xk : system state at instant K,. Mk,k+1: transition operator mapping xk to xk+1, uk : dynamical error at instant k. Observer yk = Hk (xk ) + vk . (2) yk : observation at ...
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
State estimation in high dimensional systems: the method of the ensemble unscented Kalman filter Xiaodong Luo, and Irene Moroz OCIAM, Mathematical Institute, Oxford
SCHW05, June 18, 2008
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Outline 1
Introduction Problem statement General solution by Bayesian recursive relations (BRR) BRR in linear/Gaussian systems: The Kalman filter (KF) Applying the KF to nonlinear and (or) non-Gaussian systems
2
The ensemble Kalman filter
3
The ensemble unscented Kalman filter The unscented transform (UT) Incorporating the UT into the EnKF for large-scale problems
4
Numerical results The testbed Numerical results of the EnUKF and an ordinary EnKF
5
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Problem statement
State estimation in discrete dynamical systems System model xk+1 = Mk,k+1 (xk ) + uk .
(1)
xk : system state at instant K , Mk,k+1 : transition operator mapping xk to xk+1 , uk : dynamical error at instant k. Observer yk = Hk (xk ) + vk . yk : observation at instant K , Hk : Observation operator,
vk : observational error at instant k.
(2)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
General solution by Bayesian recursive relations (BRR)
Bayesian recursive relations [1/2]
Some concepts Accumulative set of the observations: Yk = {yi }k−∞ Prior pdf of the system state: px (xk |Yk−1 )
Posterior pdf of the system state: px (xk |Yk )
Observation pdf: pv (yk |xk ) = pvk (yk − Hk (xk )), with pvk (•) being the pdf of the observational error vk Transition pdf: pu (xk |xk−1 ) = puk xk − Mk−1,k (xk−1 ) , with puk (•) being the pdf of the dynamical error uk
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
General solution by Bayesian recursive relations (BRR)
Bayesian recursive relations [2/2]
Sequential Bayesian filtering Prediction step: Compute the prior pdf px (xk |Yk−1 ) according to the following formula: Z px (xk |Yk−1 ) = pu (xk |xk−1 ) px (xk−1 |Yk−1 ) dxk−1
(3)
Filtering step: Update the prior pdf px (xk |Yk−1 ) to the posterior px (xk |Yk ) based on the Bayes’ theorem px (xk |Yk ) = R
pv (yk |xk ) px (xk |Yk−1 ) pv (yk |xk ) px (xk |Yk−1 ) dxk
(4)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
BRR in linear/Gaussian systems: The Kalman filter (KF)
The Kalman filter (KF) algorithm
Basic assumptions in the KF Mk−1,k and Hk are linear
Dynamical and observational errors uk & vk follow certain Gaussian distributions, say, N(0, Qk ) and N(0, Rk ) respectively.
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
BRR in linear/Gaussian systems: The Kalman filter (KF)
Conventions in the community of data assimilation Background: The prior estimation of a system state x before the observation y is obtained, conventionally denoted by xb . Analysis: The posterior estimation of the background xb after incorporating the additional information of the observation, denoted by xa . T Background error covariance: Pb = E xb − x xb − x . Analysis error covariance: Pa = E (xa − x) (xa − x)T .
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
BRR in linear/Gaussian systems: The Kalman filter (KF)
Schematic procedures
xak−1 , Pak−1
Mk−1,k
xbk , Pbk
Update
yk , Rk
xak , Pak
Mk,k+1
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
BRR in linear/Gaussian systems: The Kalman filter (KF)
Schematic procedures
xak−1 , Pak−1
Mk−1,k
Background at instant k
xbk , Pbk
Update
yk , Rk
xak , Pak
Mk,k+1
Analysis at instant k
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
BRR in linear/Gaussian systems: The Kalman filter (KF)
Mathematical formation Prediction step:
Numerical results
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
BRR in linear/Gaussian systems: The Kalman filter (KF)
Mathematical formation Prediction step: State estimation: xbk = Mk −1,k xak −1 Covariance: Pbk = Mk −1,k Pak −1 (Mk −1,k )T + Qk
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
BRR in linear/Gaussian systems: The Kalman filter (KF)
Mathematical formation Prediction step: State estimation: xbk = Mk −1,k xak −1 Covariance: Pbk = Mk −1,k Pak −1 (Mk −1,k )T + Qk
Filtering (or update) step:
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
BRR in linear/Gaussian systems: The Kalman filter (KF)
Mathematical formation Prediction step: State estimation: xbk = Mk −1,k xak −1 Covariance: Pbk = Mk −1,k Pak −1 (Mk −1,k )T + Qk
Filtering (or update) step:
State update: xak = xbk − Kk yk − Hk xbk . T Covariance update: Pak = Pbk − Kk (Hk ) Pbk .
