Calibration and selection of stochastic palaeoclimate models

Transcription

1 Calibration and selection of stochastic palaeoclimate models Michel Crucifix, Jake Carson, Simon Preston, Richard Wilkinson, Jonathan Rougier IRM - Internal Seminar - Uccle 2 April 2014

2 Posing the problem Principles and methods of model calibration and selection Applications

3 2 2.5 Benthic d 18 O (normalised) time (ka) time (ka) L. E. Lisiecki and M. E. Raymo, Paleoceanography, 20, PA

4 External factors? t t + 1 Internal structure / parameters

5

6 Astronomical forcing as quasi-periodic signal Precession : , , , , ,... years Obliquity : 41090, , , , 41811,... years Eccentricity : , , , ,... years from A. Berger and M.F. Loutre, Quaternary Science Reviews, 10,

7 The Langevin model (1 dimension) dx = ( U 0 (x)+f (t))dt + dw derivative of a cimate potential stochastic forcing deterministic forcing Nicolis and Nicolis, Tellus (1981) ; Nicolis, Tellus, (1982)

8 Competing paradigm : Oscillators (or oscillator-like) dx = ( + F (t) y)dt dy = (y y 3 + x) dt

9 X Y V D Forcing LR time (ka) M. Crucifix, How can a glacial inception be predicted?, The Holocene, 21,

10 Forced stochastic van der pol Model dx = ( + F (t) y)dt dy = (y y 3 + x) dt + dw

11 V D model with stochastic perturbation X V Y D time (ka) Figure 3. Same model as Figure 2 but with a small stochastic M. Crucifix, How can a glacial inception be predicted?, The Holocene, 21,

12 Ice age models founded on the hypothesis of a synchronised limit cycle feature a strange non-chaotic attractor Lyapunov stable BUT structurally unstable M. Crucifix, Why could ice ages be unpredictable?, Climate of the Past, 9, Takahito Mitsui and Kazuyuki Aihara, Dynamics between order and chaos in conceptual models of glacial cycles, Climate Dynamics, see also :

13 Posing the problem Identification of structure and parameters (using information emerging from observation and simulations with GCMs, ice sheet models, etc. ) in non-autonomous systems that may be structurally unstable... with sparse-data, and uncertain dating

14 Posing the problem Principles and methods of model calibration and selection Applications

15 / Z One observes Z = z

16 Prior : P ( ) Likelihood : P (z ) Model Evidence : P (Z = z) = R 2 P (z )P ( ) Posterior : P ( z) = P (z )P ( ) P (Z=z)

17 The latent variable as an intermediate step / X (hidden or latent variable ) Z P (z ) = Z X P (z, (X))P (X)dX z is a shorthand for Z = z

18 Dynamical system : many latent variables (F t 1, ) (F t, ) (F t+1, ) X 0 /... / X t 1 / X t / X t+1 /... Z t 1 Z t Z t+1

19 Intractable and unsolved : some thoughts on statistical data assimilation with uncertain static parameters Jonathan Rougier Three years ago, I started a collaboration with Michel Crucifix at the Université catholique de Louvain, on glacial cycles. Our intention was to use simple phenomenological models [1], which would be calibrated against measurements from ice core and benthic datasets, and used to address questions such as Ruddiman s Early Anthropogenic hypothesis (reviewed in [2]). I thought we would use an approach suggested by Liu & West [3], which was a sophisticated treatment of data assimilation with uncertain static parameters (defined below), in which the state vector is augmented with the static parameters. One year later, I was discussing our lack of progress with Prof. Christophe Andrieu, an authority on statistical data assimilation. He was not at all surprised, because in his view the problem was, in general, intractable and unsolved. Regardless, I simply upped the computational sophistication of our methods, including switching to his own recent development, particle Markov chain Monte Carlo (P-MCMC, [4]) and running on a faster computer /rsta Phil. Trans. R. Soc. A 28 May 2013 vol. 371 no

20 stochastic dynamical system implemented numerically as a map (numerical scheme) random vector (X t 1,! t 1 ) deterministic! X t inference on the! i equivalent to inference on the X i

21 State inference : P (X z, ) Joint state-parameter inference : P (,X z) Parameter inference : R X P ( z,x)

22 State inference : P (X z, ) Sequential algorithms Kalman Filter (exact if linear and Gaussian) Extended, Unscented, Ensemble Kalman Filters Sequential Monte Carlo (Particle Filter) Variational (bit by bit) Global approaches Variational (global) Annealing

23 Kalman filters Sequential estimation of state (assimilation of data) The information being propagated is The mean and the variance Their dynamics are estimated using the Linear Tangeant (Extended Kalman Filter) an ensemble of points (Ensemble of Points) the dynamics of a well chosen set of points : the Sigma Points (Unscented Kalman Filter, Julier and Uhlman) EKF and UKF are deterministic and biased

