Ensemble Filter and Smoother
Quantifying Uncertainty
Sai Ravela
M. I. T
2012
1
Quantifying Uncertainty
Ensemble Filter and Smoother
Uncertainty Propagates in Time-Depdendent
Processes
X0f
X1f
X2f
Xnf
M ≡ M(x n ; αn )
x n+1 = M(x n ; αn ) + ω n
M: -Physical or Statistical Model
2
Quantifying Uncertainty
Ensemble Filter and Smoother
Uncertainty Propagates in Bayesian Networks
L
T
P
R
F
Q
Found in Hierarchical Bayes, Graphical Models.
3
Quantifying Uncertainty
Ensemble Filter and Smoother
Uncertainty Propagates in Spatial Processes
Grid nodes
Interactions
?
4
Quantifying Uncertainty
Ensemble Filter and Smoother
Inference Problems
1. Two-point boundary value problems, incl. uncertainty estimation
propagation. Fixed Point Smoother.
2. Recursive Bayesian Estimation for Sequential Filtering and
Smoothing.
3. Nonlinearity and Dimensionality and Uncertainty: Ensemble
Filter & Smoother.
5
Quantifying Uncertainty
Ensemble Filter and Smoother
Inference Problems
1. Two-point boundary value problems, incl. uncertainty estimation
propagation. Fixed Point Smoother.
2. Recursive Bayesian Estimation for Sequential Filtering and
Smoothing.
3. Nonlinearity and Dimensionality and Uncertainty: Ensemble
Filter & Smoother.
Propagating Uncertainty, a first step.
6
Quantifying Uncertainty
Ensemble Filter and Smoother
Variational Inference
1
−1
(x − x b )T C00
(x 0 − x b )+
2 0
�
�
m
�
1
T
T −1
(y − Hx i ) R (y i − Hx i ) + λi [x i − M(xi−1 ; α)]
2 i
J(x 0 ) :=
i=1
Cannot deal with stochastic model (i.e. model error). Needs a
Bayesian formalism.
7
Quantifying Uncertainty
Ensemble Filter and Smoother
Filters and Smoothers
Sequential Filtering:
P(x n |y 1 . . . y n )
∝
P(y n |x n )
�
P(x n |x n−1 )P(x n−1 |y 1 . . . y n−1 )(1)
xn−1
=
P(y n |x n )P(x n |y 1 . . . y n−1 )
(2)
=
P(y n |x n )P(x nf )
(3)
The recursive form is simple when a perfect model is assumed, but
the Kolmogorov-Chapman equation has to be used in the presence of
model error. P(x n |y 1 . . . y n−1 ) is the forecast distribution or prior
distribution also seen as P(x fn )
8
Quantifying Uncertainty
Ensemble Filter and Smoother
Write the Objective
Sequential Filtering:
J(x n )
:=
1
(x − x fn )T Pf−1 (x n − x fn ) +
2
n
1
(y − Hx n )T R −1 (y n − Hx n )
2 n
(4)
We have assumed a linear observation operator y n = Hx n + η, with
η ∼ N(0, R).
9
Quantifying Uncertainty
Ensemble Filter and Smoother
Find the Stationary Point
Sequential Filtering:
x̂ n
=
x fn + Pf H T (HPf H T + R)−1 (y n − Hx fn )
a
Pa
= x
= (H T R −1 H + Pf−1 )−1
=
Pf − Pf H T (HPf H T + R)−1 HPf
(5)
(6)
(7)
(8)
Then, launch a new prediction x fn+1 = M(x n ) and the new uncertainty
∂M
(predicted) is Pf = LPa LT , where L = ∂x
when the model is
n
nonlinear. Propagating produces the moments of P(x n+1 |y 1 . . . y n ).
10
Quantifying Uncertainty
Ensemble Filter and Smoother
Smoother
We are interested in the state estimates at all points in an interval,
that is:
(9)
P(x 1 . . . x n |y 1 . . . y n )
The joint distribution can account for model errors, state and
parameter errors within its framework.
