## 5 Weighted Least-Squares Approach to State Estimation
### The Origin of the Non-Equivalence of the two Kalman Filters  
In Section 3, the covariance Kalman and information filter algorithms were reviewed for the case of correlated noise terms, both for the contemporaneous-form and future-form state space models, and it was stated that they are not equivalent. The reason for this lack of equivalence is not hard to find: depending on whether the contemporaneous form or future form is used, the statement that the cross-correlation matrix, \(\mathsf{S}_{t}\), does not vanish has different meaning. If in the contemporaneous form, \(\mathsf{S}_{t}\) is defined by  
\[\mathsf{S}_{t}=\mathbb{E}(\mathbf{\eta}_{\tau}\mathbf{e}_{\tau}^{\mathsf{T}}),\]  
then in future form, it would have to be defined by  
\[\mathsf{S}_{t}=\mathbb{E}(\mathbf{\eta}_{\tau-1}\mathbf{e}_{\tau}^{\mathsf{T}}),\]  
and this is precisely the definition for \(\mathsf{S}_{t}\) chosen by [PERSON] and [PERSON] (2012) since these authors use the future form but at the same time present the algorithm in Equations (12)-(17), which in the present paper, is listed as a contemporaneous-form algorithm. Then, it becomes obvious that actually two cross-correlation matrices should be defined, one for lag-zero (instantaneous) correlation and one for lag-one correlation:  
\[\mathsf{S}_{t}^{(0)}=\mathbb{E}(\mathbf{\eta}_{\tau}\mathbf{e}_{\tau}^{\mathsf{T}})\, \quad\mathsf{S}_{t}^{(1)}=\mathbb{E}(\mathbf{\eta}_{\tau-1}\mathbf{e}_{\tau}^{\mathsf{T}})\]  
Swapping the two noise terms in the definition of \(\mathsf{S}_{t}^{(0)}\) would give  
\[\big{(}\mathsf{S}_{1}^{(0)}\big{)}^{\mathsf{T}}=\mathbb{E}(\mathbf{e}_{\tau}\mathbf{ \eta}_{\tau}^{\mathsf{T}})\]  
However, in the definition of \(\mathsf{S}_{t}^{(1)}\), the two noise terms may not be swapped since  
\[\mathbb{E}(\mathbf{\eta}_{\tau-1}\mathbf{e}_{\tau}^{\mathsf{T}})\
eq\mathbb{E}(\mathbf{e}_ {\tau-1}\mathbf{\eta}_{\tau}^{\mathsf{T}})\]From the viewpoint of the contemporaneous form, the Kalman filter in Equations (12)-(17) is valid for the case  
\[S_{t}^{(0)}\
eq 0\,\quad S_{t}^{(1)}=0\]  
(with \(S_{t}\) to be replaced by \(S_{t}^{(0)}\)), while the Kalman filter in Equations (27)-(34) is valid for the case  
\[S_{t}^{(0)}=0\,\quad S_{t}^{(1)}\
eq 0\]  
(with \(S_{t}\) to be replaced by \(S_{t}^{(1)}\)). Naturally, both filters are valid for the case  
\[S_{t}^{(0)}=0\,\quad S_{t}^{(1)}=0.\]  
**6. Merging the Contemporaneous Form and Future Form**  
In the discussion of the previous section, one case has not yet been addressed:  
\[S_{t}^{(0)}\
eq 0\,\quad S_{t}^{(1)}\
eq 0.\]  
Is there a Kalman filter for this case? In order to investigate this case, the covariance matrices in Equations (63) and (75) have to be merged as follows:  
\[\mathcal{R}= \begin{pmatrix}Q_{0}&S_{1}^{(0)}&0&\dots&0&0&0\\ (S_{1}^{(0)})^{\mathsf{T}}&R_{1}&\left(S_{1}^{(1)}\right)^{\mathsf{T}}&\dots &0&0&0\\ 0&S_{1}^{(1)}&Q_{1}&\dots&0&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&0&\dots&R_{N-1}&\left(S_{N-1}^{(1)}\right)^{\mathsf{T}}&0\\ 0&0&0&\dots&S_{N-1}^{(1)}&Q_{N-1}&S_{N}^{(0)}\\ 0&0&0&\dots&0&\left(S_{N}^{(0)}\right)^{\mathsf{T}}&R_{N}\end{pmatrix} . \tag{96}\]  
This covariance matrix is no longer block-diagonal, and as a consequence, the matrix \(\mathcal{A}^{\mathsf{T}}\mathcal{R}^{-1}\mathcal{A}\) is not a one-block-banded matrix; rather, it will generically be a full matrix, such that its inverse will also be a full matrix. Consequently, no recursive algorithm for optimally solving this state estimation problem can be derived through the approach presented above in Sections 5.1 and 5.2. However, recursive algorithms for _non-optimally_ solving this problem may exist. As an example, the Kalman filter algorithms in Equations (12)-(17) and Equations (27)-(34) are considered again.