Nonlinear Stability of Ensemble Kalman filter

Transcription

1 Noliear Stability of Esemble Kalma filter Xi T Tog Joit work with Adrew J, Majda ad David, Kelly CIMS Wedesday 30 th September, 2015 Xi Tog Stability of EKF 1 / 34

2 Roadmap 1 What are esemble Kalma filters (EKF)? 2 Whe is EKF oliear stable? 3 Is it possible for EKF to diverge? 4 Ca we fix this problem? 5 Ergodicity of EKF. 6 Extesio to esemble square root filters. Xi Tog Stability of EKF 2 / 34

3 Filterig Filter: estimatio of a stochastic process/sequece through oisy observatios. Discrete time versio: Truth: U = Ψ(U 1 ) + ζ, Observatio: Z = HU + ξ. Goal: estimate U based o Z 1,..., Z Easily applicable to cotiuous systems. Lorez 96: ẋ i,t = x i 1,t (x i+1,t x i 2,t ) x i,t + F, i = 1,..., 40. Time discretizatio with itervals: U = x h. Applicatio: weather forecast. Xi Tog Stability of EKF 3 / 34

4 Kalma filter Use Gaussia distributios N (m, C ) Forecast step: N (m 1, C 1 ) N ( ˆm, Ĉ) Assimilatio step: apply the Kalma update rule m = ˆm ĈH T (I + HĈH T ) 1 (H ˆm Z ). C = Ĉ ĈH T (I + HĈH T ) 1 HĈ. Posterior at t = 1 Prior+Obs at t = forecast assimilate Posterior at t = Xi Tog Stability of EKF 4 / 34

5 Esemble Kalma filter (EKF) Use a esemble {V } K k=1 to represet N (m, C ) Forecast step: propagate each member V = Ψ(V 1 ) + ζ Assimilatio step: apply the Kalma update rule V = V ĈH T (I + HĈH T ) 1 (H V Z ξ ). Posterior at t = 1 Prior+Obs at t = V forecast: M.C. V 1 assimilate Posterior at t = V Xi Tog Stability of EKF 5 / 34

6 Applicatio ad theory of EKF Pros: Cos: Successful applicatios i weather forecast ad oil reservoir maagemet. 50 esembles ca forecast 10 6 dimesioal systems. No systematic theoretical uderstadig: why does it work? Literature: focus o oe assimilatio step error aalysis, or K with liear dyamics. Dyamical properties: almost empty kowledge. Xi Tog Stability of EKF 6 / 34

7 Catastrophic filter divergece Strage divergece to machie ifiity. (Majda, Harlim 08; Gottwald, Majda 13). Takes place whe 1 observatio is sparse, e.g. of the 5 dimesios, turbulece is very strog, e.g. F = 16 for Lorez 96. Horrible corruptios for filterig. Possible reaso: error reachig breakig poit of explicit itegrators. Possible solutio: stable implicit itegrators, but ot practical. Xi Tog Stability of EKF 7 / 34

8 Noliear stability of EKF Noliear stability: If U is bouded i probability, will the filter esemble be so as well? K e.g : sup E V 2 <? k=1 Moreover, is it ergodic, i.e. P(V ) π? Reverse: could V grow expoetially fast, or reach machie ifiity i short time? Xi Tog Stability of EKF 8 / 34

9 Stability theory for oliear systems Fulfilled by eergy priciples/ absorbig ball/ Lyapuov fuctio, e.g. ẋ i,t = x i 1,t (x i+1,t x i 2,t ) x i,t + F. defie the eergy E t = i x2 i,t E t = 2E t + 2F x i,t E t + NF 2. By Gröwall s iequality, U = x h, E 1 E(U ) e h E(U 1 ) + hnf 2. Xi Tog Stability of EKF 9 / 34

10 Stability theory for oliear systems Fulfilled by eergy priciples/ absorbig ball/ Lyapuov fuctio, e.g. ẋ i,t = x i 1,t (x i+1,t x i 2,t ) x i,t + F. I geeral: E 1 E(U defie the eergy E t = ) (1 β)e(u 1 ) + k γ i x2 i,t EE(U ) (1 β) E(U 0 ) + β 1 k γ E t = 2E t + 2F x i,t E t + NF 2. By Gröwall s iequality, U = x h, E 1 E(U ) e h E(U 1 ) + hnf 2. Xi Tog Stability of EKF 10 / 34

11 Iheritace of dissipatio Idea: ca the filter esemble iherits the eergy priciple? E 1 U 2 = Ψ(U 1 ) 2 + E( ζ 2 U 1 ) (1 β) U k γ. leads to E 1 V 2 (1 β ) V 1 + k γ? Xi Tog Stability of EKF 11 / 34

12 Esemble Kalma filter (EKF) Forecast step: propagate each member V = Ψ(V 1 ) + ζ Assimilatio step: apply the Kalma update rule V = V ĈH T (I + HĈH T ) 1 (H V Z ξ ). Posterior at t = 1 Prior+Obs at t = V forecast: M.C. V 1 assimilate Posterior at t = V Xi Tog Stability of EKF 12 / 34