We break it down via Bayes Rule, Conditional Independence and
Markov assumption, and marginalization and perfect
modelassumption, leading to a coupled set of equations that are
recursively solved.
11
Quantifying Uncertainty
Ensemble Filter and Smoother
Uncertainty Propagation is Expensive
Forward(you’ll need this in the end)
Cii =
∂M
∂M T
Ci−1i−1
0<i ≤m
∂xi−1
∂xi−1
Backward via information form:
ˆImm = H T R −1 H
∂M T
∂M
Ii+1i+1
+ H T R −1 H
∂xi
∂xi
�
�−1
∂M T ˆ ∂M
−1
= C00
+
I11
∂x0
∂x0
Îii =
ˆ 00
C
12
Quantifying Uncertainty
Ensemble Filter and Smoother
The Dimensionality and Nonlinearity Challenges
Monte-Carlo
�
Reduced-rank approximation
�
Particle Filter
Domain Decomposition
�
Localization, Localized Filters
�
Scale-recursive Spatial Inference
Model Reduction & Interpolation
� Snapshots & POD
Krylov Subspace
Response Surface Models
�
�
Deterministic Equivalent Modeling Method
�
Stochastic Response Surface Methodology
Polynomial Chaos Expansions
13
� Generalized Polynomial Chaos
Quantifying Uncertainty
Ensemble Filter and Smoother
Monte-Carlo
• •
• • •
• • •
••
•
•
• •
•
• • •
•
Leading
Lyapunov
vectors
Isotropic
Initial Perturbation
14
Quantifying Uncertainty
time
Ensemble Filter and Smoother
Filter-Updating
Af = [x f1 . . . x fs ] ⇒ All at time T
f
f
Ãf = [x˜ 1 . . . x˜ s ]
NOTE THAT
Pf =
1
Ãf ÃfT ⇐ Uncertainty
s−1
So, propagate uncertainty through Samples “Integrated” forward.
Model is not linearized.
15
Quantifying Uncertainty
Ensemble Filter and Smoother
No Linearization
y = h(x) + η,
η ∼ N(0, R)
Z = [y + η 1 , . . . , y + η s ] ← Perturbed Observations
R≈
1 ˜
Z · Z̃ T
s−1
Also, let
Ωf = h(Af ) = [h(x f1 ) . . . h(x fs )]
Ω̃f defined similarly
16
Quantifying Uncertainty
Ensemble Filter and Smoother
Uncorrelated Noise
Note
˜ fT + Z̃ T ) = (Ω
˜ f Ω̃fT + Z̃ Z̃ T )
(Ω̃f + Z̃ )(Ω
When observation noise is uncorrelated with state ≡ an assumption
Let
x a be the estimate, analysis, ‘posterior’ rv.
Aa and Ãa similarly, defined.
17
Quantifying Uncertainty
Ensemble Filter and Smoother
Easy Formulation
Aa = Af + Ãf Ω̃fT Ω̃f Ω̃fT + Z̃ Z̃ T
−1
Z − Ωf
Identical to KF/EKF in linear/linearized case
⇒ No linearization of the model
⇒ No explicit uncertainty (covariance) propagation
˜f Ω
˜ fT + Z̃ Z̃ T
Ω
−1
˜ fT + Z̃ T ]
= [Ω̃f + Z̃ ][Ω
= (CC T )−1
18
Quantifying Uncertainty
−1
Ensemble Filter and Smoother
Solution
Let
C = [U
S
VT]
[CC T ]−1 = US −2 U T
= (US −1 )(US −1 )T
√ √ T
= D D
=D
19
Quantifying Uncertainty
Ensemble Filter and Smoother
Fast Calculation
Aa = Af + Ãf Ω̃fT [US −2 U T ][Z − Ωf ]
(n, s)
(n, s)
(n, s)(s, n)(n, s)(s, s)(s, n)(n, s)
(n, s)
Return by right to left, multiply; FAST, low-dimensional
Aa =Af + Ãf X5
=Af (Is + X4 )
=Af X5
A “weakly” nonlinear transformation (X5 ≡ X5 (Af ))
20
Quantifying Uncertainty
Ensemble Filter and Smoother
Time Dependent Example
Lorentz
ẋ = −xi−2 xi−1 + xi−1 xi+1 − xi + u
+ uU
= xi−1 [xi+1 − xi−2 ] − xi
U
U
Dissipative
Advective
Chaotic
Filter
21
Quantifying Uncertainty
Forcing
Ensemble Filter and Smoother
Need to Get multimedia WORKING
Play ENKFLP.wmv!