−1 Kk ≡ Pbk (Hk )T Hk Pbk (Hk )T + Rk is the Kalman gain at the time instant k.
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Applying the KF to nonlinear and (or) non-Gaussian systems
Difficulties encountered in practice and remedies
Difficulties
Possible remedy
Example
Nonlinearity in the system model and the observer
Linearizing the nonlinear functions
The extended Kalman filter
Non-Gaussian distribution(s) in the dynamical and (or) observational errors
Distribution proximation
ap-
The Gaussian sum Kalman filter
High computational cost for large-scale problems
Monte Carlo simulation (subspace approximation)
The ensemble Kalman filter
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
How large a large-scale problem could be
Example: Weather forecasting models Degree-of-freedom (DOF): 108 - 109 Number of components in the observations: 106 - 107
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
The idea of the Ensemble Kalman filter (EnKF) Use only an ensemble of the system states to estimate the mean and covariance.
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Schematic procedures
xak−1,1 ··· xak−1,n
Mk−1,k
xbk,1 ··· xbk,n
Update
yk , Rk
xak,1 ··· xak,n
Mk,k+1
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Schematic procedures
xak−1,1 ··· xak−1,n
Mk−1,k
Background ensemble
xbk,1 ··· xbk,n
Update
yk , Rk
xak,1 ··· xak,n
Mk,k+1
Analysis ensemble
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Mathematical formation[1/3] Prediction step [1/2]: Forecast n the background ensemble o b Xk = xbk,i : xbk,i = Mk−1,k xak−1,i , i = 1, · · · , n
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Mathematical formation[1/3] Prediction step [1/2]: Forecast n the background ensemble o b Xk = xbk,i : xbk,i = Mk−1,k xak−1,i , i = 1, · · · , n
Evaluate the sample mean of the background (the prior): n 1P ˆbk = xb x n i=1 k,i
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Mathematical formation[1/3] Prediction step [1/2]: Forecast n the background ensemble o b Xk = xbk,i : xbk,i = Mk−1,k xak−1,i , i = 1, · · · , n
Evaluate the sample mean of the background (the prior): n 1P ˆbk = xb x n i=1 k,i h Define two square roots Sxb k and Sk , which are given by
h i 1 ˆbk , · · · , xbk,n − x ˆbk , xbk,1 − x n−1 h 1 ˆbk , · · · , Shk = √ Hk xbk,1 − Hk x n−1 i ˆbk . Hk xbk,n − Hk x Sbk = √
(5)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Mathematical formation [2/3] Prediction step [2/2]: Compute the following covariances: T ˆ b = Sb Sb + Qk , P k k k T ˆ xh = Sb Sh , P k k k T ˆ h = Sh Sh . P k k k
(6)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Mathematical formation [2/3] Prediction step [2/2]: Compute the following covariances: T ˆ b = Sb Sb + Qk , P k k k T ˆ xh = Sb Sh , P k k k T ˆ h = Sh Sh . P k k k −1 ˆ xh P ˆ h + Rk The Kalman gain: Kk ≡ P k k
(6)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Mathematical formation [3/3] Filtering step: Update the analysis ensemble mean: ˆak = x ˆbk + Kk yk − Hk x ˆbk x
Numerical results
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Mathematical formation [3/3] Filtering step: Update the analysis ensemble mean: ˆak = x ˆbk + Kk yk − Hk x ˆbk x
Update the square root Sbk to Sak by introducing a transform matrix Tk such that Sak = Sbk Tk (the choice of Tk will not be covered in this talk)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Mathematical formation [3/3] Filtering step: Update the analysis ensemble mean: ˆak = x ˆbk + Kk yk − Hk x ˆbk x
Update the square root Sbk to Sak by introducing a transform matrix Tk such that Sak = Sbk Tk (the choice of Tk will not be covered in this talk) ˆ a = Sa Sa T if necessary Compute P k
