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

Abstract

Kalman filter (KF) and its variants are widely used for real-time state updating and prediction in environmental science and engineering. Whereas in many applications the most important performance criterion may be the fraction of the times when the filter performs satisfactorily under different conditions, in many other applications estimation and prediction specifically of extremes, such as floods, droughts, algal blooms, etc., may be of primary importance. Because KF is essentially a least squares solution, it is subject to conditional biases (CB) which arise from the error-in-variable, or attenuation, effects when the model dynamics are highly uncertain, the observations have large errors and/or the system being modeled is not very predictable. In this work, we describe conditional bias-penalized KF, or CBPKF, based on CB-penalized linear estimation which minimizes a weighted sum of error variance and expectation of Type-II CB squared and comparatively evaluate with KF through a set of synthetic experiments for one-dimensional state estimation under the idealized conditions of normality and linearity. The results show that CBPKF reduces root mean square error (RMSE) over KF by 10–20% or more over the tails of the distribution of the true state. In the unconditional sense CBPKF performs comparably to KF for nonstationary cases in that CBPKF increases RMSE over all ranges of the true state only up to 3%. With the ability to reduce CB explicitly, CBPKF provides a significant new addition to the existing suite of filtering techniques for improved analysis and prediction of extreme states of uncertain environmental systems.

Appendices

Appendix 1

1.1 Derivation of conditional bias-penalized Fisher-like linear estimator

Here we derive the Fisher-like CB-penalized linear estimator in the context of CBPKF (see Appendix 2) with additional details that were not presented in Seo (2013). The estimator sought is of the form, $X^{*} = WZ$ where $X^{*}$ denotes the (m × 1) vector of the estimated states, W denotes the (m × (n + m)) weight matrix and Z denotes the ((n + m) × 1) observation vector. The particular choice of the dimensionality of Z is to relate to CBPKF. We assume the following linear observation equation:

$${\text{Z}} = {\text{HX}} + {\text{V}}$$

(31)

where X denotes the (m × 1) vector of the true state with E[X] = M_X and Cov[X, X^T] = Ψ_XX, H denotes the ((n + m) × m) linear observation equation matrix, and V denotes the ((n + m) × 1) zero-mean measurement error vector with Cov[V,V] = R. Assuming Cov[X,V] = 0, the Bayesian estimator for X, or $X^{*}$, is given by (Schweppe 1973):

$$X^{*} = M_{X} + W(Z - M_{Z} )$$

(32)

where W denotes the (m × (n + m)) weight matrix that minimizes the error covariance. The error covariance matrix for $X^{*}$, $\Sigma_{EV} = E_{{X,X^{*} }} [(X - X^{*} )(X - X^{*} )^{T} ]$, where the variables subscripted denote the random variables on which the expectations operate, is given by:

$$\Sigma_{EV} = \Psi_{XX} - WH\Psi_{XX} - \Psi_{XX} H^{T} W^{T} + W(H\Psi_{XX} H^{T} + R)W^{T}$$

(33)

The quadratic penalty due to Type-II CB, $X - E_{{X^{*} }} [X^{*} |X]$, is given by:

$$\Sigma_{CB} = E_{X} [(X - E_{{X^{*} }} [X^{*} |X])(X - E_{{X^{*} }} [X^{*} |X])^{T} ]$$

(34)

Using (32), we may rewrite the CB in (34) as:

$$X - E[X^{*} |X] = (X - M_{X} ) - WE[(Z - HM_{X} )|X]$$

(35)

We model $E[(Z - HM_{X} )|X]$ in (35) using the Bayesian estimator again as:

$$E[(Z - HM_{X} )|X] = \Psi_{ZX} \Psi_{XX}^{ - 1} (X - M_{X} )$$

(36)

where $\Psi_{ZX} = Cov(Z,X)$. With the above, we may now write $\Sigma_{CB}$ as:

$$\Sigma_{CB} = \Psi_{XX} - WA - A^{T} W^{T} + WCW^{T}$$

(37)

where $C \equiv \Psi_{ZX} \Psi_{XX}^{ - 1} \Psi_{XZ}$ and $A \equiv \Psi_{ZX}$. In CB-penalized estimation, we minimize $\Sigma = \Sigma_{EV} + \alpha \Sigma_{CB}$ where α is some positive weighting coefficient:

