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Table 1 Comparison of the recursive equations for the classical and the proposed optimal Bayesian Kalman smoothers

From: A Bayesian robust Kalman smoothing framework for state-space models with uncertain noise statistics

Forward step (k=1,...,L)
Classical Kalman smoother \(\widetilde {\mathbf {z}}_{k}=\mathbf {y}_{k}-\mathbf {H}_{k}\widehat {\mathbf {x}}_{k|k-1}\)
  \(\mathbf {K}_{k}=\mathbf {P}^{\mathbf {x}}_{k|k-1}\mathbf {H}^{T}_{k}\left (\mathbf {H}_{k}\mathbf {P}^{\mathbf {x}}_{k|k-1}\mathbf {H}_{k}^{T}+\mathbf {R}\right)^{-1}\)
  \(\widehat {\mathbf {x}}_{k|k} =\widehat {\mathbf {x}}_{k|k-1} +\mathbf {K}_{k}\widetilde {\mathbf {z}}_{k}\)
  \(\widehat {\mathbf {x}}_{k+1|k}=\mathbf {\Phi }_{k}\widehat {\mathbf {x}}_{k|k-1}+\mathbf {\Phi }_{k}\mathbf {K}_{k}\widetilde {\mathbf {z}}_{k}\)
  \(\mathbf {P}^{\mathbf {x}}_{k|k} =(\mathbf {I}-\mathbf {K}_{k}\mathbf {H}_{k})\mathbf {P}_{k|k-1}^{\mathbf {x}}\)
  \(\mathbf {P}^{\mathbf {x}}_{k+1|k}=\mathbf {\Phi }_{k}\left (\mathbf {I}-\mathbf {K}_{k}\mathbf {H}_{k}\right)\mathbf {P}^{\mathbf {x}}_{k|k-1}\mathbf {\Phi }^{T}_{k} +\mathbf {\Gamma }_{k}\mathbf {Q}\mathbf {\Gamma }^{T}_{k}\)
Optimal Bayesian Kalman smoother \( \widetilde {\mathbf {z}}^{\boldsymbol {\theta }}_{k}=\mathbf {y}^{\boldsymbol {\theta }}_{k}-\mathbf {H}_{k}\widehat {\mathbf {x}}^{\boldsymbol {\theta }}_{k|k-1}\)
  \(\mathbf {K}^{\Theta }_{k}=\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}^{\mathbf {x},\boldsymbol {\theta }}_{k|k-1}\right ]\mathbf {H}^{T}_{k}\mathrm {E}_{\boldsymbol {\theta }^{\ast }}^{-1}\left [\mathbf {H}_{k}\mathbf {P}^{\mathbf {x},\boldsymbol {\theta }}_{k|k-1}\mathbf {H}_{k}^{T}+\mathbf {R}^{\theta _{2}}\right ]\)
  \( \widehat {\mathbf {x}}_{k|k}^{\boldsymbol {\theta }}=\widehat {\mathbf {x}}_{k|k-1}^{\boldsymbol {\theta }} +\mathbf {K}_{k}^{\Theta }\widetilde {\mathbf {z}}^{\boldsymbol {\theta }}_{k}\)
  \(\widehat {\mathbf {x}}^{\boldsymbol {\theta }}_{k+1|k} =\mathbf {\Phi }_{k}\widehat {\mathbf {x}}^{\boldsymbol {\theta }}_{k|k-1}+\mathbf {\Phi }_{k}\mathbf {K}^{\Theta }_{k}\widetilde {\mathbf {z}}^{\boldsymbol {\theta }}_{k}\)
  \(\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}_{k|k}^{\mathbf {x},\boldsymbol {\theta }}\right ] =(\mathbf {I}-\mathbf {K}_{k}^{\Theta }\mathbf {H}_{k})\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}_{k|k-1}^{\mathbf {x},\boldsymbol {\theta }}\right ]\)
  \(\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}^{\mathbf {x},\boldsymbol {\theta }}_{k+1|k}\right ] =\mathbf {\Phi }_{k}\left (\mathbf {I}-\!\mathbf {K}^{\Theta }_{k}\mathbf {H}_{k}\right)\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}^{\mathbf {x},\boldsymbol {\theta }}_{k|k-1}\right ] \mathbf {\Phi }^{T}_{k}\)
  \(\qquad \qquad \qquad +\mathbf {\Gamma }_{k}\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {Q}^{\theta _{1}}\right ]\mathbf {\Gamma }^{T}_{k}\)
Backward step (k=L−1,L−2,...,0)
Classical Kalman smoother \(\mathbf {A}_{k}=\mathbf {P}_{k|k}^{\mathbf {x}}\mathbf {\Phi }_{k}^{T}\left (\mathbf {P}_{k+1|k}^{\mathbf {x}}\right)^{-1}\)
  \(\widehat {\mathbf {x}}_{k|L}=\widehat {\mathbf {x}}_{k|k}+\mathbf {A}_{k}\left (\widehat {\mathbf {x}}_{k+1|L}-\widehat {\mathbf {x}}_{k+1|k}\right)\)
  \(\mathbf {P}_{k|L}^{\mathbf {x}}=\mathbf {P}_{k|k}^{\mathbf {x}}+\mathbf {A}_{k}\left (\mathbf {P}_{k+1|L}^{\mathbf {x}}-\mathbf {P}_{k+1|k}^{\mathbf {x}}\right)\mathbf {A}^{T}_{k}\)
Optimal Bayesian Kalman smoother \(\mathbf {A}^{\Theta }_{k}=\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}_{k|k}^{\mathbf {x},\boldsymbol {\theta }}\right ]\mathbf {\Phi }^{T}_{k}\mathrm {E}_{\boldsymbol {\theta }^{\ast }}^{-1}\left [\mathbf {P}^{\mathbf {x},\boldsymbol {\theta }}_{k+1|k}\right ]\)
  \(\widehat {\mathbf {x}}^{\boldsymbol {\theta }}_{k|L}=\widehat {\mathbf {x}}^{\boldsymbol {\theta }}_{k|k}+\mathbf {A}^{\Theta }_{k}\left (\widehat {\mathbf {x}}^{\boldsymbol {\theta }}_{k+1|L}-\widehat {\mathbf {x}}^{\boldsymbol {\theta }}_{k+1|k}\right)\)
  \(\begin {aligned} \mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}_{k|L}^{\mathbf {x},\boldsymbol {\theta }}\right ]&=\mathrm {E}_{\boldsymbol {\theta }^{\ast }} \left [\mathbf {P}_{k|k}^{\mathbf {x},\boldsymbol {\theta }}\right ]+\mathbf {A}^{\Theta }_{k}\left (\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}^{\mathbf {x},\boldsymbol {\theta }}_{k+1|k}\right ]\right.\\ &\quad -\left.\mathrm {E}_{\boldsymbol {\theta }^{\ast }}\left [\mathbf {P}^{\mathbf {x},\boldsymbol {\theta }}_{k+1|L}\right ]\right)\left (\mathbf {A}^{\Theta }_{k}\right)^{T} \end {aligned}\)