k
k
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Mathematical formation [3/3] Filtering step: Update the analysis ensemble mean: ˆak = x ˆbk + Kk yk − Hk x ˆbk x
Update the square root Sbk to Sak by introducing a transform matrix Tk such that Sak = Sbk Tk (the choice of Tk will not be covered in this talk) ˆ a = Sa Sa T if necessary Compute P k
k
k
Generate the analysis ensemble in the following way: √ ˆak + n − 1 (Sak )i , i = 1, · · · , n, xak,i = x where Sak
i
denotes the i-th column vector of Sak
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
The unscented transform (UT)
Problem statement x
F
y
Given an m-dimensional random vector x following a Gaussian ˆ and distribution, with the mean and covariance estimated by x ˆ Px respectively, we are interested in the problem of estimating the mean and covariance of the transformed random variable y = F (x), where F is a nonlinear transform function.
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
The unscented transform (UT)
Problem statement x
F
y
ˆ and Pˆx x are known Given an m-dimensional random vector x following a Gaussian ˆ and distribution, with the mean and covariance estimated by x ˆ Px respectively, we are interested in the problem of estimating the mean and covariance of the transformed random variable y = F (x), where F is a nonlinear transform function.
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
The unscented transform (UT)
Problem statement x
ˆ and Pˆx x are known
F
y
ˆ and y Pˆy ?
Given an m-dimensional random vector x following a Gaussian ˆ and distribution, with the mean and covariance estimated by x ˆ Px respectively, we are interested in the problem of estimating the mean and covariance of the transformed random variable y = F (x), where F is a nonlinear transform function.
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
The unscented transform (UT)
The unscented transform for the estimation problem[1/3] Generation of the sigma points First of all, a set of 2L + 1 system states (L ≥ m), called the sigma points, are generated as follows ˆ, X0 = x ˆ+ Xi = x
q
ˆx (L + λ)P
ˆ− Xi = x
q
ˆx (L + λ)P
q
i
, i = 1, · · · , L,
i−L
(7)
, i = L + 1, · · · , 2L,
ˆ (L + λ)Px denotes the i-th column of the square where i q ˆ root matrix (L + λ)Px , and the parameter λ is chosen as λ = 3 − L if x follows a Gaussian distribution [2].
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
The unscented transform (UT)
The unscented transform for the estimation problem [2/3] Weights of the sigma points Next, a set of weights, λ , L+λ 1 , i = 1, · · · , 2L, Wi = 2(L + λ)
W0 =
are allocated to the sigma points in Eq. (7).
(8)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
The unscented transform (UT)
The unscented transform for the estimation problem [3/3] Estimated mean and covariance of the transformed random variable Finally, the sample and covariance of the transformed random variable y are estimated as follows ˆ= y
2L X
Wi f (Xi ) ,
(9a)
i=0
ˆy = P
2L X i=0
ˆ) (f (Xi ) − y ˆ) Wi (f (Xi ) − y
T
(9b)
ˆ) (f (X0 ) − y ˆ)T . + β (f (X0 ) − y Note that in Eq. (9b), the second term on the rhs is introduced to reduce approximation error. β = 2 is shown to be optimal if x follows a Gaussian distribution [1].
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
The difficulty Given an m dimensional system, the UT requires that the number of the sigma points should be larger than 2m, which is infeasible if m itself is very large. The remedy To prevent the number of the sigma points getting too large, some sigma points have to be discarded, in the spirit of principal components analysis (PCA).