Chalk Talk: Method 2.
Demo: Matlab.
Demo: PI Bottle.
22
Quantifying Uncertainty
Ensemble Filter and Smoother
Plug and Play
So,
•
••
•
x0
•
• ••
•
x1
y1
Aa0 = Af0 Is ← No measurement
Aa1 = Af1 X 51 ← Filter, same as X5
As1 = Aa1 Is ← No future measurement
−2 T
˜ fT
As = Aa + Ãa Ω
U1 ][Z1 − Ωf1 ]
1 [U1 S
0
0
0
1
= Aa0 X 51
Note: X 5 here is same as X5 in earlier slide.
23
Quantifying Uncertainty
Ensemble Filter and Smoother
Send me a message
On the graph
y1
x0
x1
X 51
Message sent from x 1 to x 0 (X 51 )
x 0 smoothed by y 1 i.e As0 ∼ Pr (x 0 |y 1
)
24
Quantifying Uncertainty
Ensemble Filter and Smoother
Fixed Interval & Fixed Lag
Fixed Interval
Y1
x0
x1
x2
x3
Yn−1
Yn
xn−1
xn
P(x 0 |y 1 · · · y n )
P(x 1 |y 1 · · · Yn )
..
.
⎫
⎪
⎬
Smoother
⎪
⎭
P(x n |y 1 · · · y n ) ← Filter
25
Quantifying Uncertainty
Ensemble Filter and Smoother
Fixed Interval & Fixed Lag
Fixed Lag
Y1
x0
x1
x2
x3
Y1
x0
x1
x2
x3
Yn−1
Yn
xn−1
xn
Yn−1
Yn
xn−1
xn
∼ P(x |y . . . y ),
P(x 0 |y 1 . . . y n ) =
0 1
L
P(x i |Y0 . . . y i+L )
Smothed up to a “window”
26
Quantifying Uncertainty
(L < n)
Ensemble Filter and Smoother
Fixed Interval: The Dumb Way
Fixed Interval: The Dumb Way
27
Quantifying Uncertainty
Ensemble Filter and Smoother
Backward Recursion
Key Assumption: Jointly Gaussian Distributions.
Aks
=
N
N
a
Ak
X 5j
j=k +1
Ck =
N
N
X 5j = X 5k +1 Ck +1
j=k +1
28
Quantifying Uncertainty
Ensemble Filter and Smoother
Interval:
New
Normal
FixedFixed
Interval:
TheThe
New
Normal
29
Quantifying Uncertainty
and T=[0, Ensemble
1, 2,...,
Filter and99].
SmootherLorenz and Emanuel (1998)
associate the dimensionless time step 0.01 with a real time
ofInterval
1.2 h,
implying
that our measurement interval
Fixed
Lorenz
Fixed
Interval
onOn
Lorenz
corresponds to 6 h and our fixed-interval smoothing
window is 120 h. Figure 1 compares the estimation error
(over all states) obtained from an ensemble Kalman filter
algorithms s
tional time
smoothing p
(N/n) is ty
efficient very
expected as
Fig. 1 Comparison of V1 and FBF with ensemble Kalman filter
(EnKF). The error is computed between the analysis ensemble mean
and truth. Observations are spaced
every five model steps, the
30
interval length is 100 and smoothed analyses are sought at every
model time step. As expected, V1 and FBF give identical estimates;
they only differ in computational requirements
Quantifying Uncertainty
Fig. 2 Comp
smoothing est
parameters are
give identical
Short fixed la
while longer
smoother (com
Ensemble Filter and Smoother
CostsCosts
of130Inference,
ToyToy
Problem
of Inference,
Problem
35
Computational Time (s)
30
25
20
15
10
5
0
90
50
100
FB
Fig. 4 Comp
throughout fix
interval smoot
option (which
lag option is sh
faster than V1
31
Quantifying Uncertainty
that
V1 will require more computauel (1998) algorithms suggests
Ensemble Filter and
Smoother
tional
time
when
R>1+2N/sn.
