Multimodality in the Kalman Filter and Ensemble Kalman Filter - PowerPoint PPT Presentation

Multimodality in the Kalman Filter and Ensemble Kalman Filter Maxime Conjard, Henning Omre Department of Mathematical Sciences NTNU 28/5/2018 Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Kalman Model . . . d 1 d 2 d 3 d t . . . x T +1 x 1 x 2 x 3 x t Process model’s assumptions: 1 Gaussian initial distribution f ( x 1 ) 2 Single site dependence and conditional independence 3 Gauss-linear forward and likelihood model: f ( x t +1 | x t ) = ϕ p ( x t +1 , Bx t , Σ x | x ) f ( d t | x t ) = ϕ p ( d t , Hx t , Σ d | x ) [Kalman(1960)],[Myrseth and Omre(2010)] Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Kalman Model Properties Properties: 1 Analytically tractable, conjugate prior 2 Models linear unimodal processes Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Selection Gaussian distribution Let A ⊂ R q ,and � x 0 � �� x 0 � � µ x 0 � � Σ 11 �� Σ 12 ∼ ϕ p + q ; µ = , Σ = ν ν µ ν Σ 21 Σ 22 then x 0 , A = [ x 0 | ν ∈ A ] is Selection Gauss. Flexibility Bimodality 1 Skewness 2 Multimodality 3 Conjugate prior to a Gauss-linear likelihood and forward model [Azzalini and Valle(1996)],[Rimstad and Omre(2014)] Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Selection Gaussian Kalman Model Model’s assumptions: 1 Selection Gaussian initial distribution f ( x 1 ) 2 Single site response and conditional independence 3 Gauss-linear forward and likelihood model: f ( x t +1 | x t ) = ϕ p ( x t +1 ; Bx t , Σ x | x ) f ( d t | x t ) = ϕ p ( d t ; Hx t , Σ d | x ) [Naveau et al.(2005)] Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Selection Gaussian Kalman Model Properties Marginal smoothing distribution 1 Analytically tractable 2 Models multimodality 3 Easy to implement Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Implementation � x 1 � 1 We start with that is Gaussian ν   x 1  that is still Gaussian 2 We increment (update) to ν  d 1   x 2 x 1 3 We increment (forward) to     ν   d 1 4 etc . . . Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Implementation 1 Access to Kalman filtering x t | d 1 , ..., d t , smoothing x s | d 1 , ..., d t , s ≤ t and inversion x 1 | d 1 , ..., d T . 2 Fast computation 3 Conserve a Gaussian structure Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Example: Backtracking the 2D Heat equation The heat equation: ∂ T ∂ t − ∇ 2 T =0 ∇ T . n =0 Modelled using finite differences on [0 , 1] × [0 , 1], it gives the following Gauss-linear forward model: f ( T t +1 | T t ) = ϕ p ( T t +1 , BT t , Σ T | T ) (1) Data is collected at 5 different locations using the following Gauss-linear likelihood model: f ( d t | T t ) = ϕ p ( d t , HT t , Σ d | T ) (2) Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Example: Backtracking the 2D Heat equation Initial Heat map Facts 1 Discontinuous initial conditions 2 5 data collection points Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Example: Backtracking the 2D Heat equation Data collected Vs True process Parameters 1 dt = 1 s 2 Σ d | x = 0 . 01 I . Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Initial model: A reflection of our a priori knowledge Scenario 1: Sel-Gauss initial Scenario 2:Gaussian initial model model Properties Properties 1 Two lobes. 1 E ( x 1 ) = 20. 2 Var ( x 1 ) = 100. Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Initial model: A reflection of our a priori knowledge Realizations from the initial distribution: Sel-Gauss initial distribution Gaussian initial distribution Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Exhibit [ x 1 , i | d 1 , ..., d T ] at 2 different locations Initial Heat map Facts 1 Compare the marginal distribution at two different point 2 One inside, one outside Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Exhibit [ x 1 , i | d 1 , ..., d T ] at 2 different locations Sel-Gauss initial model Gaussian initial model Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Global behavior We define LR ( x ) as: LR i ( x ) = P ( x i > 28 , 75) ∀ i ∈ [1 , p ] LR ( x 1 | d 1 , ..., d t ) for different values of t Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Algorithm:EnKF for the Sel-Gauss (EnKF(SG)) Initiate n e = no. of ensemble members � � x u ( i ) 0 , Σ u 0 , i = 1 , ..., n e iid. f ( x u 0 , ν u 0 ) = N ( µ u 0 ) ν u ( i ) 0 d ( i ) 0 = Hx u ( i ) 0 , i = 1 , ..., n e with η 0 ∼ N (0 , Σ d | x + η i 0 ) 0 Iterate t = 0 , ..., T Estimate Σ x ,ν, d from { ( x u ( i ) , ν u ( i ) , d i t ) , i = 1 , ..., n e } t t � � � � x c ( i ) x u ( i ) + Γ x ,ν, d Σ − 1 d ( d t − d i t t = t ), i = 1 , ..., n e ν c ( i ) ν u ( i ) t t � � � � x u ( i ) g ( x c ( i ) � � ) δ t , i = 1 , ..., n e with δ t ∼ N (0 , Σ x | x t t +1 = + ) ν u ( i ) ν c ( i ) t 0 t t +1 d ( i ) t +1 = Hx u ( i ) t +1 , i = 1 , ..., n e with η t +1 ∼ N (0 , Σ d | x t +1 + η i t +1 ) Estimate µ u T +1 , Σ u T +1 and assess f ( x T +1 | d 0 , ..., d T , ν ∈ A ) Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Algorithm:EnKF for the Sel-Gauss (EnKF(SG)) 1 Non gaussian output: Ensemble of x , ν rather than x | ν ∈ A 2 Forward step made easy by : g ( x t | ν ∈ A ) = g ( x t ) | ν ∈ A Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Test on a linear forward model: Previous example Consider now: � � � � � x u ( i ) x c ( i ) � B � δ t � 0 t t +1 = + i = 1 , ..., n e ν u ( i ) ν c ( i ) 0 I 0 t +1 t We ”show” that the EnKF(SG) converges numerically to the Selection Gauss Kalman Filter as n e → ∞ when the forward model is linear. Expected Value: Norm of Covariance matrix: Norm of the difference for x 2 | d 1 , d 2 the difference for x 2 | d 1 , d 2 Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

Ongoing work: Use EnKF for parameter estimation 1 Idea: Put a Sel-Gauss prior on the parameter, one lobe per possible value for the parameter (diffusivity coefficient, but also porosity). 2 Use the EnKF to estimate the parameters. Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter

A. Azzalini and A. Dalla Valle. The multivariate skew-normal distribution. Biometrika , 83(4):715–726, 1996. E Kalman. A new approach to linear filtering and prediction problems. Transactions of the ASME–Journal of Basic Engineering , 82(Series D):35–45, 1960. I. Myrseth and H. Omre. The Ensemble Kalman Filter and Related Filters , pages 217–246. John Wiley and Sons, Ltd, 2010. P. Naveau, M. Genton, and X. Shen. A skewed kalman filter. Journal of Multivariate Analysis , 94(2):382 – 400, 2005. K. Rimstad and H. Omre. Skew-gaussian random fields. Spatial Statistics , 10:43 – 62, 2014. Maxime Conjard, Henning Omre Multimodality in the Kalman Filter and Ensemble Kalman Filter