Data assimilation with and without a model

Transcription

1 Data assimilation with and without a model Tim Sauer George Mason University Parameter estimation and UQ U. Pittsburgh Mar. 5, 2017 Partially supported by NSF

2 Most of this work is due to: Tyrus Berry, Postdoc, GMU Franz Hamilton, Postdoc, NC State

3 OUTLINE OF TALK 1. EnKF and applications Parameters Network edge detection 2. Problem: Don t know noise covariances Adaptive filtering 3. Problem: Model error Multimodel DA 4. Problem: Model unknown Kalman-Takens filter

4 DATA ASSIMILATION x k = f (x k 1 ) + η k η k N(0, Q) y k = h(x k ) + ν k ν k N(0, R) Main Problem. Given the model above plus observations y k, Filtering: Estimate the current state p(x k y 1,..., y k ) Forecasting: Estimate a future state p(x k+l y 1,..., y k ) Smoothing: Estimate a past state p(x k l y 1,..., y k ) Parameter estimation We ll apply the ensemble Kalman filter (EnKF) to attempt to achieve these goals.

5 DATA ASSIMILATION x k = f (x k 1 ) + η k η k N(0, Q) y k = h(x k ) + ν k ν k N(0, R) Possible obstructions. The observations y k mix system noise η k with obs noise ν k. Model error Q and R may be unknown Known model with unknown parameters Wrong model, even with best fit parameters Have model for some, not all of the variables

6 EXAMPLE 1. LORENZ 96 dx i dt = x i 2 x i 1 + x i 1 x i+1 x i + F Time

7 EXAMPLE 2. MEA RECORDINGS

8 TWO STEPS TO FIND p(x k y 1,..., y k ) Assume we have p(x k 1 y 1,..., y k 1 ) Forecast Step: Find p(x k y 1,..., y k 1 ) Assimilation Step: Perform a Bayesian update, p(x k y 1,..., y k ) p(x k y 1,..., y k 1 )p(y k x k, y 1,..., y k 1 ) Posterior Prior Likelihood

9 BEST POSSIBLE SCENARIO x k = f (x k 1 ) + η k η k N(0, Q) y k = h(x k ) + ν k ν k N(0, R) f and h are linear, all parameters known. x k = F k 1 x k 1 + η k η k N(0, Q) y k = H k x k + ν k ν k N(0, R)

10 KALMAN FILTER Assume linear dynamics/obs and additive Gaussian noise x k = F k 1 x k 1 + η k η k N (0, Q) y k = H k x k + ν k ν k N (0, R) observability condition for linear systems: H k H k l = H k+1 F k. H k+l+1 F k+l F k Must be full rank for some l KF guaranteed to work

11 KALMAN FILTER Assume linear dynamics/obs and additive Gaussian noise x k = F k 1 x k 1 + η k η k N (0, Q) y k = H k x k + ν k ν k N (0, R) Assume current estimate is Gaussian: p(x k 1 y 1,..., y k 1 ) = N (x a k 1, Pa k 1 ) Forecast: Linear combinations of Gaussians Prior: p(x k y 1,..., y k 1 ) = N (x f k, Pf k ) x f k = F k 1 x a k 1 P f k = F k 1 P k 1 F k 1 + Q Likelihood: p(y k x k, y 1,..., y k 1 ) = N (y f k, Py k ) y f k = H k x f k P y k = H kp f kh k + R

12 KALMAN FILTER Forecast: Linear combinations of Gaussians Prior: p(xk y 1,..., y k 1 ) = N (x f k, Pf k ) x f k = F k 1 x a k 1 P f k = F k 1 P k 1 F k 1 + Q Likelihood: p(y k x k, y 1,..., y k 1 ) = N (y f k, Py k ) y f k = H k x f k P y k = H kpkh f k + R Assimilation: Product of Gaussians (complete the square) p(x k y 1,..., y k ) = N (xk f, Pf k ) N (y k f, Py k ) = N (x k a, Pa k ) Define the Kalman gain: K k = Pk f HT k (Py k ) 1 x a k = xk f + K k(y k yk f ) P a k = (I K kh k )P f k

