Organization. I MCMC discussion. I project talks. I Lecture.

Transcription

1 Organization I MCMC discussion I project talks. I Lecture.

2 Content I Uncertainty Propagation Overview I Forward-Backward with an Ensemble I Model Reduction (Intro)

3 Uncertainty Propagation in Causal Systems X f 0 X f 1 X f 2 X f n M M(X t ; t ) X t+1 = M(X t ; t )+! t M :- Physical or Statistical Model.

4 Inference in Bayesian Networks T L P R F Q Hierarchical Bayes Found in Hierarchical Bayes, Grahical Models.

5 Belief Propagation in Undirected Graphs Grid nodes Interactions? Graphical Models

6 Inference Problems 1. Updating States 2. Estimating Parameters 3. Learning Structure Propagating Uncertainty, a first step.

7 Propagating Uncertainty Monte-Carlo Model Reduction & Interpolation I Snapshots & POD I Krylov Subspace Response Surface Models I Deterministic Equivalent Modeling Method I Stochastic Response Surface Methodology Polynomial Chaos Expansions I Generalized Polynomial Chaos

8 From Class 1 & 2 J(x 0 ):= 1 2 (x 0 x b ) T C00 1 (x 0 x b ) + mx 1 2 (y i Hx i ) T R 1 (y i Hx i )+ T i [x i M(x i 1 ; )] i=1

9 Forward Backward Forward (from ) x i = M(x i 1 ; ) 0 < i < m Backward (from x): m = H T R 1 (y m Hx m ) i i+1 + H T R 1 (y i Hx i ) ˆx 0 = x b + C 0 1

10 Uncertainty? Via Linearization Forward (you ll need this in the end) C i 1 C i 1i i 1 0 < i apple m What about backward? Convenient via information form: Î mm = H T R 1 H Î ii i Ĉ 00 = i + H T R 1 H apple C Dimensionality, Linearization challenges. 1

11 Monte-Carlo Isotropic Initial Perturbation Leading Lyapunov vectors time

12 Filter Updating NOTE THAT A f =[x f 1...x f s ] ) All at time T à f =[x f 1 x f...x f s x f ] P f = 1 s 1Ãf à ft ( Uncertainty So, propagate uncertainty via samples integrated forward.

13 No Linearization Y = h(x)+, N(0, R) Z =[Y + 1,...,Y + s ] R 1 s 1 Z Z T Perturbed Observations Also, let! f = h(a f )=[h(x f 1)...h(x f s )]! f defined similarly

14 Uncorrelated Noise Notice (! f + Z)(! ft + Z T )=(! f! ft + Z Z T ) When observation noise is uncorrelated with state an assumption Let X a be the estimate, analysis, posterior rv. A a and à a similarly, defined.

15 Easy Formulation A a = A f + Ã f! ft h! f! ft + Z Z T i 1 h Z! f i Identical to KF/EKF in linear/linearized case ) No linearization of the model ) No explicit uncertainty (covariance) propagation h! f! ft + Z Z i T 1 = [! f + Z][! ft + Z T 1 ] =(C C T ) 1

16 Solution Let C =[u s v T ] [C C T ] 1 = u s 2 u T =(u s 1 )(u s 1 ) T q = D qd T = D

17 Fast Calculation A a = A f + Ã f! ft [u s 2 u T ][Z! f ] (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 memory A a =A f + Ã f =A f (I s + ) =A f T (ENKF) A weakly nonlinear transformation (T T (A))

18 Time Dependent Example Lorenz ẋ i = x i 2 x i 1 + x i 1 x i+1 x i + u i = x i 1 [x i+1 x i 2 ] {z } Advective x i {z} Dissipative + u {z} Forcing Filter

19 Back in Time, How? Recall: A a = A f T Also, written as A a = A f All at same time t

20 Plug and Play So, x 0 x 1 y 1 A a 0 = Af 0 I s A a 1 = Af 1 1 A s 1 = A a 1I s A s 0 = Aa 0 = A a 0 1 No measurement Filter No future measurement + Ãa 0!fT 1 [u 1s 2 1 ut 1 ][Z 1! f 1 ]

21 Send me a message

22 Fixed Interval & Fixed Lag Fixed Interval Y 1 Y n 1 Y n x 0 x 1 x 2 x 3 x n 1 x n P(X 0 Y 1 Y n ) P(X 1 Y 1 Y n ). P(X n Y 1 Y n ) Smoother Filter

23 Fixed Lag Fixed Lag Y 1 Y n 1 Y n x 0 x 1 x 2 x 3 x n 1 x n Y 1 Y n 1 Y n x 0 x 1 x 2 x 3 x n 1 x n P(X 0 Y 1...Y n ) = P(X 0 Y 1...Y L ), (L < n) P(X i Y 0...Y i+l ) Smothed up to a window

24 Fixed Interval: The Dumb Way

25 Backward Recursion

26 Fixed Interval: The New Normal

27 Fixed Interval On Lorenz

28 Costs of Inference, Toy Problem

29 Fixed Lag Fixed Lag Smoother

30 Fixed Lag: The Dumb Way Computational Time (s) FBF V1-lag--->

31 Fixed Lag is FIFO A s k = A a k k+w Y j=k+1 X 5 j = A a k C k C k = X 5 1 k C k 1X 5 k+w

32 Fixed Lag: The New Normal Computational Time (s) FBF FIFO-Lag

33 Where does ensemble come from? singular vectors P f 0 P f 1 Low dimensional Subspace Span {S 0...S n } S (0) 0! L S (1) x=x0

34 Things get tough...the Tough linearize X 1 = M(X 0 ) = M( X 0 + X 0 ) = M( X X = X 0 X 0 X 1 = L X 0 S (k) 1 = LS (k) 0

35 Eigenvalue Problem Now,let C 1 be a metric on S 1 let C 0 be a metric on S 0 Maximize k: k = < LS 0, C 1 LS 0 > < S 0, C 0 S 0 > = < S 0, L # C 1 LS 0 > < S 0, C 0 S 0 > )L # C 1 LS (k) 0 = k C 0 S (k) 0 Ageneralizedeigenvalueproblem When C 1 = I &C 0 = P f 0! S 1 loading directions of P f 1

36 SV approach Notes I Adjoint & TLM not easy to calculate but one of the most robust methods available. I L may be really large too! I How can we reduce L? I Sensitivity to norm.

37 Breeding

38 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?

39 Two ways to simplify models 1. Truncate the Model: Linearize and reduce order, construct from snapshots. 2. Sample Input-Output pairs, create an auxiliary (simple) model. In Class 6

40 Truncation/Linearization

41 Another