In practical intervala real time
Fixed Lag
smoothing problems the ratio of ensemble to state size
nt Fixed
interval Lag
smoothing (N/n) is typically very small so FBF becomes more
mation error efficient very rapidly. The superior performance of FBF is
alman filter expected as V1 computation time grows quadratically with
Fixed Lag Smoother
Fixed Lag Smoother
Kalman filter
semble mean
el steps, the
ght at every
cal estimates;
Fig. 2 Comparison of V1-lag and FIFO-lag fixed-lag ensemble
smoothing estimates for different lag window lengths. The other
parameters are identical to those used in Fig. 1. V1-lag and FIFO-lag
give identical estimates although there
can be numerical differences.
32
Short fixed lags give results closer to the ensemble Kalman filter,
while longer fixed lags give results closer to the fixed interval
smoother (compare to Fig. 1)
Quantifying Uncertainty
Ensemble Filter and Smoother
FixedFixed
Lag:Lag:
The The
Dumb
Way
Dumb
Way
35
Computational Time (s)
30
25
20
15
10
5
0
90
50
100
FBF
1
2
3
4
5
6 7 8
V1-lag--->
9 10 11 12 13
Fig. 4 Computational times required to estimate model states
33 500, and 900 for FBF (fixedthroughout fixed intervals of 100,
interval smoothing) and V1-lag (fixed-lag smoothing). The FBF
option (which does not depend on lag value) is shown at far left. V1Quantifying Uncertainty
lag option is shown for a range of lags
from one through 13. FBF is
Ensemble Filter and Smoother
Fixed Lag is FIFO
Ask = Aka
kN
+w
X 5j
j=k +1
= Aak Ck
Ck = X 5k−1 Ck −1 X 5k +w
34
Quantifying Uncertainty
han V1-lag,
ween FIFOe expensive
dent of the
lengths are
ur conservahe threshold
shows.
othing as an
s computan the two is
ive than V1
mparison of
g is more
FBF is faster
h and it is
or an intermputational
n=100 and
Computational Time (s)
Ensemble Filter and Smoother
nly a modest observed time,
i.e. s<1, these bounds can worsen, but not
g, with the
in Lag:
most practical
situations
the ratio N/sn can still be
Fixed
The
New
Normal
FixedatLag:
Theto New
length
expected
remain Normal
much smaller than 1.
.
12
10
8
6
4
2
0
900
900
500
500
100
FBF
100
1
2
3
4
5 6 7 8
FIFO-Lag
9 10 11 12 13
Fig. 5 Computational times required to estimate model states
throughout fixed intervals of 100, 500, and 900 for FBF (fixedinterval smoothing) and FIFO-lag (fixed-lag smoothing). The FBF
option (which does not depend on lag value) is shown at far left. The
FIFO-lag option is shown for a range of lags from one through 13.
FBF is faster than FIFO-lag by 35
a fixed factor. FIFO-lag computational time is nearly independent of lag (small fluctuations are
related to random differences in time required to perform singular
Quantifying Uncertainty
value decompositions at different lags)
Ensemble Filter and Smoother
We need to fix the multimedia!
Watch FLKSO.wmv!
Reading: Ravela and McLaughlin, Fast Ensemble Smoothing, Ocean
Dynamics, 2007
Schneider 2001: Analysis of incomplete climate data: Estimation of
mean values and covariance matrices and imputation of missing
values, Journal of Climate
36
Quantifying Uncertainty
Ensemble Filter and Smoother
Where does ensemble come from?