13 KALMAN FILTER SUMMARY forecast x f k = F k 1 x a k 1 covariance update P f k = F k 1 P a k 1 F T k 1 + Q k 1 P y k = H k P f k HT k + R k 1 Kalman gain K k = P f k HT k (Py k ) 1 ɛ k = y k y f k = y k H k x f k assimilation step x a k = x f k + K kɛ k P a k = (I K k H k )P f k

14 WHAT ABOUT NONLINEAR SYSTEMS? Consider a system of the form: x k+1 = f (x k ) + η k+1 η k+1 N (0, Q) y k+1 = h(x k+1 ) + ν k+1 ν k+1 N (0, R) More complicated observability condition (Lie derivatives) Extended Kalman Filter (EKF): Linearize F k = Df (ˆx a k ) and H k = Dh(ˆx f k ) Problem: State estimate ˆx a k may not be well localized Solution: Ensemble Kalman Filter (EnKF)

15 ENSEMBLE KALMAN FILTER (ENKF) + P xx P xx x t + x t i F x t ~ x i t F(x t i ) Generate an ensemble with the current statistics (use matrix square root): x i t = sigma points on semimajor axes xt f = 1 F(x i t ) 2n P f xx = 1 2n 1 (F(x i t ) x f t )(F (x i t ) x f t ) T + Q

16 ENSEMBLE KALMAN FILTER (ENKF) + P xx P xx x t + x t i F x t ~ x i t F(x t i ) Calculate yt i = H(F (xt i )). Set yt f = 1 2n i y t i. P yy = (2n 1) 1 (y i t y f t )(y i t y f t ) T + R P xy = (2n 1) 1 (F(x i t ) x f t )(y i t y f t ) T K = P xyp 1 yy x a t+1 = x f t + K (y t y f t ) and P a xx = P f xx KP yyk T

17 PARAMETER ESTIMATION When the model has parameters p, x k+1 = f (x k, p) + η k+1 Can augment the state x k = [x k, p k ] Introduce trivial dynamics for p x k+1 = f (x k, p k ) + η k+1 p k+1 = p k + η p k+1 Need to tune the covariance of η p k+1

18 EXAMPLE OF PARAMETER ESTIMATION Consider the Hodgkin-Huxley neuron model, expanded to a network of n equations V i = g Na m 3 h(v i E Na ) g K n 4 (V i E K ) g L (V i E L ) n +I + Γ HH (V j )V j j i ṁ i = a m (V i )(1 m i ) b m (V i )m i ḣ i = a h (V i )(1 h i ) b h (V i )h i ṅ i = a n (V i )(1 n i ) b n (V i )n i Γ HH (V j ) = β ij /(1 + e 10(V j +40) ) Only observe the voltages V i, recover the hidden variables and the connection parameters β

19 EXAMPLE OF PARAMETER ESTIMATION Can even turn connections on and off (grey dashes) Variance estimate statistical test (black dashes)

20 LINK DETECTION FROM NETWORKS OF MODEL NEURONS Network of Hindmarsh-Rose neurons, modeled by Hindmarsh-Rose Network of Hodgkin-Huxley neurons, modeled by Hindmarsh-Rose Sensitivity = true positive rate Specificity = true negative rate

21 LINK DETECTION FROM MEA RECORDINGS

22 LINK DETECTION FROM MEA RECORDINGS

23 ENKF: WRONG Q AND R 3.2 Simple example with full observation and diagonal noise covariances Red indicates RMSE of unfiltered observations Black is RMSE of optimal filter (true covariances known) RMSE RMSE Variance used in R k Variance used in Q k

