Time-series analysis for palaeoclimate data

Transcription

1 Time-series analysis for palaeoclimate data Michel Crucifix acknowledgements: Guillaume Lenoir, Carlos Almeida (ULouvain), Jonathan Rougier (Bristol), Richard Wilkinson, Jake Carson (Nottingham)

2 Disclaimer (!) Time series analysis is a very vast subject, with hundreds of books and thousands of publications. Here : necessarily subjective and incomplete but consistent with 15 years experience of using palaeoclimate data and an understanding of the needs of the palaeoclimate community Non-linear time series analysis (as from non-linear physics) will not be covered. For the latter, see Kantz and Schreiber, 2003 Holger Kantz and Thomas Schreiber, Nonlinear Time Series Analysis, 2003

3 Forcing Climate System Sensor (e.g : δ18o) Archive (sediment) Postprocessing Observation

4 e.g.: reasonably raw data ODP 927 benthic forams d18o Depth [m] d13c Bickert, T., Curry, W.B. & Wefer, G. Late Pliocene to Holocene (2.6 0 Ma) western equatorial Atlantic deep-water circulation: Inferences from benthic stable isotopes, Proc. Ocean Drilling Program, Sci. Results 154, , 1997.

5 a more post-processed series d18o (per mil) LR04 benthic stack of 57 records Age [ka] L. E. Lisiecki and M. E. Raymo, A Pliocene-Pleistocene stack of 57 globally distributed benthic d^{18}o records, Paleoceanography, 20, PA

6 Palaeoclimate data Generally low sampling rate, Measurement is not direct and what they tell us about climate with system not always well understood (a few exceptions : e.g.: greenhouse gas) Dating uncertainties

7 The climate system Complex system Multi-scale Non-autonomous (i.e.: external forcing) Non-stationary (slow changes : e.g.: tectonics ) Physics partly known and system partly biologically controlled laws of emergence, self-regulation

8 Forcing Climate System Sensor (e.g : δ18o) Archive (sediment) Postprocessing Observation

9 Forcing Climate System Sensor (e.g : δ18o) Archive (sediment) Postprocessing Observation

10 Time series analysis = navigation upstream the causal flow identify patterns (regularities) test hypotheses (e.g.: leads/lags; influence of a factor) estimate parameters select models IMPLY A CERTAIN PRE-CONCEPTION OF WHAT WE ARE PREPARED TO FIND Luke Skinner: we are all modellers

11 Outline Projection methods Frequentist regression Bayesian approach State-space time models

12 ~v = x~e x + y~e y + z~e z

13 chose a basis estimate the coefficients, from a finite-sampled series 1. Fourier basis sines and cosines different estimators, generally parametric 2. Optimal basis : Basis estimated from slide-windowing over the time series (singular spectrum analysis) Allows to efficiently capture the main modes Implicit assumption of stationarity next slide : SSA, 200-element window; P. Yiou et al., Nonlinear variability of the climatic system from singular and power spectra of late Quaternary records, Climate Dynamics, 9, Rssa package by Anton Korobeynikov

14 r.benthic LR04 = D$F r.times trend + D$F2 + D$F r.times 100 ka D$F4 + D$F r.times 40 ka r.times time [ka] + remainder

15 LR04 SSA log_10 (Lambda) Index

16 Hypothesis testing Beware : Out-of-the box packages generally assume red noise as null hypothesis Power density (variance units) (στ) 2 1 T 1 (2πτ) 1 (2δt) Frequency (time units 1 )

17 but hang on : do you really believe your data is AR1? LR04 spectrum using MTM MTM spectrum 99% confidence e-05 1e frequency (cycles / ka) Multitaper spectral estimation, with 99 % confidence limit. Percival, D.B., and A.T. Walden, 1993: Spectral analysis for physical applications - Multitaper and conventional univariate techniques. Cambridge University, 580 p., software implementation by Dettinger, M.D.,et al., 1995: Software expedites singular-spectrum analysis of noisy time series, Eos, Trans. AGU, v. 76(2), p. 12, 14, 21.

