Correction to: Conditional bias-penalized Kalman filter for improved estimation and prediction of extremes

1 Correction to: Stoch Environ Res Risk Assess (2018) 32:183–201 https://doi.org/10.1007/s00477-017-1442-8

In the original publication, Eqs. (3) and (4) should read:

$$ \widehat{H}^{T} = H^{T} + \alpha C^{T} $$

(3)

$$ \Lambda = R + \alpha \left( {1 - \alpha } \right)C\Psi_{XX} C^{T} - \alpha H\Psi_{XX} C^{T} - \alpha C\Psi_{XX} H^{T} $$

(4)

In the above, the (m × n) conditional bias (CB) gain matrix for the observation vector, $C_{1,k}^{T}$, in $C^{T} = \left[ {C_{1,k}^{T} C_{2,k}^{T} } \right],\quad C_{2,k}^{T} = 0,$ is given by:

$$ C_{1,k} = \left[ {\left( {H_{k} \Psi_{XX} H_{k}^{T} + R_{k} } \right)G_{1,k} + H_{k} \Psi_{XX} G_{2,k} } \right]L_{k}^{ - 1} $$

where

$$ G_{2,k}^{T} = \left( {H_{k}^{T} H_{k} + I} \right)^{ - 1} $$

$$ G_{1,k}^{T} = G_{2,k}^{T} H_{k}^{T} $$

$$ \begin{aligned} L_{k} &= G_{2,k}^{T} \left[ H_{k}^{T} \left( H_{k} \Psi_{XX} H_{k}^{T} + 2R_{k} \right)H_{k} + H_{k}^{T} H_{k} \Psi_{XX} \right.\\ &\left.\quad + \Psi_{XX} H_{k}^{T} H_{k} + \Psi_{XX} + 2\Sigma_{k|k - 1} \right]G_{2,k} \end{aligned}$$

With the above, Eqs. (5), (7) and (8) in the original publication do not apply. In the original publication, Eqs. (10–12) should read:

$$ \Lambda_{21,k} = - \alpha \Psi_{XX} C_{1,k}^{T} $$

(10)

$$ \Lambda_{11,k} = R_{k} + \alpha \left( {1 - \alpha } \right)C_{1,k} \Psi_{XX} C_{1,k}^{T} + \Lambda_{12,k} H_{k}^{T} + H_{k} \Lambda_{21,k} $$

(11)

$$ \Lambda_{22,k} = \Sigma_{k|k - 1} $$

(12)

In the original publication, Eqs. (19) and (20) should read:

$$ \begin{aligned} \Sigma_{k|k} &= \left[ {\varpi_{1,k} H_{k} + \varpi_{2,k} } \right]^{ - 1} \left( {\varpi_{1,k} R_{k} \varpi_{1,k}^{T} + \varpi_{2,k} \Sigma_{k|k - 1} \varpi_{2,k}^{T} } \right)\\&\quad \; \left[ {\varpi_{1,k} H_{k} + \varpi_{2,k} } \right]^{ - 1} \end{aligned} $$

(19)

$$ K_{k} = \left[ {\varpi_{1,k} H_{k} + \varpi_{2,k} } \right]^{ - 1} \varpi_{1,k} $$

(20)

where

$$ \varpi_{1,k} = \widehat{H}_{1,k}^{T} \Gamma_{11,k} + \Gamma_{21,k} $$

$$ \varpi_{2,k} = \widehat{H}_{1,k}^{T} \Gamma_{12,k} + \Gamma_{22,k} $$

In the above, the Γ matrices are as defined in the original publication.

The above correction results from Bayesian estimation of $E[(Z - HM_{X} )|X]$ [see Eq. (36) of the original publication] for which the linear observation equation is given by X = G^TZ − G^TV where $G^{T} = \left( {U^{T} H} \right)^{ - 1} U^{T}$ with U^T being some (m × (n + m)) nonzero matrix.

