Filtering Turbulent Signals Using Gaussian and non-gaussian Filters with Model Error

Transcription

1 Filering Turbulen Signals Using Gaussian and non-gaussian Filers wih Model Error June 3, 3 Nan Chen Cener for Amosphere Ocean Science (CAOS) Couran Insiue of Sciences New York Universiy /

2 I. Ouline Use es models o sudy he skills of filering he. These es models have exacly solvable saisics. They mimic he amosphere and ocean behaviors. Main feaures of Amosphere and Ocean Sciences: Inermien insabiliy. Unresolved/unobserved variable/process. 3 Model error. Goal Generae he signals from he sysem wih inermien insabiliy and unresolved process. Sudy he filering skill of using imperfec forecas models. Compare he filering skill using Gaussian and non-gaussian filers in a perfec model seing. /

3 Differen ypes of filer (w.r.. he prior/background disribuion). Assume linear observaion operaor and Gaussian observaion noise. Kalman Filer: linear and Gaussian sysem u m+ = Fu m + σ m+ wih consan F. Exended Kalman Filer: nonlinear non-gaussian sysem wih linear angenial approximaion u m+ = f (u m)+σ m+ = u m+ f (ū m m )+ f (ū m m )(u m ū m m )+σ m+. Two momens filer (or Nonlinear Exended Kalman Filer): nonlinear non-gaussian sysem wih exacly solvable mean and covariance. Noe ha model error sill exiss when using wo momens filer o filer he non-gaussian signal! Non-Gaussian filer: The informaion in he higher order momens is included in filering he signals. 3/

4 II. A. Tes Models du = r(, )u + l(, ) + σu(, )Ẇ u (), d d = F(, ) + σ (, )Ẇ (), d Resolved/unresolved variable: u is he resolved variable. is he unresolved variable. Inermien insabiliy in u: Posiive r(, ) corresponds o insabiliy. Negaive r(, ) corresponds o sabiliy. 3 Model error: A complicaed dynamics is used o generae he rue signal. Some simplified dynamics are used as he forecas model. To obain he saisics of (), here s no need o solve he -D Fokker-Planck equaion nor use he Mone Carlo simulaion. Insead, he saisics can be solved wih some cheap ways using he condiional momens. () 4/

5 & Coupled Generalized Fokker-Planck Eqns Disribuion and Momen Join disribuion: p(u, ). Marginal disribuion: π() = p(u, )du, π(u) = p(u, )d. Condiional disribuion: p(u ). Condiional Momen A one fixed : M N( ) = u N p(, u)du, As a funcion of : M N() = u N p(, u)du. Remark: M () = p(u, )du = π(). 5/

6 Tes Models Proposiion du = r(, )u + l(, ) + σu(, )Ẇ u (), d d = F(, ) + σ (, )Ẇ (), d The vecor of condiional momens M N(, ) of order N associaed wih he probabiliy densiy of () saisfies he coupled generalized Fokker-Planck equaions (CGFPE) MN(, ) =LFPMN(, ) + r(z, )NMN(, ) + Nl(, )M N (, ) + N(N )σ u(, )M N (, ), () () where M = π(, ) and we prescribe M and M and L FPM N(, ) = (F(, )MN(, )) + (σ (, )M N(, )). The momen equaion () can be solved recursively. 6/

7 Wih he condiional momens, M (, ) = p(, u, )du, M (, ) = up(, u, )du, M (, ) = u p(, u, )du, he mean and covariance of sysem () are solved via ( ) ū() = up(u,, )dud = up(u,, )du d = M (, )d, () = Var(u)() = ( = Var()() = ( = Cov(u, )() = = p(u,, )dud = ( u p(u,, )dud ū () ) u p(u,, )du p(u,, )dud () d ū () = ) p(u,, )du d () = up(u,, )dud ū ( ) up(u,, )du d ū = ) p(u,, )du d = M (, )d ū (), M (, )d, M (, )d (), M (, )d ū. 7/

8 III. Assume he rue signal is generaed from he following sysem, u du = ( u + f u)d + σ udw u, d = ( a + b c 3 + f )d + (A B)dW c + σdw, Equilibrium PDF of u True PDF Gaussian fi 4 p Equilibrium PDF of log scale u (3).5 p 5 log scale Assume linear observaions wih g = [, ], and herefore v m+ = u m+ + σ o m+. Observaion ime sep: =.5, less han he averaged decorrelaion ime τ u corr =.36 of u and decorrelaion ime τ corr = 3.64 of. Observaion noise level: R o equals.5% of he averaged energy of he rue signal. 8/

9 Measuremen of he filering skill. Le u m be he rue signal and ū m m be he filer esimae. Roo mean square (RMS) error RMSE = T (u m ū m m ) T T. m=t Paern correlaion (PC) (u m u m )(ū m m ū m m ) PC = (u m u m ) (ū m m ū m m ) PC= 3 PC= 3 9/