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

(38)

The weighting coefficient, α, may be made into an (m × m) matrix if it is necessary to prescribe different weights to the CB penalty for different state variables. In this work, it is assumed for simplicity that the weights are the same for all state variables. Differentiating Σ with respect to W and setting it to 0, we have:

$$W = (\Psi_{XX} H^{T} + \alpha A^{T} )[H\Psi_{XX} H^{T} + R + \alpha C]^{ - 1}$$

(39)

Replacing W in (38) with (39), we have for the estimation variance for the CB-penalized estimate:

$$\Sigma = (1 + \alpha )\Psi_{XX} - \Psi_{XX} \hat{H}^{T} [\hat{H}\Psi_{XX} \hat{H}^{T} + \Lambda ]^{ - 1} \hat{H}\Psi_{XX}$$

(40)

where

$$\hat{H}^{T} = H^{T} + \alpha \Psi_{XX}^{ - 1} \Psi_{XZ}$$

(41)

$$\Lambda = R + \alpha (1 - \alpha )\Psi_{ZX} \Psi_{XX}^{ - 1} \Psi_{XZ} - \alpha H\Psi_{XZ} - \alpha \Psi_{ZX} H^{T}$$

(42)

Using the matrix inversion lemma, we may rewrite $\Sigma$ in (40) as:

$$\Sigma = \alpha \Psi_{XX} + [\hat{H}^{T} \Lambda ^{ - 1} \hat{H} + \Psi_{XX}^{ - 1} ]^{ - 1}$$

(43)

Replacing W in (32) with (39) and after some matrix manipulations, we have for $X^{*}$:

$$X^{*} = [\hat{H}^{T} \Lambda ^{ - 1} \hat{H} + \Psi_{XX}^{ - 1} ]^{ - 1} \{ \hat{H}^{T} \Lambda ^{ - 1} Z + \Psi_{XX}^{ - 1} M_{X} \} + \Delta$$

(44)

where

$$\Delta = \Psi_{XX} \hat{H}^{T} [\hat{H}\Psi_{XX} \hat{H}^{T} + \Lambda ]^{ - 1} \Psi_{ZX} \Psi_{XX}^{ - 1} M_{X}$$

(45)

To render the Bayesian solution to a Fisher solution, we let $\Psi_{XX}^{ - 1}$ vanish in (42) and (44) in the brackets only which are associated exclusively with the penalty for error covariance in (33) and arrive at the following intermediate solution for CB-penalized Fisher-like linear estimation:

$$\Sigma = B[\hat{H}^{T} \Lambda ^{ - 1} \hat{H}]^{ - 1}$$

(46)

$$X^{*} = [\hat{H}^{T} \Lambda ^{ - 1} \hat{H}]^{ - 1} \hat{H}^{T} \Lambda ^{ - 1} Z + \Delta$$

(47)

where

$${\rm B} = \alpha \Psi_{XX} \hat{H}^{T} \Lambda ^{ - 1} \hat{H} + I$$

(48)

To obtain the estimator of the form, $X^{*} = WZ$, we impose the unbiasedness condition, $E[X^{*} ] = X$, or equivalently:

$$WH = I$$

(49)

It is readily seen in (47) that the above unbiasedness condition is satisfied by replacing $[\hat{H}^{T} \Lambda ^{ - 1} \hat{H}]^{ - 1}$ with $[\hat{H}^{T} \Lambda ^{ - 1} H]^{ - 1}$ and dropping ∆. The Fisher-like solution for CB-penalized linear estimation is hence given by:

$$\Sigma = {\rm B}[\hat{H}^{T} \Lambda ^{ - 1} H]^{ - 1}$$

(50)

$$X^{*} = [\hat{H}^{T} \Lambda ^{ - 1} H]^{ - 1} \hat{H}^{T} \Lambda ^{ - 1} Z$$

(51)

