76}\]  
Also, in this case, the matrix product \(\mathcal{A}^{\mathsf{T}}\mathcal{R}^{-1}\mathcal{A}\) yields a one-block-banded matrix, according to Equation (44), with blocks given by \[ \begin{split} M_{t}&=\begin{cases}\mathbf{\Theta}_{0}+\mathbf{\Lambda}_{ 2}^{\mathsf{T}}\mathbf{\Sigma}_{1}\mathbf{C}_{1}+(\mathbf{\Lambda}_{2}^{\mathsf{T}}\mathbf{ \Sigma}_{1}\mathbf{C}_{1})^{\mathsf{T}}+\mathbf{C}_{1}^{\mathsf{T}}\mathbf{\Psi}_{1}\mathbf{C} _{1}+\mathbf{\Lambda}_{2}^{\mathsf{T}}\mathbf{\Omega}_{1}\mathbf{\Lambda}_{2}&\text{if }\ t=1\\ \mathbf{\Omega}_{t-1}+\mathbf{\Lambda}_{t+1}^{\mathsf{T}}\mathbf{\Sigma}_{t}\mathbf{C}_{t}+( \mathbf{\Lambda}_{t+1}^{\mathsf{T}}\mathbf{\Sigma}_{t}\mathbf{C}_{t})^{\mathsf{T}}+\mathbf{C}_ {t}^{\mathsf{T}}\mathbf{\Psi}_{t}\mathbf{C}_{t}\\ +\mathbf{\Lambda}_{t+1}^{\mathsf{T}}\mathbf{\Omega}_{t}\mathbf{\Lambda}_{t+1}& \text{if }\ t=2,\ldots,N-1\\ \mathbf{\Omega}_{N-1}+\mathbf{C}_{N}^{\mathsf{T}}\mathbf{\Phi}_{N}\mathbf{\
abla}_{N}\mathbf{ \
abla}_{N}&\text{if }\ t=N\end{cases}\\ \text{W}_{t}&=-\mathbf{C}_{t-1}^{\mathsf{T}}\mathbf{\Sigma}_{t-1}^{\mathsf{ T}}-\mathbf{\Lambda}_{t}^{\mathsf{T}}\mathbf{\Omega}_{t-1}\end{split}  \tag{77}\]  
The vector \(\mathbf{\mathcal{A}}^{\mathsf{T}}\mathcal{R}^{-1}\mathcal{Y}\) consists of stacked \(m\)-dimensional subvectors, given by  
\[\mathbf{\mathsf{v}}_{t}= \begin{cases}\left(\mathbf{C}_{t}^{\mathsf{T}}\mathbf{\Psi}_{1}+ \mathbf{\Lambda}_{2}^{\mathsf{T}}\mathbf{\Sigma}_{1}\right)\mathbf{\mathsf{y}}_{1}+\mathbf{ \Theta}_{0}\mathbf{\mathrm{x}}_{1|0}&\text{if }\ t=1\\ \left(\mathbf{C}_{t}^{\mathsf{T}}\mathbf{\Psi}_{t}+\mathbf{\Lambda}_{t+1}^{\mathsf{T}}\bm {\Sigma}_{t}\right)\mathbf{\mathsf{y}}_{t}-\mathbf{\Sigma}_{t-1}\mathbf{\mathsf{y}}_{t-1}& \text{if }\ t=2,\ldots,N-1\\ \mathbf{C}_{N}^{\mathsf{T}}\mathbf{\Phi}_{N}\mathbf{\mathsf{y}}_{N}-\mathbf{\Sigma}_{N-1} \mathbf{\mathsf{y}}_{N-1}&\text{if }\ t=N\end{cases}