Appendices

Appendix 1

In the original publication, Eqs. (37), (38), (39) and (45) should read:

$$ \Sigma_{CB} = \Psi_{XX} - WC\Psi_{XX} - \Psi_{XX} C^{T} W^{T} + WC\Psi_{XX} C^{T} W^{T} $$

(37)

$$ \Sigma = \left( {1 + \alpha } \right)\Psi_{XX} - W\left( {H + \alpha C} \right)\Psi_{XX} - \Psi_{XX} \left( {H^{T} + \alpha C^{T} } \right)W^{T} + W\left( {H\Psi_{XX} H^{T} + R + \alpha C\Psi_{XX} C^{T} } \right)W^{T} $$

(38)

$$ {\text{W}} = \Psi_{XX} \widehat{H}^{T} \left[ {\widehat{H}\Psi_{XX} \widehat{H}^{T} + \Lambda } \right]^{ - 1} $$

(39)

$$ \Delta = \alpha \Psi_{XX} \widehat{H}^{T} \left[ {\widehat{H}\Psi_{XX} \widehat{H}^{T} + \Lambda } \right]^{ - 1} CM_{X} $$

(45)

where $\widehat{H}^{T}$, C and $\Lambda$ are described in Eqs. (3) and (4).

Appendix 2

In the original publication, Eqs. (52) through (57) should read:

$$ \widehat{H}_{1}^{T} = H_{k}^{T} + \alpha C_{1,k}^{T} $$

(52)

$$ \widehat{H}_{2}^{T} = I + \alpha C_{2,k}^{T} = I $$

(53)

$$ \Lambda_{11} = R_{k} + \alpha \left( {1 - \alpha } \right)C_{1,k} \Psi_{XX} C_{1,k}^{T} + \Lambda_{12,k} H_{k}^{T} + H_{k} \Lambda_{21,k} $$

(54)

$$ \Lambda_{12} = - \alpha C_{1,k} \Psi_{XX} $$

(55)

$$ \Lambda_{21} = - \alpha \Psi_{XX} C_{1,k}^{T} $$

(56)

$$ \Lambda_{22} = \Sigma_{k|k - 1} $$

(57)

In the original publication, Eq. (58) should be ignored, and Eqs. (59), (60) and (63) should read:

$$ \varpi_{1,k} = \widehat{H}_{1,k}^{T} \Gamma_{11} + \Gamma_{21} $$

(59)

$$ \varpi_{2,k} = \widehat{H}_{1,k}^{T} \Gamma_{12} + \Gamma_{22} $$

(60)

$$ \widehat{H}^{T} \Lambda^{ - 1} H = \varpi_{1} H_{k} + \varpi_{2} $$

(63)

In the original publication, Eqs. (64) and (65) should be ignored, and the apparent updated error covariance $\Sigma_{a,k|k}$ in Eq. (66), which reflects both error covariance and expectation of Type II CB squared, and the scaling matrix B in Eq. (67) should read:

$$ \Sigma_{a,k|k} = B\left[ {\widehat{H}^{T} \Lambda^{ - 1} H} \right]^{ - 1} = \alpha \Sigma_{k|k - 1} + \left[ {\varpi_{1,k} H_{k} + \varpi_{2,k} } \right]^{ - 1} $$

(66)

$$ B = \alpha \Sigma_{k|k - 1} \widehat{H}^{T} \Lambda^{ - 1} \widehat{H} + I $$

(67)

In the original publication, Eq. (68) should be ignored, and Eqs. (69) and (71) should read:

$$ \widehat{X}_{k|k} = \left[ {\varpi_{1,k} H_{k} + \varpi_{2,k} } \right]^{ - 1} \left[ {\varpi_{1,k} Z_{k} + \varpi_{2,k} \widehat{X}_{k|k - 1} } \right] $$

(70)

$$ K_{k} = \left[ {\varpi_{1,k} H_{k} + \varpi_{2,k} } \right]^{ - 1} \varpi_{1,k} $$

(71)