The development above indicates that CB-penalized Fisher-like linear estimation is analogous to Fisher estimation in which the observation matrix, H, and the measurement error covariance matrix, R, are modified by the a priori knowledge of Ψ_XX and Ψ_XZ, and the estimation variance is scaled by a factor of B. Because (50) and (51) are not based on explicit constrained minimization, they may not represent the optimal solution in the least squares sense. It can be shown, however, that for m = 1 and α = 0.5 with perfect observations (51) is identical to the conditional bias-penalized kriging (CBPK) estimate which is based on explicit constrained minimization (Seo 2013; Seo et al. 2014; Kim et al. 2016), and that (50) converges to the CBPK estimation variance as $n \to \infty$.

Appendix 2

2.1 Derivation of conditional bias-penalized Kalman filter (CBPKF)

Here we derive CBPKF from the Fisher-like solution of Appendix 1 for estimation of the (m × 1) true state, $X_{k}$, using the (n × 1) observation, Z_k, and (m × 1) model prediction, $\hat{X}_{k|k - 1}$, and their (n × n) and (m × m) error covariances, $R_{k} = E[V_{k} V_{k}^{T} ]$, and $\Sigma_{k|k - 1}$, respectively. The observation equation is given by $Z_{k} = H_{k} X_{k} + V_{k}$ where it is assumed that the true state, $X_{k}$, is independent of the measurement error, V_k, or the model prediction error, $\hat{X}_{k|k - 1} - X_{k}$, so that we may write, e.g., $\Psi_{{Z_{k} X_{k} }} = H_{k} \Psi_{{X_{k} X_{k} }}$. Decomposing the structure matrix H in (31) into the first submatrix that relates the observations to the true states, H₁ = H_k, and the second submatrix that relates the model-predicted states to the true states, H₂ = I, we have for the m × (n + m) modified structure matrix, $\hat{H}^{T} = [\hat{H}_{1}^{T} \begin{array}{*{20}c} {} \\ \end{array} \hat{H}_{2}^{T} ]$ in (41):

$$\hat{H}_{1}^{T} = H_{k}^{T} + \alpha \Psi_{{X_{k} X_{k} }}^{ - 1} \Psi_{{X_{k} Z_{k} }} = (1 + \alpha )H_{k}^{T}$$

(52)

$$\hat{H}_{2}^{T} = I + \alpha \Psi_{{X_{k} X_{k} }}^{ - 1} \Psi_{{X_{k} X_{k|k - 1} }} = (1 + \alpha )I$$

(53)

where the (m × n) and (m × m) covariance matrices, $\Psi_{{X_{k} Z_{k} }}$ and $\Psi_{{X_{k} X_{k|k - 1} }}$, denote $Cov[X_{k} ,Z_{k}^{T} ]$ and $Cov[X_{k} ,X_{k|k - 1}^{T} ]$, respectively. An obvious choice for $\Psi_{XX}$ in practice is $\Sigma_{k|k - 1}$ obtained from propagating $\Sigma_{k - 1|k - 1}$ using the dynamical model with model errors as appropriate. With $\Psi_{XX} = \Sigma_{k|k - 1}$, the (n × n), (n × m), (m × n) and (m × m) submatrices, Λ₁₁, Λ₁₂, Λ₂₁ and Λ₂₂, of the (n + m) × (n + m) revised error covariance matrix, $\Lambda$ in (42), are given by:

$$\Lambda _{11} = R_{k} - \alpha (\alpha + 1)H_{k} \Sigma_{k|k - 1} H_{k}^{T}$$

(54)

$$\Lambda _{12} = - \alpha (\alpha + 1)H_{k} \Sigma_{k|k - 1}$$

(55)

$$\Lambda _{21} = - \alpha (\alpha + 1)\Sigma_{k|k - 1} H_{k}^{T}$$

(56)

$$\Lambda _{22} = \{ 1 - \alpha (\alpha + 1)\} \Sigma_{k|k - 1}$$

(57)