13 Iheritace of dissipatio Yes! for the forecast step: V Ucertai for the assimilatio step: V = V = Ψ(V 1 ) + ζ. ĈH T (I + HĈH T ) 1 (H V Z ) = (I + ĈH T H) 1 V + (I + ĈH T H) 1 Ĉ H T Z. V 2 ca be bouded by V 2 oly if H = I. Xi Tog Stability of EKF 13 / 34

14 Observable eergy Algebraic trick V = V ĈH T (I + HĈH T ) 1 (H V Z ), ca be writte as HV = H V HĈH T (I + HĈH T ) 1 (H V Z = (I + HĈH T ) 1 H V ) + HĈH T (I + HĈH T ) 1 Z. HV 2 ca be bouded by H V 2. A eergy priciple based o H V 2 should work. Xi Tog Stability of EKF 14 / 34

15 Observable eergy Algebraic trick V = V ĈH T (I + HĈH T ) 1 (H V Z ), ca be writte as HV = H V HĈH T (I + HĈH T ) 1 (H V Z = (I + HĈH T ) 1 H V ) + HĈH T (I + HĈH T ) 1 Z. HV 2 ca be bouded by H V 2. A eergy priciple based o H V 2 should work. Xi Tog Stability of EKF 14 / 34

16 Observable eergy stability Theorem (T., Majda ad Kelly) If the dyamical system follows a observable eergy priciple: E 1 HU 2 (1 γ) HU k γ. the there are costats M ad D such that E 1 ( HV 2 + M HU 2 ) (1 1 γ)( HV M HU 1 2 ) + D. Xi Tog Stability of EKF 15 / 34

17 Applicatios How to use observable eergy criterio Whe H is full rak, boud i Hu is equivalet to boud i u. For geeral H I E 1 HU 2 (1 γ) HU k γ E 1 U 2 (1 γ) U k γ Useful whe the cotractio is strog: E 1 HU 2 (1 γ) HU k γ, C 1 Hu 2 C u 2 E 1 U 2 (1 γ)c 2 HU Ck γ. Higher order dampig: du t = f(u t )dt ɛu 3 t dt + dw t. Xi Tog Stability of EKF 16 / 34

18 Restrictios For geeral H I E 1 HU 2 (1 γ) HU k γ E 1 U 2 (1 γ) U k γ Sharpess from a couterexample : a cocrete catastrophic filter divergece. First cocrete example with rigorous proofs. Xi Tog Stability of EKF 17 / 34

19 Eergy iflatio from assimilatio step V = (I + ĈH T H) 1 V + (I + ĈH T H) 1 Ĉ H T Z. Geesis of cata. filter divergece (Gottwald & Majda 13 ): Aligmet of the esemble. Sparse observatios. [ ] { Cosider V 1 0 ± x = (m, ±ɛ) ad H = ɛ 2 1 ɛ 2 x + y Xi Tog Stability of EKF 18 / 34

20 Cycle of eergy blow up Use a rotatio-lockig map to secure blowup Ψ( x) = Ψ lock (Ψ rot u) : [ ] [ ] [ x ρ cos θ ρ si θ x Ψ rot = y ρ si θ ρ cos θ y] Ψ rot (x, y) = (r(x), 2ɛr(y + ɛ)/2ɛ + ɛ). Xi Tog Stability of EKF 19 / 34

21 Cocrete catastrophic divergece Theorem (Kelly, Majda ad T.) For ay fixed p < 1,N ad K, there is a dyamical system with eergy priciple, such that if a EKF with 2K esemble size is started from certai locatios, the with probability at least p, the eergy blowup cycle cotiues for at least N steps ε= ε=0.005 V ε=0.02 ε=0.1 ε=0.05 ε= Xi Tog Stability of EKF 20 / 34

22 Ca we fix this? Flaw: both sigal ad observatios stayed bouded, yet the assimilatio step keeps iflatig eergy. Strategy: desig a adaptive mechaism that pulls the esemble back. Covariace iflatio: istead of usig Ĉ, uses a Ĉ = Ĉ + ρi V = (I + ĈH T H) 1 V Stroger iflatio gives better stability. Stroger iflatio sacrifices more accuracy. + (I + ĈH T H) 1 Ĉ H T Z. Xi Tog Stability of EKF 21 / 34

23 Adaptive iflatio Iflatio stregth is adaptive, stroger iflatio whe ustable. Idicators: Error measured by the averaged iovatio Θ = 1 K H V Z K 2 k=1 [ ] Partially observed case: V = ( X, Ŷ), H0 H =. Possible 0 istability from aligmet Ξ = ( X X ) (Ŷ Ŷ ) Possible iflatio: C = Ĉ + Θ (Ξ + 1)I. Xi Tog Stability of EKF 22 / 34

24 Bechmarks Iflatio impairs accuracy, istead uses C = Ĉ + 1 Θ>M 1 or Ξ >M 2 Θ (Ξ + 1)I. M 1 ad M 2 set up thresholds to tur o iflatio. Tured o oly whe the filter is malfuctioig. Malfuctio worse tha bechmarks. Bechmark: use statioary dist. as prior, coditioed o the observatio Z. 1 V U K 2 Bechmark R.M.S.Error. Theorem (T., Majda ad Kelly) EKF with adaptive covariace iflatio stroger tha above is oliearly stable. Xi Tog Stability of EKF 23 / 34