18 A time-frequency filter : the Continuous Wavelet Transform E. W. Bolton, K. A. Maasch, and J.~M. {Lilly}, A wavelet analysis of Plio-Pleistocene climate indicators a new view of periodicity evolution, Geophysical Research Letters, 22, [; S. Mallat, A Wavelet tour of signal processing, 1998 ; D. Maraun, J. Kurths, and M. Holschneider, Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significance testing, Physical Review E, 75, Here: Morlet wavelet, w_0=9., normalisation proportional to 1/scale)

19 original signal ridge reconstruction G. Lenoir, Thesis in preparation

20 Outline Projection methods Frequentist regression Bayesian approach State-space time models

21 The climate equation (after M. Mudelsee) X(t) =X trend (t)+x outlier (t)+x noise (t) parametric models example : X trend (t) = t X outlier (t) =0 X noise (t) =AR1(, 1)

22 Another example of trend model : the ramp M. Mudelsee and M. E. Raymo, Slow dynamics of the Northern Hemisphere glaciation, Paleoceanography, 20, PA

23 Application to marine data to detect glacial inception in the Northern Hemisphere M. Mudelsee and M. E. Raymo, Slow dynamics of the Northern Hemisphere glaciation, Paleoceanography, 20, PA

24 Frequentist regression Parameter estimators least-square least median square maximum likelihood (to be defined later) consistency (converge to right result if infinite series with this model) small bias for finite time series robustness to outliers (small variance) Confidence intervals

25 Confidence interval: interval that frequently (e.g.: 95%) includes the parameter of interest if the observation is repeated indefinitely,... and if the model is correct depends on the observations estimator true value in practice : approximate confidence interval

26 In general : you only know the confidence interval if you already know the parameters of the distribution of the data... approximate CI, based on theory of large samples simulate a sample of estimators, using only a fraction of the data and repeating the experiment many times BOOTSTRAPPING METHODS M. Mudelsee, Climate Time Series Analysis, Springer

27 Outline Projection methods Frequentist regression Bayesian approach State-space time models

28 remember : frequentist approach defined a... Confidence interval: observed interval, that frequently (e.g.: 95%) includes the parameter of interest if the observation is repeated indefinitely, given a number of hypotheses about the process being observed... :but hang on : you won t repeat the experiment indefinitely...

29 Bayesian reasoning P ( )

30 Bayesian reasoning P ( )

31 Bayesian reasoning P ( )

32 Bayesian reasoning P ( ) P ( z) {z } posterior P ( ) {z } prior P (z ) {z } likelihood

33 The prior Bayesian statistics derive from Aristotelean logic Formalised by TJ Bayes ( ) and PS Laplace ( ) Chosing a prior is an art may want to express ignorance rules about invariance wrt to unit changes (Jeffrey s prior) minimise information content = maximise information entropy (Jaynes MaxENT principle to derive distributions) use symmetries or, to the contrary, express knowledge e.g. : effect of astronomical forcing on temperature must be positive -> parameter positive E. T. Jaynes, Probability Theory : The Logic of Science, Cambridge Univerisity Press, 2003 (posthumous)

34 The likelihood The likelihood is a function of the parameter... but it may also be seen as a probability : the probability of observing what you have observed, if your model is correct and parameter has value theta

35 The posterior The posterior combines prior and likelihood Designing a prior and a likelihood involve subjective choices Deriving the posterior from these premises must follow the objective framework of Bayesian statistics Hence, Bayesian statistics are sometimes called a paradigm of restricted inference Credible interval: interval within with the true value of the quantity being estimated lie within x probability, according to its posterior probability distribution

36 Example of Bayesian time series analysis X(t i )=X trend (t i )+X noise (t i ) X trend (t i )=B 1 sin(!t i )+B 2 sin(!t i ) X noise (t i ) i.i.d. 8 express ignorance >< >: X noise N (0, & 2 ) P (&) 1/& P (B 1 ) P (B 2 ) uniform

