Probabilistic Models of Past Climate Change

Transcription

1 Probabilistic Models of Past Climate Change Julien Emile-Geay USC College, Department of Earth Sciences SIAM International Conference on Data Mining Anaheim, California April 27, 2012

2 Probabilistic Models of Past Climate Change Julien Emile-Geay (USC) with: Dominique Guillot (USC) Bala Rajaratnam (Stanford) Tapio Schneider (CalTech) acknowledgements: Jianghao Wang (USC) Outline: 1. Mathematical Problem 2. Gaussian Graphical Models 3. Some results 4. Error estimates

3 Reconstructing Past Climates Why paleoclimatology? Is it warmer now than in AD 1000? ( Hockey Stick problem) Is the rate of warming anomalous? What are the spatiotemporal characteristics of natural climate variability? How (un)certain is all this? Statistical challenges Short training set (calibration) Very high-dimensional (p >n) Noisy, autocorrelated predictors No straightforward spatial model Temperature Anomaly ( C) Temperature Anomaly ( C) 0 CPS land with uncertainties EIV land with uncertainties EIV land+ocn with uncertainties Mann and Jones (2003) Esper et al. (2002) Moberg et al. (2005) HAD Instrumental Record CRU Instrumental Record Northern Hemisphere CPS land with uncertainties EIV land with uncertainties EIV land+ocn with uncertainties Mann and Jones (2003) Esper et al. (2002) Moberg et al. (2005) HAD Instrumental Record CRU Instrumental Record CPS land+ocn with uncertainties Briffa et al. (2001) Crowley and Lowery (2000) Mann et al. (1999) Jones et al. (1998) Oerlemans (2005) Mann et al. (2003) Optimal Borehole Huang et al. (2000) Borehole Mann et al., PNAS, 2008 Julien Emile-Geay

4 High-resolution paleoclimate proxies Speleothems Tree Rings Ice cores Sediment cores Corals

5 High-precision dating Ruddiman, 2006

6 Climate Field Reconstruction A missing data problem backcast T from proxy observations multivariate inference A high-dimensional problem e.g. Mann et al [2008] database p = p i +p p = n =150 Instrumental Temperature T 1,, T pi Proxies P 1,,P pp Covariance matrix captures relationship between temperature and proxies sample covariance matrix is rank-deficient unknown P 1,,P pp estimation is impossible from the sample covariance matrix: it must be regularized 0

7 Regularized Covariance Estimation (1) Maximum Likelihood Estimation using l2-penalized likelihood and Expectation-Maximization [Dempster, Laird and Rubin, 1977] L 2 (Σ,h)= n 2 tr(σ 1 S) n 2 Σ 1 + h i<j σ ij 2 2 (2) Regularized Solution: ˆΣ aa = VΛ 2 V T F Λ V T ˆΣam (Fourier coefficients) ˆB = V Diag(f j ) Λ F with fj the filter factors Tikhonov Regularization ( Ridge regression ) f j = λ j 2 /(λ j 2 + h 2 ) Julien Emile-Geay USC 2012

8 Explicit Spatial Modeling C ij = f(d i j ) e.g. kriging Limitations: isotropic rigid subjective Julien Emile-Geay USC 2012

9 Graphical models: spatial modeling Land/Ocean boundaries Mountain ranges Teleconnections / climate patterns 5/21

10 Marginal vs Conditional Independence Marginal independence: X i X j Σ ij =0 Conditional independence: X i X j {rest of variables} Ω ij =0. Example: Ω = Σ =

11 Discovering conditional independence relations Exploiting conditional independence relations: Conditional independence relations are inherent to climate fields: Knowledge of such relations zeros in Ω dimension reduction; Can be discovered using 1 type optimization methods. Graphical lasso: Friedman, Hastie & Tibshirani [2008] max Ω>0 log-likelihood(ω)+ρ Ω ij. i,j Ω 1 Graphical maximum likelihood estimate: ˆΣ G = max likelihood(σ) Σ 1 P + G Best covariance matrix compatible with the CI structure. Benefits: Adding a 1 penalty favors sparse estimates of Ω; Sparse estimates achieve the necessary dimension reduction for proper estimation of Σ. 7/21

12 Flexible covariance representation Temperature neighbors of selected point 80 o N 40 o N 0 o 40 o S 80 o S 180 o W 120 o W 60 o W 0 o 60 o E 120 o E 180 o W HadCRUT3v data, Julien Emile-Geay USC 2012