24 ENKF: WRONG Q AND R Standard Kalman Update: Pk f = F k 1 Pk 1 a F k 1 T + Q k 1 RMSE P y k = H k P f k HT k + R k 1 K k = P f k HT k (Py k ) 1 P a k = (I K k H k )P f k Variance used in R k ɛ k = y k y f k = y k H k x f k x a k = x f k + K kɛ k RMSE Variance used in Q k

25 ADAPTIVE FILTER: ESTIMATING Q AND R Innovations contain information about Q and R ɛ k = y k yk f = h(x k ) + ν k h(xk f ) = h(f (x k 1 ) + η k ) h(f (x a k 1 )) + ν k H k F k 1 (x k 1 x a k 1 ) + H kη k + ν k IDEA: Use innovations to produce samples of Q and R : E[ɛ k ɛ T k ] HPf H T + R E[ɛ k+1 ɛ T k ] HFPe H T HFK E[ɛ k ɛ T k ] P e FP a F T + Q In the linear case this is rigorous and was first solved by Mehra in 1970

26 ADAPTIVE FILTER: ESTIMATING Q AND R To find Q and R we estimate H k and F k 1 from the ensemble and invert the equations: E[ɛ k ɛ T k ] HPf H T + R E[ɛ k+1 ɛ T k ] HFPe H T HFK E[ɛ k ɛ T k ] This gives the following empirical estimates of Q k and R k : P e k Pk e = (H k+1 F k ) 1 (ɛ k+1 ɛ T k + H k+1f k K k ɛ k ɛ T k )H T k Qk e = Pk e F k 1Pk 1 a F k 1 T R e k = ɛ k ɛ T k H kp f k HT k is an empirical estimate of the background covariance

27 ADAPTIVE ENKF We combine the estimates of Q and R with a moving average Original Kalman Eqs. P f k = F k 1 P a k 1 F T k 1 + Q k 1 P y k = H k P f k HT k + R k 1 Our Additional Update P e k 1 = F 1 k 1 H 1 k ɛ k ɛ T k 1 H T k 1 + K k 1 ɛ k 1 ɛ T k 1 H T k 1 K k = P f k HT k (Py k ) 1 P a k = (I K k H k )P f k Q e k 1 = P e k 1 F k 2P a k 2 F T k 2 R e k 1 = ɛ k 1 ɛ T k 1 H k 1P f k 1 HT k 1 ɛ k = y k y f k x a k = x f k + K kɛ k Q k = Q k 1 + (Q e k 1 Q k 1)/τ R k = R k 1 + (R e k 1 R k 1)/τ

28 ADAPTIVE FILTER: APPLICATION TO LORENZ-96 We will apply the adaptive EnKF to the 40-dimensional Lorenz96 model integrated over a time step t = 0.05 dx i dt = x i 2 x i 1 + x i 1 x i+1 x i + F We augment the model with Gaussian white noise x k = f (x k 1 ) + η k η k = N (0, Q) y k = h(x k ) + ν k ν k = N (0, R) The Adaptive EnKF uses F = 8 We will consider model error where the true F i = N (8, 16)

29 RECOVERING Q AND R, PERFECT MODEL True Covariance Initial Guess Final Estimate Difference 8 x 10 3 Q RMSE of Q Q k Filter Step k 0.8 R RMSE Filter Steps RMSE (bottom right) for the initial guess covariances (red) the true Q and R (black) and the adaptive filter (blue)

30 COMPENSATING FOR MODEL ERROR The adaptive filter compensates for errors in the forcing F i True Covariance Initial Guess Final Estimate Difference Q R RMSE (Q f ) ii Relative Change Site Number F i Model Error (%) Filter Steps x 10 RMSE (bottom right) for the initial guess covariances (red) the perfect model (black) and the adaptive filter (blue)

31 ESTIMATING UNMODELED DYNAMICS Multiple models approach: Run m subfilters in parallel ẇ = F (w) + η t y... y S = H(w) + ν t where w = x 1... x m c 1... c m d, F = f (x, p 1 )... f (x, p m) , H(w) = h(x 1 )... h(x m ) i,j c i j xi j + d Learn c i j and d from training data S, then can predict unmodeled S.