10 u u. The perfec forecas model, du = ( u + f u)d + σ udw u, d = ( a + b c 3 + f )d + (A B)dW c + σdw, f u =.5 Perfec model: RMSE=.643, PC= True signal Observaion Poserior mean f u =.5 sin(.5) Perfec model: RMSE=.75, PC=.9979 True signal Observaion Poserior mean Perfec in u Obs error f u =.643 (.977).864 f u =.5 sin(.5).76 (.9979).864 /

11 u u. Mean sochasic model (MSM), in which he damping in u is se o be he averaged value of and herefore no unresolved dynamics is included, du = ( M u + f u)d + σ udw u..5.5 f u = MSM: RMSE=.79, PC= f u =.5 sin(.5) MSM: RMSE=.39, PC= MSM in u Perfec in u Obs error f u =.79 (.968).643 (.977).864 f u =.5 sin(.5).39 (.995).76 (.9979).864 /

12 u u 3. The imperfec forecas model of is a linear Gaussian model, such ha he forecas model becomes he simplified SPEKF-ype model, du = ( u + f u)d + σ udw u, d = d M ( ˆ M )d + σ M dw. Remark: The evoluion of he mean and covariance of SPEKF model can be solved analyically. f u =.5.5 SPEKF: d =., σ = 3σu, RMSE=.645, PC= f u =.5 sin(.5) SPEKF: d =., σ = 3σu, RMSE=.75, PC= SPEKF in u Perfec in u Obs error f u =.645 (.974).643 (.977).864 f u =.5 sin(.5).75 (.9978).76 (.9979).864 /

13 Robusness of he parameers in SPEKF model wih f u =..8 Damping Noise RMS error SPEKF Perfec reference Obs MSM reference du = ( u + f u)d + σ udw u,.6 3 d M / d = d M ( ˆ M )d + σ M dw Paern correlaion d M /.6 σ M /σu.966 σ M /σu Figure : Robusness of he parameers in SPEKF. Top: filering skill dependence of d M wih fixed raio σ M /σu = 3. Boom: filering skill dependence of σm wih fixed raio dm / =.. 3/

14 B. Special ype of non-gaussian filer for he es model. Approximaion of he prior disribuion p (u,, ): Complee recovery of he marginal disribuion in via M (, ) = p(, u, )du. Condiional Gaussian for each fixed by making use of M (, ) = up(, u, )du, M (, ) = u p(, u, )du. 4/

15 Comparison of he full PDF and is non-gaussian and Gaussian approximaion a he equilibrium in he dynamical regime of filering. 5/

16 Comparison of wo momens filer and non-gaussian filer. Two momens filer Perfec model wo momens filer: RMSE=.643, PC= u.5 True signal Observaion Poserior sae Non-Gaussian filer Perfec model non Gaussian filer: RMSE=.64, PC=.979 u RMSE (PC) in u Obs error Two Momens.643 (.977).864 Non-Gaussian.64 (.979).864 6/

17 Filering skill as a funcion of observaion ime sep (op) and observaion noise (boom) using differen filers wih respec o he resolved variable u (unforced case: f u = ) RMS error Perfec model wo momens filer Perfec model non Gaussian filer SPEKF MSM Obs noise Obs ime sep obs Paern correlaion Obs ime sep obs.3.. RMS error Perfec model wo momens filer Perfec model non Gaussian filer SPEKF MSM Obs noise.98 Paern correlaion e.e.5e.e.5e Obs noise r o.88.5e.e.5e.e.5e Obs noise r o 7/

18 Filering skill as a funcion of observaion ime sep (op) and observaion noise (boom) using differen filers wih respec o he resolved variable u (unforced case: f u =.5 sin(.5))..4. RMS error MSM Obs noise. Paern correlaion Perfec model wo momens filer Perfec model non Gaussian filer SPEKF Obs noise Obs ime sep obs Obs ime sep obs.4. MSM Obs noise RMS error Paern correlaion Perfec model wo momens filer Perfec model non Gaussian filer SPEKF Obs noise e.e.5e.e.5e Obs noise r o.5e.e.5e.e.5e Obs noise r o 8/

19 Comparison of he unresolved variable. Equilibrium PDF of Perfec model wo momens filer.5 p Equilibrium PDF of.5 p Perfec model non Gaussian filer True Poserior Pah-wise soluion. Two momens filer Non-Gaussian filer RMS error Paern correlaion Informaion crieria. The Shannon enropy, S(U m), of he residual U m u m ū m m, U p, is given by S(U m) = p(u m) ln p(u m)du m. I expresses he uncerainy in he filer esimae ū m m abou he rue sae u m a ime m. Two momens filer Non-Gaussian filer Shannon enropy /

20 IV. In his work, we sudied Filering skill using wo momens filer wih differen predicion models. Filering skill using wo momens and non-gaussian filer in a perfec model seing. Main conclusions are summarized as follows: For he wo momens filer, SPEKF has comparable filering skill wih he perfec model filer while MSM filer has low filering skill. The special non-gaussian filer has very lile improvemen of he filering skill wih respec o he resolved variable u. 3 Filer esimaes of he unresolved variable are quie differen using wo momens filer and non-gaussian filer. /

21 Thank you /