CBPKF requires that Λ₁₁ and Λ₂₂ to be positive semidefinite (see Appendix 3) which yields the following necessary condition for α from (54) and (57):

$$0 \le \alpha \le \hbox{min} \left\{ {\sqrt {\frac{{Tr[R_{k} ]}}{{Tr[H_{k} \Sigma_{k|k - 1} H_{k}^{T} ]}} + \frac{1}{4}} - \frac{1}{2},\frac{{\sqrt[{}]{5} - 1}}{2}} \right\}$$

(58)

where Tr[ ] denotes the trace of the symmetric matrix bracketed and the second term in the upper bound is ~0.618. Note in (58) that, if the states are perfectly observed so that we have $Tr[R_{k} ] = 0$, α is reduced to zero and hence CBPKF becomes KF. Similarly, if the model forecast is diffuse so that we have $\Sigma_{k|k - 1} \to \infty$, CBPKF is again reduced to KF. The (m × (n + m)) non-normalized weight matrix, $\varpi = [\varpi_{1} \begin{array}{*{20}c} {} \\ \end{array} \varpi_{2} ] = \hat{H}^{T} \Lambda ^{ - 1}$ in (51), where $\varpi_{1}$ and $\varpi_{2}$ are the (m × n) and (m × m) non-normalized weight submatrices for Z_k and $\hat{X}_{k|k - 1}$, respectively, may be evaluated by:

$$\varpi_{1} = \hat{H}_{1}^{T} \Gamma_{11} + \hat{H}_{2}^{T} \Gamma_{21} = (1 + \alpha )[H_{k}^{T} \Gamma_{11} + \Gamma_{21} ]$$

(59)

$$\omega_{2} = \hat{H}_{1}^{T} \Gamma_{12} + \hat{H}_{2}^{T} \Gamma_{22} = (1 + \alpha )[H_{k}^{T} \Gamma_{12} + \Gamma_{22} ]$$

(60)

In the above, the inverse of the (n + m) × (n + m) modified error covariance matrix Γ is given by:

$$\Lambda ^{{ - 1}} = \Gamma = \left[ {\begin{array}{*{20}c} {\Gamma _{{11}} } & {\Gamma _{{12}} } \\ {\Gamma _{{21}} } & {\Gamma _{{22}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\Lambda _{{11}}^{{ - 1}} + \Lambda _{{11}}^{{ - 1}} \Lambda _{{12}} \Gamma _{{22}} \Lambda _{{21}} \Lambda _{{11}}^{{ - 1}} } & { - \Lambda _{{11}}^{{ - 1}} \Lambda _{{12}} \Gamma _{{22}} } \\ { - \Gamma _{{22}} \Lambda _{{21}} \Lambda _{{11}}^{{ - 1}} } & {\Gamma _{{22}} } \\ \end{array} } \right]$$

(61)

where

$$\Gamma_{22}^{ - 1} = \Lambda _{22} - \Lambda _{21} \Lambda _{11}^{ - 1} \Lambda _{12}$$

(62)

The (m × m) matrix, $\hat{H}^{T} \Lambda ^{ - 1} H$ in (50) and (51), is given by:

$$\hat{H}^{T} \Lambda ^{ - 1} H = \varpi_{1} H_{k} + \varpi_{2} = (1 + \alpha )[(H_{k}^{T} \Gamma_{11} + \Gamma_{21} )H_{k} + \Gamma_{21} H_{k} + \Gamma_{22} ]$$

(63)

Positive semidefiniteness of $\hat{H}^{T} \Lambda ^{ - 1} H$ and nonnegativity of the CBPK gain (see 71) yield the following additional necessary conditions:

$$Tr[(H_{k}^{T} \Gamma_{11} + \Gamma_{21} )H_{k} ] \ge 0$$

(64)

$$Tr[\Gamma_{22} + H_{k}^{T} \Gamma_{12} ] \ge 0$$

(65)