13 Example of graph (HadCRUT3v data) Julien Emile-Geay USC 2011

14 e Graphical EM (GraphEM) algorithm (2) Compute Dempster, Laird & Rubin, 1977 Schneider, 2001 (1) µ 0, Σ 0 CI structure ˆΣ = ˆΣ G (Regularization) Update Compute regression coefficients (5) {µ, Σ} ˆB = ˆΣ + aa ˆΣ am (3) ˆx m = ˆµ m +(x a ˆµ a )ˆB Guillot et al, JASA, submitted (4) Estimate unknown values from the available ones

15 A virtual climate laboratory Standard deviation of CSM1.4 millennial run CSM1.4 specs: - Coupled General Circulation Model - Plausible surrogate climate - Generate pseudoproxies as statistically-degraded, subsampled version of the temperature field σ (K) Julien Emile-Geay USC 2012

16 Pseudoproxy Tests The GraphEM methodology is tested on synthetic proxies derived from a forced simulation of the NCAR CSM1.4 model (including volcanic and solar forcing) P p (s, t) =T (s, t)+ξ(s, t)/snr where ξ is a standard, uncorrelated Gaussian process Signal to noise ratio: SNR = ρ 1 ρ 2 MSE = i Test Statistic: (ŷ i y i ) 2 Case SNR = SNR = 1 SNR = 0.5 SNR = 0.25

17 Error reduction % MSE reduction, GraphEM - RegEM TTLS SNR = 0.5 >= <= 40 Julien Emile-Geay USC 2012

18 North American reconstructions (100 noise realizations) GGM+TTLS MSE improvement (%) 65 1 Global mean reconstructions TTLS GGM+TTLS Target Temperature deviation (C) Time (years) Substantial reduction in MSE Much improved risk properties Julien Emile-Geay USC 2012

19 Coral-based sea-surface temperature reconstructions 40 o N Proxy representation by age (47 records) 30 o N 20 o N 10 o N 0 o 10 o S 20 o S 30 o S 18 O Sr/Ca other 40 o S Most ancient age resolved Proxy availability over time # proxies O Sr/Ca other Time Julien Emile-Geay USC 2012

20 Bootstrap error estimates NINO3.4 reconstruction Temperature deviation (oc) Narrower confidence intervals, but 2 regularization problems persist RegEM TTLS 95% CI GraphEM TTLS 95% CI 6 # proxies available Time (year) Time Julien Emile-Geay USC 2012

21 Coral-based SST uncertainties: GraphEM TTLS Spatial estimate of the uncertainties t = 1650 t = 1675 t = 1700 t = t = 1750 t = 1775 t = 1800 t = t = 1850 t = 1875 t = 1900 t = t = 1950 t = 1975 t = 1995

22 Conclusions Paleoclimate Reconstructions High-dimensional, multivariate inference problem Should benefit from latest advances in statistics Gaussian Graphical Models Enable flexible covariance estimation, reduce errors Model selection, not regularization ( 2 still needed) Choice of graph: spatial truncation? Current and future developments GraphEM: Analysis of uncertainties (bootstrap interval coverage rate) Application to up-to-date proxy databases (coral, multiproxy) Bayesian Modeling Closed-form Bayes estimators with graphical covariance structure Incorporation into Bayesian hierarchical models

23 questions, comments, data, preprints:

24 Bayes Bayesian Hierarchical Models Pr(par obs) = Pr(obs par) Pr(par) Theorem Posterior Likelihood Prior 3 levels of conditioning: Y : Climate Process (s,t) W : Latent Process (s,t) (unobserved) Z : Observed Process (s,t) Scientific understanding can be encoded at the appropriate level p Y; W I;j ; W P;j ; qj Z I;j ; Z P;k ff ðyjqþg WI;j ; W P;k Y; q " Y N I h I;j ZI;jWI;j ; q #" Y N P h P;k ZP;kWP;k ; q # pðqþ: j ¼ 1 k ¼ 1 Posterior Likelihood Prior Tingley et al, 2012 Julien Emile-Geay IMSC11

25 Full Proxy Network for Climate Field Reconstruction (1138 records) 80 o N 60 o N 40 o N 20 o N 0 o 20 o S 40 o S 60 o S 80 o S 180 o W 120 o W 60 o W 0 o 60 o E 120 o E 180 o W # proxies Tree Ring Width Tree Ring MXD Ice core Coral Speleothem Documentary Sediment Composite Proxy availability over time Time