32 SELECTING THE ASSIMILATION MODEL Assume a generic spiking neuron model (Hindmarsh-Rose) V = y av 3 + bv 2 z + I ẏ = c dv 2 y ż = τ(s(v + 1) z) V neuron potential, y fast scale dynamics, z slow scale dynamics Goal: Use multiple parameterized Hindmarsh-Rose models to reconstruct unmodeled neuronal quantities while assimilating recorded neuron potential J. Hindmarsh and R. Rose, Proc. Roy. Soc., 221, 87 (1984).

33 RECONSTRUCTING UNMODELED GATING VARIABLES Observing Hodgkin-Huxley voltage, reconstruct unmodeled gating variables (assimilation model is Hindmarsh-Rose) A. Hodgkin and A. Huxley, J. Physiology 117, 500 (1952). Actual Observed Predicted

34 RECONSTRUCTING UNMODELED GATING VARIABLES Observing Hodgkin-Huxley voltage, reconstruct unmodeled gating variables (assimilation model is Hindmarsh-Rose) A. Hodgkin and A. Huxley, J. Physiology 117, 500 (1952). Actual Observed Predicted

35 RECONSTRUCTING UNMODELED GATING VARIABLES Observing Hodgkin-Huxley voltage, reconstruct unmodeled gating variables (assimilation model is Hindmarsh-Rose) A. Hodgkin and A. Huxley, J. Physiology 117, 500 (1952). Actual Observed Predicted

36 RECONSTRUCTING UNMODELED IONIC DYNAMICS Observing seizure voltage, reconstruct unmodeled potassium and sodium dynamics (assimilation model is Hindmarsh-Rose) J. Cressman, G. Ullah, J. Ziburkus, S. Schiff, and E. Barreto, Journal of Comp. Neuroscience 26, 159 (2009). Actual Observed Predicted

37 RECONSTRUCTING UNMODELED IONIC DYNAMICS Observing seizure voltage, reconstruct unmodeled potassium and sodium dynamics (assimilation model is Hindmarsh-Rose) J. Cressman, G. Ullah, J. Ziburkus, S. Schiff, and E. Barreto, Journal of Comp. Neuroscience 26, 159 (2009). Actual Observed Predicted

38 RECONSTRUCTING UNMODELED IONIC DYNAMICS Observing seizure voltage, reconstruct unmodeled potassium and sodium dynamics (assimilation model is Hindmarsh-Rose) J. Cressman, G. Ullah, J. Ziburkus, S. Schiff, and E. Barreto, Journal of Comp. Neuroscience 26, 159 (2009). Actual Observed Predicted

39 RECONSTRUCTING EXTRACELLULAR POTASSIUM FROM AN In Vitro NETWORK We want to track extracellular potassium dynamics in a network but measurements are difficult and spatially limited Extracellular potassium is an unmodeled variable

40 RECONSTRUCTING EXTRACELLULAR POTASSIUM FROM AN In Vitro NETWORK The local extracellular potassium in an MEA network can be reconstructed and predicted using our approach (assimilation model is Hindmarsh-Rose) Actual Observed Predicted

41 RECONSTRUCTING EXTRACELLULAR POTASSIUM FROM AN In Vitro NETWORK The local extracellular potassium in an MEA network can be reconstructed and predicted using our approach (assimilation model is Hindmarsh-Rose) Actual Observed Predicted

42 RECONSTRUCTING EXTRACELLULAR POTASSIUM FROM AN In Vitro NETWORK The local extracellular potassium in an MEA network can be reconstructed and predicted using our approach (assimilation model is Hindmarsh-Rose) Actual Observed Predicted