37 ... the maths (just for fun) G.L. Bretthorst, Bayesian Spectrum Analysis and Parameter Estimation, Springer Lecture Notes in Statistics

38 example with a twin experiment t i true value X(t i ) P(ω) ωp(ω)dω ω ω

39 ... and trying data in which there is no periodic component (this is a case of model mispecification) t i X(t i ) P(ω) 0e+00 4e 04 8e ωp(ω)dω ω the prior on omega (here: uniform between 0 and 10) has not been updated ω

40 ... though beware : model mispecification may sometimes fool you P ( ) P ( z) {z } posterior P ( ) {z } prior P (z ) {z } likelihood

41 ... talking from experience: An untrained Bayesian is one who, vaguely expecting a horse, and catching a glimpse of a donkey, strongly believes he has seen a mule The better-trained Bayesian will assess his model, check its predictive power and consider the behaviour of the marginal likelihood

42 Outline Projection methods Frequentist regression Bayesian approach State-space time models

43

44 forcing hidden states observations depends is here taken in a statistical sense but it is more natural to make it correspond to the causal (physical) direction

45 forcing and parameter hidden states observations Estimate (X_t θ, F) is the state estimation problem (reconstruct climate given a known model) Estimate (X_t, θ F) is the joined state-estimation problem infer about the properties of climate and reconstruct it

46 Which model? Can in principle be non-linear and complex but hang-on : if many parameters, can be untractable and / or meaningless! A stochastic model better reflects our beliefs that it is imperfect, i.e. :

47 One natural way to obtain a model is as a stochastic differential equation same idea as classic differential equation (Newton s law, Navier-Stokes, etc. )... with a random component Mathematics formalised by Itô and Stratanovitch, introduced by Hasselman and Frankignoul (1976) in climate science K. Hasselmann, Stochastic climate models Part {I}. Theory, Tellus, 28, ; C. Frankignoul and K. Hasselmann, Stochastic climate models, Part II: Application to sea-surface temperature anomalies and thermocline variability., Tellus, 29, ; special issue : Stochastic physics and climate modelling, Editors: Tim Palmer and Paul Williams, Phil. Trans. Royal Society, vol. 366, 2008.

48 What about ice ages? (Pleistocene) Several of conceptual models of ice ages in the literature... that s the right level of complexity to begin with One example (home-made) dx = [ x + 2 x(x 2 1) y + + ]dt + 1 dw 1 dy = [ (y y 3 /3+x) ]dt + 2 dw 2 Michel Crucifix, Oscillators and relaxation phenomena in Pleistocene climate theory, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370,

49 Consider three data samples LR04 stack Time [ka] benthic d18o [per mil] Huybers 04 stack Time [ka] benthic d18o [std] ODP929 Time [ka] benthic d18o [per mil]

50 Resolution by PMCMC 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 LR04 Huy04 ODP C. Andrieu, A. Doucet, and R. Holenstein, Particle Markov chain Monte Carlo methods, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72,

51 Be honest: where are we with this? The idea of Bayesian inference with state-space models on palaeoclimate data is recent (Haslett, 2006) There are good reasons to anticipate it being tricky (Rougier, 2013) Our experience is that it is probably feasible, but need deal with complex (and complicated) algorithms of model-dependent robustness which can be time consuming... and the problem of dealing with uncertain time scales is only at its beginning so : it is not yet ready for user-friendly interfaces J. Haslett et al., Bayesian palaeoclimate reconstruction, Journal of the Royal Statistical Society: Series A (Statistics in Society), 169, J.C. Rougier, 'Intractable and unsolved': some thoughts on statistical data assimilation with uncertain static parameters, Phil. Trans. R. Soc. A , 2013

52 Summary : what for what? Decomposition / Wavelets qualitative investigation detection of early-warning signals / signature of regime changes requires some hand-tuning (time-window length, nr of tapers, wavelet etc.) Regression (frequentist or Bayesian) detection / timing of events estimate / characterise signal vs noise Bayesian calibration of state-space models climate reconstruction link with physical, more complex models hypothesis testing / model selection joint inference on climate and chronology