singular vectors
•
• ••
•
P1f
•
••
•
P0f
Low dimensional
Subspace
Span {u (0) . . . u (N) }
(0)
u 0 → L0 ≈
∂M
∂x
(0)
→ u1
x=x0
37
Quantifying Uncertainty
Ensemble Filter and Smoother
Things get tough... the Tough linearize
Thus
x 1 = M(x 0 )
= M(x̄ 0 + x̃ 0 )
∂M = M(x̄ 0 ) +
x̃
∂x X =x̄ 0
0
x̃1 = Lx̃0
(k )
(k )
u 1 = Lu 0
38
Quantifying Uncertainty
Ensemble Filter and Smoother
Eigenvalue Problem
Now,let C1 be a metric on vector u 1 and let C0 be a metric on u 0
λ=
< Lu 0 , C1 Lu 0 >
< u 0 , L# C1 Lu 0 >
=
< u 0 , C0 u 0 >
< u 0 , C0 u 0 >
Maximize ratio for the k th perturbation: λk :
(k )
(k )
⇒ L# C1 Lu 0 = λk C0 u 0
Which is a generalized eigenvalue problem. Note that when C1 = I,
(k )
and C0 = P0f then u 1 are leading directions of P1f
39
Quantifying Uncertainty
Ensemble Filter and Smoother
SV aproach
Notes
� Adjoint & TLM not easy to calculate but robust.
I
I
�
L may be really large too! How can we reduce L?
I
�
Sensitivity to norm.
40
Quantifying Uncertainty
Ensemble Filter and Smoother
Breeding
•
••
•
•
• ••
•
Initial
Perturbation
Align with leading
directions of error growth
(Lyapunov vectors)
Q R
| i+1{z i+1}
Q R decomposition
Q0 ≡ I
= |{z}
L Qi
TLM
Q0 → Qi · · ·
Qk
|{z}
forgetsQ0
41
Quantifying Uncertainty
Ensemble Filter and Smoother
It’s easy to breed
1. Generate “random” initial perturbation
2. Let it grow; renormalize. (i.e propagate it)
3. Repeat
⇒ Breeding vectors
How many bred vectors ?
⇒ Size of L?
42
Quantifying Uncertainty
Ensemble Filter and Smoother
Ways to simplify Models for Uncertainty Propagation
1. Spectral Truncation: Find a few leading directions of Covariance
or Model and propagate them. Breed Vectors. Calculate a
reduced local linear model from ensemble.
2. Localization: Localize filtering and smoothing, use
scale-recursive decomposition.
3. Model Reduction: Reduce order of linearized model, construct a
reduced model from snapshots.
4. Sample Input-Output pairs to create a simple auxiliary model.
43
Quantifying Uncertainty
Ensemble Filter and Smoother
Model Reduction
Model Reduction
Structure
Krylov
SVD
Nonlinear
POD
Empirical Grammian
Linear
Balanced Truncation
Hankel approx.
SVD Krylov
44
Quantifying Uncertainty
Lanczos
Arnoldi etc.
Ensemble Filter and Smoother
Model Bypass – Non-Intrusive Approaches
X0
M
X1
−1
C10 C00
Response Surface
Modeling, Polyno­
mial Chaos.
45
Quantifying Uncertainty
Ensemble Filter and Smoother
Extra-Special on Covariance Representations
If we have a large covariance matrix C nonetheless representable by
computer and if we know it is a block-circulant matrix, then the
Fourier Transform can be used to diagonalize it:
D = UCU T
(10)
For the unitary transform U, and D is diagonal. So, subsequent
processing with covariance is simplified, provided the model and
state can be also expressed in fourier domain.
Uδxn+1
=
δξn+1
T −1
δxn Cn δxn
=
=
∂M T
U Uxn
∂x
LF δξn
δξ T Dn−1 δξ
U
Spectral truncation to a few wave numbers in U also leads to a
reduced order model. Incidentally, similar process for wavelet
46
decomposition.
DO MATLAB EXAMPLE
Quantifying Uncertainty
(11)
(12)
(13)
Ensemble Filter and Smoother
Iterative calculation
If a covariance C has eigen vectors U and eigenvalues λ, i.e.