24 Particle Filter X w i,j = p(! i,j )p(z i X i,j )/q(! i,j )... Lik j = P w i,j Lik( ) = P j (Lik j)! i,j q(! i,j ) t j t j+1

25 Joint state-parameter inference : P (,X z) 1. Augmented state approach ( Liu and West, 2001) add =0 2. PMCMC (Andrieux et al. 2010) 3. SMC 2 (Chopin et al. 2012)

26 SMC-2 (Chopin et al. 2012) z (1) (2) x2 x2 x3 x1 x2 x1 x2 x1 t 0 z t 1 t 1 t 0 t 2 (1) Sample (,X 0 ) and propagate forward until filter degenerates (e.g. : at time t 1 ) (2) Resample, perturb leaving the posterior distribution at t 1 invariant, and continue formard

27 SMC-2 (Chopin et al. 2012) z (1) (2) x2 x2 x3 x1 x2 x1 x2 x1 t 0 z t 1 t 1 t 0 t 2 (1) Sample (,X 0 ) and propagate forward until filter degenerates (e.g. : at time t 1 ) (2) Resample, perturb leaving the posterior distribution at t 1 invariant, and continue formard

28 Posing the problem Principles and methods of model calibration and selection Applications

29 application 1 : Liu and West Algorithm M. Crucifix and J. Rougier, European Physics Journal Special Topics, 174,

30 The Satlzman & Maash model Ice Sheets (ζ) [1] B. Saltzman and K. A. Maasch. A first-order global model of late Cenozoic climate. Trans. R. Soc. Edinburgh Earth Sci, 81: , Ocean Circulation (ψ) Carbon Dioxide (theta) = 1 (c 2 )µ 3 k 2 k R 2 R(t) + I = K ( 5 6 ) + F µ (t) + = 1 2 I 3 + F (t) + K = 1 ( )

31 Augmented-state strategy : handle filter degeneracy 2e+13 9e+13 Parameter: a1 Parameter: b1 4e Parameter: b2 Parameter: b3 Parameter: b4 0 3e 04 2e 05 Parameter: b Years (ky, CE) M. Crucifix and J. Rougier, European Physics Journal Special Topics, 174,

32 application 2 : Estimating likelihood: Kalman vs Particle unpublished note

33 linear model linear model dx = ( + F (t) y)dt + 1 dw 1 dy = (y + x)dt + 2 dw 2 with observations z t = x t + t, t N (0, obs)

34 Log lik estimator : linear model as a function of TAU log likelihood KF PF (n=200) PF (n=1500) tau

35 Lik. estimator : linear model i log likelihood KF PF (n=100) log of mean(lik) PF

36 Forced van der pol Model non- linear model dx = ( + F (t) y)dt + 1 dw 1 dy = (y y 3 + x)dt + 2 dw 2 with observations z t = x t + t, t N (0, obs)

37 Log lik estimator : NON linear model as a function of TAU log likelihood UKF UKF PF (n=200) PF (n=1500) tau

38 application 3 : Model Calibration : PMCMC vs SMC2 poster, EGU Vienna, 2013

39 Forced slow-fast model (CR12) Dynamics dx = [ x + 2 (x 2 1) y + + ]dt + 1 dw 1 dy = [ (y y 3 /3) + x) ]dt + 2 dw 2 state Observations determinstic drift = 30 (slow-fast system) astronomical forcing additive di usion ((w 1,w 2 ) =Wiener) z t = x t + t, t N (0, obs)

40 LR04 stack Time [ka] Huybers 04 stack Time [ka] benthic d18o [std] ODP929 Time [ka] benthic d18o [per mil]

41 Calibration on LR04 β 0 β 1 β 2 δ γ π Density Density Density Density Density γ ε σ x σ y σ z Density Density Density Density PMCMC SMC2 prior

42 application 4 : Model selection with SMC2 Jake Carson s thesis : in prep.

43 Bayes Factor = Model Evidence (M1) Model Evidence (M2)

44 Observations = data simulated with the Forced SM91 SS-F Model Evidence (Forced Model) Evidence (Unforced Model) SM T PP

45 Observations = LR04 stack (astronomically tuned) LR04 Model Evidence (Forced Model) Evidence (Unforced Model) SM T PP

46 Observations = non-astronomically tuned data ODP982b Model Evidence (Forced Model) Evidence (Unforced Model) SM T PP12 * -

47 Particle Filter... is an art choice of number of particles, proposal, comparison between algorithm, ofter problem dependent Augmented space approach for parameter estimation may be deceptive. We no longer recommend it. PMCMC was proposed as a solution may have poor mixing : stationary solutions not trusted SMC^2 conceptually more advanced, and gives model evidence as a bonus : model selection is possible [ The practical added value of PF in additive, Gaussian noise problem still needs to be demonstrated ]