## 3 The Statistical Regression Adaptive Kalman Filtering Algorithm
### The Expectation Approach  
\(\log p_{\Phi_{k}}(\mathbf{X}_{k|k},\mathbf{Z}_{1:k})\) is factored using Equation (11):  
\[ \begin{split}\log p_{\[PERSON]_{1}}(\mathbf{X}_{k|k},\mathbf{Z}_{1:k} )&=\log[p_{\Phi_{k}}(\mathbf{Z}_{k|k}|\mathbf{X}_{k|k},\mathbf{Z} _{1:k-1})p_{\Phi_{k}}(\mathbf{X}_{k|k}|\mathbf{Z}_{1:k-1})p(\mathbf{Z}_{1:k-1 })]\\ &=\log[p_{\Phi_{k}}(\mathbf{Z}_{k|k}|\mathbf{X}_{k|k},\mathbf{Z}_{1 :k-1})]+\log[p_{\Phi_{k}}(\mathbf{X}_{k|k}|\mathbf{Z}_{1:k-1})]+\log[p(\mathbf{ Z}_{1:k-1})]\end{split}  \tag{11}\]  
where \(p_{\Phi_{k}}(\mathbf{Z}_{k}|\mathbf{X}_{k|k},\mathbf{Z}_{1|k-1})=p_{\Phi_{k}}( \mathbf{Z}_{k}|\mathbf{X}_{k|k})\) because \(\mathbf{X}_{k|k}\) includes all of the information in \(\mathbf{Z}_{1:k-1}\). **Remark 2:**_The predicted PDF \(p_{\Phi_{k}}(\mathbf{X}_{k|k}|\mathbf{Z}_{1:k-1})\) is approximated to follow the Gaussian distribution; then, the \(2n+1\) sigma points are selected, and the moment estimation of \(p_{\Phi_{k}}(\mathbf{X}_{k|k}|\mathbf{Z}_{1:k-1})\) is calculated using the unscented transformation. The mean and covariance matrix of \(p_{\Phi_{k}}(\mathbf{X}_{k|k}|\mathbf{Z}_{1:k-1})\) with higher accuracy are acquired for the nonlinear state equation._  
\[ \begin{split} p_{\Phi_{k}}\Big{(}\mathbf{X}_{k|k}|\mathbf{Z}_{1 :k-1}\Big{)}&=\text{N}\Big{(}\mathbf{X}_{k|k};\mathbf{X}_{k|k-1},\mathbf{P}_{k|k-1}\Big{)}\\ &=\text{N}\Big{(}\mathbf{X}_{k|k};\sum\limits_{i=0}^{2n}\omega_ {m}(i)\mathbf{F}_{k|k-1}\chi_{i,k-1|k-1},\mathbf{P}_{k|k-1}\Big{)}\end{split}  \tag{12}\]  
_where \(N(\bullet;\mu,\sum)\) is the Gaussian distribution with the mean \(\mu\) and covariance \(\sum\)._  
The predicted error covariance matrix and \(\mathbf{P}_{k|k-1}\) predicted state vector \(\mathbf{X}_{k|k-1}\) are calculated as follows:  
The sigma points are given using Equation (13):  
\[ \begin{cases}\chi_{i,k-1|k-1}=\mathbf{X}_{k-1|k-1},&i=0\\ \chi_{i,k-1|k-1}=\mathbf{X}_{k-1|k-1}+\big{(}\sqrt{(n+\rho)\mathbf{P}_{k-1|k-1} }\big{)}_{i},&i=1,\ldots,n\\ \chi_{i+n,k-1|k-1}=\mathbf{X}_{k-1|k-1}-(\sqrt{(n+\rho)\mathbf{P}_{k-1|k-1}} \big{)}_{i},&i=1,\ldots,n\end{cases}  \tag{13}\]  
where \(n\) means the dimension of \(\mathbf{X}_{k|k}\); \(\rho\) means the scaling parameter; \((\bullet)_{i}\) is the \(i\)th column of the matrix; \(\mathbf{P}_{k-1|k-1}\) is the error covariance matrix at the time \(k-1\). The weight functions of the mean \(W_{m}(i)\) and covariance matrix \(W_{C}(i)\) are defined as follows:  
\[W_{m}(i)=W_{C}(i)= \begin{cases}\rho/(n+\rho),&i=0\\ 1/2(n+\rho),&i=1,\ldots,2n\end{cases}  \tag{14}\]  
The predicted state vector \(\mathbf{X}_{k|k-1}\) and predicted error covariance matrix \(\mathbf{P}_{k|k-1}\) are given:  
\[\left\{ \begin{array}{l}\mathbf{X}_{k|k-1}=\sum\limits_{i=0}^{2n}\omega_{m}(i )\mathbf{F}_{k|k-1}\chi_{i,k-1|k-1}+\mathbf{u}_{k-1}\Delta t\\ \mathbf{P}_{k|k-1}=\sum\limits_{i=0}^{2n}\omega_{C}(i)\Big{(}\mathbf{F}_{k|k-1} \chi_{i,k-1|k-1}-\mathbf{X}_{k|k-1}\Big{)}\Big{(}\mathbf{F}_{k|k-1}\chi_{i,k-1 |k-1}-\mathbf{X}_{k|k-1}\Big{)}^{T}+\mathbf{Q}_{k-1}\end{array} \right. \tag{15}\]Similarly, \(p_{\Phi_{k}}(\mathbf{Z}_{k}|\mathbf{X}_{k|k})\) is approximated as  
\[ \begin{split} p_{\Phi_{k}}\Big{(}\mathbf{Z}_{k}|\mathbf{X}_{k|k} \Big{)}&=\mathrm{N}\Big{(}\mathbf{Z}_{k};\mathbf{Z}_{k|k-1}, \mathbf{R}_{k}\Big{)}\\ &=\mathrm{N}\Big{(}\mathbf{Z}_{k};\sum\limits_{i=0}^{2n}\omega_{ m}(i)\mathbf{H}\mathbf{F}_{k|k-1}\chi_{i,k-1|k-1},\mathbf{R}_{k}\Big{)}\end{split}  \tag{16}\]  
where the predicted measurement vector \(\mathbf{Z}_{k|k-1}\) is calculated as follows:  
\[\left\{ \begin{array}{l}\mathbf{Z}_{k|k-1}^{*}=\mathbf{H}\mathbf{F}_{k|k-1} \chi_{i,k-1|k-1}\\ \mathbf{Z}_{k|k-1}=\sum\limits_{i=0}^{2n}W_{m}(i)\mathbf{Z}_{k|k-1}^{*}\end{array}