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
Selecting the sigma points via the truncated singular value decomposition (TSVD) Singular value decomposition (SVD) ˆ a at ˆak and covariance P Given the analysis ensemble mean x k ˆ a can be expressed as instant k, first of all, P k ˆ a = Ek Dk (Ek )T , P k 2 , · · · , σ2 where Dk = diag σk,1 k,m is a diagonal matrix
2 ’s, with σ 2 ≥ σ 2 ≥ 0, ∀i > j, consisting of the eigenvalues σk,i k,j k,i and Ek = ek,1 , · · · , ek,m is the matrix consisting of the eigenvectors ek,i ’s.
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
Selecting the sigma points via the truncated singular value decomposition (continued) Generation of the sigma points Based on SVD, a set of 2lk + 1 (lk ≪ m) sigma points can be generated in the spirit of Eq. (7) a ˆak , Xk,0 =x
a ˆak + (lk + λ)1/2 σk,i ek,i , i = 1, · · · , lk , Xk,i =x
(10)
a ˆak − (lk + λ)1/2 σk,i−lk ek,i−lk , i = lk + 1, · · · , 2lk , Xk,i =x
Allocation of the weights The associated weights are specified as follows Wk,0 =
λ 1 ; Wk,i = , i = 1, · · · , 2lk . lk + λ 2 (lk + λ)
(11)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
Schematic procedures
ˆak−1 x ˆa P k−1
TSVD +UT
a Xk−1,0 ··· a Xk−1,2l k −1
Mk−1,k
b Xk,0 ··· b Xk,2l k −1
Update
yk , Rk
ˆak x ˆa P k
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Incorporating the UT into the EnKF for large-scale problems
Mathematical formation[1/4] Prediction step: [1/3] Forecast o n the background ensemble b Xk = xbk,i : xbk,i = Mk−1,k xak−1,i , i = 0, · · · , 2lk−1
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
Mathematical formation[1/4] Prediction step: [1/3] Forecast o n the background ensemble b Xk = xbk,i : xbk,i = Mk−1,k xak−1,i , i = 0, · · · , 2lk−1
Compute the ensemble mean of the background and its projections: 2lk −1
ˆbk x
=
X i=0
2lk −1
ˆk = y
X i=0
b Wk−1,i Xk,i ,
(12)
b Wk−1,i Hk Xk,i
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
Mathematical formation [2/4] Prediction step: [2/3] h Create two square roots Sxb k and Sk , which are given by
q i ˆbk , · · · , Wk,2lk −1 xbk,2l − x ˆbk , Wk,0 + β xbk,0 − x k −1 q hq b h ˆk , Wk,1 Hk xbk,1 − y ˆk , Wk,0 + β Hk xk,0 − y Sk = q i b ˆ · · · , Wk,2lk −1 Hk xk,2lk −1 − yk . Sbk =
hq
(13)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
Mathematical formation [3/4] Prediction step: [3/3] Compute the covariance matrices: T ˆ b = Sb Sb + Qk , P k k k T ˆ xh = Sb Sh , P k k k T ˆ h = Sh Sh . P k k k
(14)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
Mathematical formation [3/4] Prediction step: [3/3] Compute the covariance matrices: T ˆ b = Sb Sb + Qk , P k k k T ˆ xh = Sb Sh , P k k k T ˆ h = Sh Sh . P k k k −1 ˆ xh P ˆ h + Rk The Kalman gain: Kk = P k k
(14)
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Incorporating the UT into the EnKF for large-scale problems
Mathematical formation [4/4] Filtering step: Compute the ensemble mean of the analysis: ˆak = x ˆbk + Kk yk − Hk x ˆbk x
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Incorporating the UT into the EnKF for large-scale problems
Mathematical formation [4/4] Filtering step: Compute the ensemble mean of the analysis: ˆak = x ˆbk + Kk yk − Hk x ˆbk x Compute the ensemble covariance of the analysis: T ˆa = P ˆ b − Kk P ˆ xh P k k k
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Incorporating the UT into the EnKF for large-scale problems
Mathematical formation [4/4] Filtering step: Compute the ensemble mean of the analysis: ˆak = x ˆbk + Kk yk − Hk x ˆbk x Compute the ensemble covariance of the analysis: T ˆa = P ˆ b − Kk P ˆ xh P k k k ˆa Applying the TSVD to P k , and generate a new set of sigma points Xk,0 , · · · , Xk,2lk and the associated weights Wk,0 , · · · , Wk,2lk , according to Eqs. (10) and (11) respectively.