25 Numerical evidece Turbulet Lorez 96 model, F = 16, ẋ i,t = x i 1,t (x i+1,t x i 2,t ) x i,t + F, i = 1,..., 5. A determiistic model with bouded attractor. Catastrophic filter divergece with obs. o x 1,t, oise N(0, 0.01). Adaptive covariace iflatio removes catastrophic filter divergece. Xi Tog Stability of EKF 24 / 34

26 Additioal beefit: tuig Apply with additioal costat iflatio C = Ĉ + ρi + 1 Θ>M 1 or Ξ >M 2 Θ (Ξ + 1)I. Optimal choice of ρ, bechmark RMSE= ρ CI-Div. 3% 8% 18% 18% 28% 42% 57% 75% RMSE Cor Xi Tog Stability of EKF 25 / 34

27 Additioal beefit: itegrator Allows the use of ustable itegrators: CAI CI Itegrator Explicit Explicit RK4 ode45 Implicit Cata. Div. 0% 18% 5% 0% 0% RMSE(12.96) NaN NaN Patter Cor NaN NaN Avg. Time Xi Tog Stability of EKF 26 / 34

28 Geometric ergodicity Covergece to a ivariat measure geometrically fast P(U, V ) π tv Ce γ. Also kow as L 1 -distace: P Q tv = p(x) q(x) dx. x Iitializatio is forgotte statistically with a expoetial rate. Log time average coverges to the average at equilibrium. Foudatio for further ivestigatio. Xi Tog Stability of EKF 27 / 34

29 A framework for ergodicity Well established i fiite-dimesio cases. (Mey, Tweedie 93). Equivalet to the existece of a couplig of X ad X : P(X ) P(X ) tv = if C P(X X ). A simple framework Lyapuov fuctio E: EE(X ) e γ E(X 0) + K. Reachability: there is a poit x, P( X x < ɛ) > 0. Cotrollability at x : the trasitio kerel P (x, dx ) is cotiuous aroud x = x ad x = x. Lyapuov fuctios from 1 Observable eergy criterio Hu u. 2 Adaptive iflatio. Xi Tog Stability of EKF 28 / 34

30 Theorem for ergodicity Assumptio The trasitioal kerel of the system oise ζ i U = Ψ(U 1 ) + ζ is odegeerate. Give ay M > 0, there is a α > 0 such that P (ζ dz U 1 ) > αλ BM (dz). Reachability. Cotrollability: V = Ψ(V 1 ) + ζ. V = (I + ĈH T H) 1 V + (I + ĈH T H) 1 Ĉ H T Z. Theorem (T., Majda ad Kelly) If the EKF process has a Lyapuov fuctio, ad o-degeerate system oise, the the EKF process {V } joitly with U is geometrically ergodic. Xi Tog Stability of EKF 29 / 34

31 Esemble Square Root filter (ESRF) Forecast step: propagate each member V = Ψ(V 1 ) + ζ Assimilatio step: achieve the Kalma covariace rule by lettig S = [V C = Ĉ ĈH T (I + HĈH T ) 1 HĈ. V ] with ETKF: S = ŜT, T = (I + (K 1) 1 Ŝ T H T HŜ) 1/2. EAKF: S = A Ŝ, A = QΛG T (I + D) 1/2 Λ Q T, QΛR = Ŝ, G T DG = (K 1) 1 Λ T Q T H T HQΛ. Xi Tog Stability of EKF 30 / 34

32 Esemble Kalma filter (EKF) Use a esemble {V } K k=1 to represet N (m, C ) Forecast step: propagate each member V = Ψ(V 1 ) + ζ Assimilatio step: apply the Kalma update rule V = V ĈH T (I + HĈH T ) 1 (H V Z ξ ). Posterior at t = 1 Prior+Obs at t = V forecast: M.C. V 1 assimilate Posterior at t = V Xi Tog Stability of EKF 31 / 34

33 Extesio to ESRF Stability/boudedess from observable criterio ad adaptive covariace iflatio: all ESRF. Example of cata. div. ad ergodicity: ETKF ad a versio of EAKF. The versio: Diagoalizatio of a diagoal matrix is itself. Xi Tog Stability of EKF 32 / 34

34 Summary EKF is stable if it follows a observable eergy priciple. EKF could grow expoetially whe filterig stable dyamics. Adaptive covariace iflatio ca resolve this issue. EKF is geometric ergodic if it is stable ad the system oise ζ is o-degeerate. All the claims ca be exteded to Esemble square root filters. Xi Tog Stability of EKF 33 / 34

35 Referece Noliear stability ad ergodicity of esemble based Kalma filters. arxiv: Cocrete esemble Kalma filters with rigorous catastrophic filter divergece. PNAS (2015) v Noliear stability of the esemble Kalma filter with adaptive covariace iflatio. accepted by Commu. Math. Sci. Liks ad slides ca be foud at tog. Thak you! Xi Tog Stability of EKF 34 / 34