CU = UΛ, then we may recursively calucate the leading modes in U
because:
N
�
X
C=
u k λkk u Tk
(14)
k =1
Where U = [u 1 . . . u N ] and Λ = diag(λ11 , . . . , λNN ), in decreasing
order. Let C11 = C, and iteratively calculate:
for k = 1 . . . N
{u k , λkk } = LeadingEig(Ckk )
Ck +1k +1 = Ckk − u k λkk u Tk
We need a procedure to calculate the leading Eigenvector and
Eigenvalue.
47
Quantifying Uncertainty
(15)
(16)
(17)
(18)
Ensemble Filter and Smoother
Basic approach: Power Iteration
WARNING: There are many advanced methods for calculating eigen
vectors and eigen values iteratively and one should use them (e.g.
from ARPACK). Here, we provide an intuition for the process.
To calculate the leading vector of C, let us consider a vector in the
basis z, which we may expand as:
z=
N
X
�
ck u k
(19)
k =1
Now, we can write for the nth power of C:
n
C z
=
N
X
�
ck C n u k
(20)
ck λnkk u k
(21)
k =1
=
N
X
�
k =1
HOW?
48
Quantifying Uncertainty
Ensemble Filter and Smoother
Power Iteration Continued
Cnz
=
c1 λn11 (u 1 +
N
�
X
ck λnkk
u )
c1 λn11 k
(22)
k =2
Defining z n = C n z, we note that
zn
→ u1
||z n ||
n→∞⇒
Algorithm PowerIteration(C):
Initialize z; z ←
z
||z||
Iterate: t ← Cz, z ←
t
||t||
49
Quantifying Uncertainty
(23)
Ensemble Filter and Smoother
LeadingEig(C)
u = PowerIteration(C)
λ = u T Cu
return(u, λ)
end
50
Quantifying Uncertainty
(24)
(25)
(26)
Ensemble Filter and Smoother
But the Covariance is too LARGE!
�
I
I
�
The preceding discussion is all fine, but often the dimensionality
is such that we have a really large covariance that cannot be
represented. Fortunately, many physical problems have only a
few modes of interest which we represent through data, e.g. an
ensemble.
So we begin with a skinny matrix X , and assume the covariance
is C = XX T . We would like a representation without explicitly
calculating C and exploiting the rank-deficiency due to a skinny
X.
51
Quantifying Uncertainty
Ensemble Filter and Smoother
Alternate form
�
I
�
I
�
I
�
I
�
I
Let X = USV T be the singular value decomposition and here
Sii ≥ Si+1i+1 . Then C = UΛU T where Λ = S 2
We will calculate only a few top left and right singular vectors and
singular values iteratively for a reduced order representation
Cd = Ud Λd UdT .
Note that because X is skinny, i.e. it is of size n × N with
N << n. We may further only pick d modes, d ≤ N.
We would like a representation of Cd without explicitly calculating
it.
Notice that D = X T X is a small matrix when X is skinny.
52
Quantifying Uncertainty
Ensemble Filter and Smoother
Alternate form
�
I
�
I
Let X = USV T be the singular value decomposition and here
Sii ≥ Si+1i+1 . Then D = X T X = V ΛV T where Λ = S 2 , a small
matrix.
We calculate the eigen vectors and eigen values of D recursively.
Let D1 = D; and for k = 1 . . . d
vk
λkk
=
=
=
PowerIteration(Dk )
v Tk Dk v k
Dk − v k λkk v Tk
Dk +1
√
Noting that Sd = Λd , we obtain Ud as a skinny nxd matrix:
Ud = XVd Sd−1
Store Ud and Λd and use them to calculate the norm in an
application. DEMO IN MATLAB 53
Quantifying Uncertainty
(27)
(28)
(29)
(30)
Ensemble Filter and Smoother
Applicable to Processes
∂θ
(x, t) = F θ(x, t) → System
∂t
∂θ
R(θ) =
− F θ → Residual
∂t
θ = uη(t) → KLT (POD or Krylov)
u T R = 0 → Galerkin Projection
∂η
= u T Fuη → ROM
∂t
54
Quantifying Uncertainty
MIT OpenCourseWare
http://ocw.mit.edu
12.S990 Quantifying Uncertainty
Fall 2012
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