From (50), we then have for the filtered variance:

$$\Sigma_{k|k} = B[\hat{H}^{T} \Lambda ^{ - 1} H]^{ - 1} = (1 + \alpha )\alpha \Sigma_{k|k - 1} + \{ (1 + \alpha )[(H_{k}^{T} \Gamma_{11} + \Gamma_{21} )H_{k} + \Gamma_{21} H_{k} + \Gamma_{22} ]\}^{ - 1}$$

(66)

where

$${\rm B} = \alpha \Sigma_{k|k - 1} \hat{H}^{T} \Lambda ^{ - 1} \hat{H} + I = \alpha \Sigma_{k|k - 1} (1 + \alpha )^{2} [(H_{k}^{T} \Gamma_{11} + \Gamma_{21} )H_{k} + \Gamma_{21} H_{k} + \Gamma_{22} ] + I$$

(67)

In (66), a necessary condition for positive semidefiniteness of $\Sigma_{k|k - 1} - \Sigma_{k|k}$ is:

$$(1 + \alpha )(\alpha^{2} + \alpha - 1) \le Tr([(H_{k}^{T} \Gamma_{11} + \Gamma_{21} )H_{k} + \Gamma_{21} H_{k} + \Gamma_{22} ]^{ - 1} )/Tr(\Sigma_{k|k - 1} )$$

(68)

Noting that the minimum for the right-hand side of (71) is zero, we may reduce (71) to $(\alpha^{2} + \alpha - 1) \le 0$ which is identical to the necessary condition for positive semidefiniteness conditions for (57). As such, (58) suffices. Because the positive semidefiniteness conditions employed above are only necessary conditions, there is no guarantee that $\Sigma_{k|k} \le \Sigma_{k|k - 1}$ holds in (66). If this inequality does not hold, it is necessary to reduce α and repeat the above process. From (51), we have for the filtered estimate:

$$\begin{aligned} \hat{X}_{k|k} &= [\hat{H}^{T} \Lambda ^{ - 1} H]^{ - 1} [\varpi_{1} Z_{k} + \varpi_{2} \hat{X}_{k|k - 1} ] \hfill \\ &= [(H_{k}^{T} \Gamma_{11} + \Gamma_{21} )H_{k} + \Gamma_{21} H_{k} + \Gamma_{22} ]^{ - 1} \{ [H_{k}^{T} \Gamma_{11} + \Gamma_{21} ]Z_{k} + [H_{k}^{T} \Gamma_{12} + \Gamma_{22} ]\hat{X}_{k|k - 1} \} \hfill \\ \end{aligned}$$

(69)

It can be easily shown using the matrix inversion lemma that (69) can be rewritten in the more familiar form:

$$\hat{X}_{k|k} = \hat{X}_{k|k - 1} + K_{k} [Z_{k} - H_{k} \hat{X}_{k|k - 1} ]$$

(70)

where the CBPK gain, $K_{k}$, is given by:

$$K_{k} = [(H_{k}^{T} \Gamma_{11} + \Gamma_{21} )H_{k} + \Gamma_{21} H_{k} + \Gamma_{22} ]^{ - 1} [H_{k}^{T} \Gamma_{11} + \Gamma_{21} ]$$

(71)

Appendix 3

3.1 Alternative form of conditional bias-penalized Kalman filter (CBPKF)

Here we express CBPKF in an alternative form for direct comparison with KF by factorizing $\Lambda ^{ - 1}$ in (50) and (51) as follows:

$$\begin{aligned} \Lambda ^{{ - 1}} = \left( {\begin{array}{*{20}c} {\Lambda _{{11}} } & {\Lambda _{{12}} } \\ {\Lambda _{{21}} } & {\Lambda _{{22}} } \\ \end{array} } \right)^{{ - 1}} = \left[ {\left( {\begin{array}{*{20}c} I & {0\quad } \\ {\Lambda _{{21}} } & {\Lambda _{{11}}^{{ - 1}} \quad I} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\Lambda _{{11}} } & 0 \\ 0 & {\Lambda _{{22}} - \Lambda _{{21}} \Lambda _{{11}}^{{ - 1}} \Lambda _{{12}} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} I & {\Lambda _{{11}}^{{ - 1}} \Lambda _{{12}} } \\ 0 & I \\ \end{array} } \right)} \right]^{{ - 1}} \hfill \\ = \left( {\begin{array}{*{20}c} I & { - \Lambda _{{11}}^{{ - 1}} \Lambda _{{12}} } \\ 0 & I \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\Lambda _{{11}}^{{ - 1}} } & 0 \\ 0 & {(\Lambda _{{22}} - \Lambda _{{21}} \Lambda _{{11}}^{{ - 1}} \Lambda _{{12}} )^{{ - 1}} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} I & 0 \\ { - \Lambda _{{21}} \begin{array}{*{20}c} {\Lambda _{{11}}^{{ - 1}} } \\ \end{array} } & I \\ \end{array} } \right) \hfill \\ \end{aligned}$$

(72)

In the above, we could have factorized $\Lambda$ such that the first and third matrices are lower and upper matrices, respectively, which would have yielded an alternative but equivalent expression. With (72), it can be easily shown that the CBPKF error covariance and estimate, $\Sigma_{k|k}$ and $\hat{X}_{k|k}$, respectively, are given by:

$$\Sigma_{k|k} = \frac{1}{1 + \alpha }\tilde{\Sigma }_{k|k} + \alpha (1 + \alpha )\Sigma_{k|k - 1}$$

(73)

$$\hat{X}_{k|k} = \tilde{X}_{k|k} - \tilde{\Sigma }_{k|k} \tilde{\Sigma }{}_{k|k - 1}^{ - 1} \Lambda _{12} \Lambda _{11}^{ - 1} Z_{k}$$

(74)

In the above, $\tilde{\Sigma }_{k|k}$, $\tilde{X}_{k|k}$ and $\tilde{\Sigma }_{k|k - 1}$ denote the “pseudo” updated error covariance, updated states and forecast error covariance, respectively, defined solely to render the CBPKF solution to resemble the familiar KF solution below:

$$\tilde{\Sigma }_{k|k} = [H_{k}^{T} \Lambda _{11}^{ - 1} H{}_{k} + \tilde{\Sigma }{}_{k|k - 1}^{ - 1} ]^{ - 1}$$

(75)

$$\tilde{X}_{k|k} = \hat{X}_{k|k - 1} + \tilde{K}_{k} [Z_{k} - H_{k}^{T} \hat{X}_{k|k - 1} ]$$

(76)

$$\tilde{K}_{k} = \tilde{\Sigma }_{k|k - 1} H_{k}^{T} [\Lambda _{11} + H_{k} \tilde{\Sigma }_{k|k - 1} H_{k}^{T} ]^{ - 1}$$

(77)

In the above, the forecast error covariance is given by:

$$\tilde{\Sigma }_{k|k - 1}^{ - 1} = (I - H_{k}^{T} \Lambda _{11}^{ - 1} \Lambda _{12} )(\Lambda _{22} - \Lambda _{21} \Lambda _{11}^{ - 1} \Lambda _{12} )^{ - 1} = (I - H_{k}^{T} \Lambda _{11}^{ - 1} \Lambda _{12} )\Gamma_{22} = \Gamma_{22} + H_{k}^{T} \Gamma_{12}$$

(78)

Nonnegativity of the CBPK gain in (77) requires that $\Lambda _{11}$ in (54), $\Lambda _{22}$ (from the alternative expression for (72)) in (57) and $\tilde{\Sigma }_{k|k - 1}^{ - 1}$ in (78), respectively, are positive semidefinite. The first two conditions are used in (58) and the third condition is used in (65). The alternative development described above indicates that CBPKF is a combination of KF with modified measurement and model forecast error covariances as shown in (75), (77) and (78), and adjustment to the resulting KF solution according to (73) and (74).