43 KALMAN-TAKENS FILTER: THROWING OUT THE MODEL... Starting with historical observations {y 0,..., y n } Form Takens delay-embedding state vectors x i = (y i, y i 1,..., y i d ) Build an EnKF: Apply analog forecast to each ensemble member Use the observation function Hxi = y i Crucial to estimate Q and R

44 NO MODEL: KALMAN-TAKENS FILTER Reasons why this might not work: Nonparametric methods that depend on local reconstructions tend to fail in presence of observation noise Takens Theorem assumes no obs noise Neighbors in reconstruction space are wrong Kalman filters depend on knowledge of model to advance dynamics Filter equations need knowledge of dynamics to advance background

45 ENSEMBLE KALMAN FILTERING (ENKF) At kth step: Form an ensemble of state vectors centered at best guess x + k 1, with spread given by estimated covariance P+ k 1 Model f is applied to this ensemble and observed with function h. Form the prior state estimate x k and the predicted observation y k Construct the resulting covariance matrices P xy k and P y k Update the state and covariance estimates with the observation y k K k = P xy k (Py k ) 1 P + k = P k Pxy k (Py k ) 1 P yx k x + k = x k + K ( k yk y ) k.

46 ENSEMBLE KALMAN FILTERING (ENKF) At kth step: Form an ensemble of state vectors centered at best guess x + k 1, with spread given by estimated covariance P+ k 1 Model f is applied to this ensemble and observed with function h. Form the prior state estimate x k and the predicted observation y k Construct the resulting covariance matrices P xy k and P y k Update the state and covariance estimates with the observation y k K k = P xy k (Py k ) 1 P + k = P k Pxy k (Py k ) 1 P yx k x + k = x k + K ( k yk y ) k.

47 KALMAN-TAKENS FILTER At kth step: Form ensemble of reconstructed states centered at delay-coordinate vector x + k 1, with spread given by estimated covariance P + k 1 Apply nonparametric prediction and observe with function h. Estimate prior state x k and observation y k Construct the resulting covariance matrices P xy k and P y k Update the state and covariance estimates with the observation y k K k = P xy k (Py k ) 1 P + k = P k Pxy k (Py k ) 1 P yx k x + k = x k + K ( k yk y ) k.

48 KALMAN-TAKENS FILTER: LORENZ x x Time Time full model Kalman- Takens

49 KALMAN-TAKENS FILTER Comparison of Kalman-Takens (red) with full model (blue) RMSE Percent Noise

50 KALMAN-TAKENS FILTER: LORENZ 96 x 1 x Time Time full model Kalman- Takens

51 KALMAN-TAKENS FILTER: MODEL ERROR Lorenz 63 Lorenz 96

52 KALMAN-TAKENS FILTER: LORENZ Time Time

53 KALMAN-TAKENS FILTER: FORECAST ERROR El Nino index Training data: Jan Dec Forecasts after RMSE Forecast Steps (months)

54 SUMMARY EnKF is a useful data assimilation technique for neurodynamics and other types of data Parameter estimation Adaptive QR is helpful when Q and R are unknown Difficulties Model error Unmodeled variables No model Multimodel data assimilation Kalman-Takens filter

55 REFERENCES T. Berry, T. Sauer, Adaptive ensemble Kalman filtering of nonlinear systems. Tellus A 65, (2013) F. Hamilton, T. Berry, N. Peixoto, T. Sauer, Real-time tracking of neuronal network structure using data assimilation. Phys. Rev. E 88, (2013) F. Hamilton, J. Cressman, N. Peixoto, T. Sauer, Reconstructing neural dynamics using data assimilation with multiple models. Europhys. Lett. 107, (2014) F. Hamilton, T. Berry, T. Sauer, Ensemble Kalman filtering without a model. Phys. Rev. X 6, (2016). F. Hamilton, T. Berry, T. Sauer, Kalman-Takens filtering in the presence of dynamical noise. To appear, Eur. Phys. J.