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
The testbed
Lorenz and Emanuel model We choose a simplified atmospheric model proposed by Lorenz and Emanuel [4], which is defined as follows dxi = (xi+1 − xi−2 ) xi−1 − xi + F , i = 1, · · · , 40. dt
(15)
The quadratic terms simulate the advection, the linear term represents the internal dissipation, while the constant F = 8 acts as the external forcing ([3]). The variables xi ’s are defined cyclically such that x−1 = x39 , x0 = x40 , and x41 = x1 .
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Numerical results of the EnUKF and an ordinary EnKF
Performance measure We adopt the time averaged relative rms error (relative rmse for short) to measure the performance of state estimation, which is defined as kmax 1 X ˆak − xtrk k2 /kxtrk k2 , kx (16) er = kmax k=1
where kmax is the maximum instant, xtrk denotes the truth (the state of a control run) at instant k, and k•k2 means the L2 norm.
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Numerical results of the EnUKF and an ordinary EnKF
Performance of the EnUKF
1
Initial ensemble size=3 Initial ensemble size=4 Initial ensemble size=5 Initial ensemble size=6
0.9
0.8
Relative rmse
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
Covariance inflation factor δ
7
8
9
10
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Numerical results of the EnUKF and an ordinary EnKF
Performance of an ordinary EnKF
Corresponding to the EnUKF with initial ensemble size = 3 Corresponding to the EnUKF with initial ensemble size = 4 Corresponding to the EnUKF with initial ensemble size = 5 Corresponding to the EnUKF with initial ensemble size = 6
0.9
0.8
Relative rmse
0.7
0.6
0.5
0.4
0.3
0.2
0
1
2
3
4
5
6
Covariance inflation factor δ
7
8
9
10
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Numerical results of the EnUKF and an ordinary EnKF
Performance of an ordinary EnKF
Table: Minima of the relative rms errors
Ensemble Filter EnUKF Ordinary EnKF
Minimum of the relative rms errors n=3 n=4 n=5 n=6 0.1719 0.1722 0.1730 0.1753 0.2074 0.2074 0.2074 0.2074
Conclusion
Introduction
The ensemble Kalman filter
Summary
We have introduced:
The ensemble unscented Kalman filter
Numerical results
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Summary
We have introduced: The ensemble Kalman filter (EnKF) for large scale problems
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Summary
We have introduced: The ensemble Kalman filter (EnKF) for large scale problems A modification scheme, called ensemble unscented Kalman filter (EnUKF), by incorporating the unscented transform
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Some possible directions in future works
Numerical results
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Some possible directions in future works
Use fast SVD algorithm to reduce computational cost in generating the sigma points
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Some possible directions in future works
Use fast SVD algorithm to reduce computational cost in generating the sigma points Explore other aspects of the EnUKF, e.g., stability of the filter, sensitivities of the EnUKF to some parameters
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
Some possible directions in future works
Use fast SVD algorithm to reduce computational cost in generating the sigma points Explore other aspects of the EnUKF, e.g., stability of the filter, sensitivities of the EnUKF to some parameters Possibility of other generation schemes of the sigma points
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Q&A
Numerical results
Conclusion
Introduction
The ensemble Kalman filter
The ensemble unscented Kalman filter
Numerical results
Conclusion
References
S. J. Julier, The scaled unscented transformation, in: The Proceedings of the American Control Conference, Anchorage, AK, 2004. S. J. Julier, J. K. Uhlmann, H. F. Durrant-Whyte, A new approach for filtering nonlinear systems, in: The Proceedings of the American Control Conference, Seattle, Washington, 1995. E. N. Lorenz, Predictability-a problem solved, in: T. Palmer (ed.), Predictability., ECMWF, Reading, UK, 1996. E. N. Lorenz, K. A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, J. Atmos. Sci. 55 (1998) 399–414.
