## 3 Materials and Methods
### Methodology
#### 3.1.3 Recursive Estimate
\tag{24}\]  
Therefore, we can solve for \(\mathbf{q}_{t}(x)\), yielding the desired minimizing weight vector  
\[\mathbf{q}_{t}(x)=(\mathbf{I}-\mathbf{M}_{t}^{T}\mathbf{F}\mathbf{W}_{t})\mathbf{L}_{t}\mathbf{W}_{t}(\mathbf{ p}_{t-1}(x)+\mathbf{k}(x))+\mathbf{M}_{t}^{T}\mathbf{f}(x), \tag{25}\]  
where  
\[\mathbf{L}_{t}=(\mathbf{W}_{t}\mathbf{P}_{t-1}\mathbf{W}_{t}^{T}+\mathbf{W}_{t}\mathbf{K} \mathbf{W}_{t}^{T}+\mathbf{R}_{t})^{-1}\] \[\mathbf{M}_{t}=(\mathbf{F}\mathbf{W}_{t}^{T}\mathbf{L}_{t}\mathbf{W}_{t}\mathbf{F}^{T})^{ -1}\mathbf{F}\mathbf{W}_{t}^{T}\mathbf{L}_{t}\]  
As all \(\mathbf{K}\) and \(\mathbf{R}_{t}\) are positive definite, and this implies that the inverse exists. \(\mathbf{P}_{t}\) is positive semi-definite covariance matrices, and \(\mathbf{L}_{t}\) is also positive definite. Then, we can prove that \(\mathbf{F}\mathbf{W}_{t}^{T}\mathbf{L}_{t}\mathbf{W}_{t}\mathbf{F}^{T}\) is full rank and has rank \(r\) because we have assumed \(\mathbf{F}\) has rank \(r\). This implies that \(\mathbf{M}_{t}\) exists. By the generalized least squares method we can obtain the unknown drift coefficients vector \(\mathbf{m}_{t}\) in Equation (10):  
\[\mathbf{m}_{t}=\mathbf{M}_{t}\tilde{\mathbf{Y}}_{t}. \tag{27}\]  
Thus, we can obtain the estimates of \(\widehat{c}_{t}\) and \(\widehat{c}_{t}(x)\) by the following equations:  
\[\widehat{c}_{t}=\widehat{\mathbf{Z}}_{t}-\mathbf{F}^{T}\mathbf{m}_{t} \tag{28}\] \[=\widehat{c}_{t-1}+(\mathbf{P}_{t-1}+\mathbf{K})\mathbf{W}_{t}^{T}\mathbf{L}_{t}( \mathbf{I}-\mathbf{W}_{t}\mathbf{F}^{T}\mathbf{M}_{t})\tilde{\mathbf{Y}}_{t},\] \[\widehat{c}_{t}(x)=\widehat{Z}_{t}(x)-\mathbf{f}^{T}(x)\mathbf{m}_{t}\] \[=\widehat{c}_{t-1}(x)+(\mathbf{p}_{t-1}^{T}(x)+\mathbf{k}^{T}(x))\mathbf{W}_{ t}^{T}\mathbf{L}_{t}(\mathbf{I}-\mathbf{W}_{t}\mathbf{F}^{T}\mathbf{M}_{t})\tilde{\mathbf{Y}}_{t}. \tag{29}\]  
Let \(\mathbf{G}_{t}=(\mathbf{P}  
\[\widehat{c}_{t}(x)=\sum_{i=1}^{T}\mathbf{g}_{i}^{T}(x)\widetilde{Y}_{i}. \tag{33}\]  
Then, we evaluate \(\mathbf{P}_{t}\), as in Appendix A:  
\[\mathbf{P}_{t}=(\mathbf{P}_{t-1}+\mathbf{K})(\mathbf{I}-\mathbf{W}_{t}^{T}\mathbf{G}_{t}^{T}). \tag{34}\]  
To achieve the unknown quantity \(\mathbf{p}_{t-1}(x)\), we discuss the relationship between \(\mathbf{p}_{t-1}(x)\) and \(\mathbf{k}(x)\) further in Appendix B, and obtain:  
\[\mathbf{L}_{t}=(\mathbf{W}_{t}\mathbf{P}_{t-1}\mathbf{W}_{t}^{T}+\mathbf{W}_{t}\mathbf{K} \mathbf{W}_{t}^{T}+\mathbf{R}_{t})^{-1}\]0  
We yield the solution between \(\mathbf{g}_{t}(x)\) and \(\mathbf{G}_{t}\) :  
\[\mathbf{L}_{t}=(\mathbf{W}_{t}\mathbf{P}_{t-1}\mathbf{W}_{t}^{T}+\mathbf{W}_{t}\mathbf{K} \mathbf{W}_{t}^{T}+\mathbf{R}_{t})^{-1}\]1  
The solution in Equation (33) we can also be written as:  
\[\mathbf{L}_{t}=(\mathbf{W}_{t}\mathbf{P}_{t-1}\mathbf{W}_{t}^{T}+\mathbf{W}_{t}\mathbf{K} \mathbf{W}_{t}^{T}+\mathbf{R}_{t})^{-1}\]2 \[\mathbf{L}_{t}=(\mathbf{W}_{t}\mathbf{P}_{t-1}\mathbf{W}_{t}^{T}+\mathbf{W}_{t}\mathbf{K} \mathbf{W}_{t}^{T}+\mathbf{R}_{t})^{-1}\]3  
Therefore,  
\[ \begin{gathered}\widehat{Z}_{t}(x)=\widehat{c}_{t-1}(x)+\mathbf{q}_{t }^{T}(x)\widetilde{Y}_{t}\\ =\widehat{c}_{t-1}(x)+\mathbf{g}_{t}^{T}(x)\widetilde{Y}_{t}+\mathbf{f}^{ T}(x)\mathbf{m}_{t}\\ =\mathbf{k}^{T}(x)\mathbf{K}^{-1}\widehat{c}_{t}+\mathbf{f}^{T}(x)\mathbf{M}_{t}( \mathbf{Y}_{t}-\mathbf{W}_{t}\widehat{c}_{t-1})\\ =\mathbf{k}^{T}(x)\mathbf{K}^{-1}\widehat{c}_{t-1}+\mathbf{k}^{T}(x)\mathbf{K}^{- 1}\mathbf{G}_{t}(\mathbf{Y}_{t}-\mathbf{W}_{t}\widehat{c}_{t-1})+\mathbf{f}^{T}(x)\mathbf{M}_{t}( \mathbf{Y}_{t}-\mathbf{W}_{t}\widehat{c}_{t-1})\end{gathered} . \tag{38}\]  
Finally, we use the equation in Equation (5) to write:  
\[\widehat{Y}_{t}(x)=\mathbf{k}^{T}(x)\mathbf{K}^{-1}\widehat{c}_{t-1}+\mathbf{k}^{T}(x)\bm {K}^{-1}\mathbf{G}_{t}(\mathbf{Y}_{t}-\mathbf{W}_{t}\widehat{c}_{t-1})+\mathbf{f}^{T}(x)\mathbf{M }_{t}(\mathbf{Y}_{t}-\mathbf{W}_{t}\widehat{c}_{t-1}). \tag{39}\]