Bayesian PalaeoClimate Reconstruction from proxies:

Transcription

1 Bayesian PalaeoClimate Reconstruction from proxies: Framework Bayesian modelling of space-time processes General Circulation Models

2 Space time stochastic process C = {C(x,t) = Multivariate climate at all locations x and at all times t} Eg 3 dims (Growing degree days, Mean temp coldest month, AET/PET) years at 20 year intervals on 50 x 50 grid = dim random variable

3 Inference on C C! proxies But other influences and meas error Forward model Pr( proxy data C) modern and ancient Inverse model Pr(C proxy data) " Pr( proxy data C)Pr(C) dim(c) = Sample from Pr(C proxy data)

4 Modules for Inference on C Decompose Pr( proxy data C) = Pr( proxy data old C)Pr( proxy data new C) Pr( proxy data old C) = Pr( pollen data old C)Pr(diatom data old C)...

5 Modules for Inference on C Decompose Pr( pollen data old C) = Pr( pollen data old,site1 C)! Pr( pollen data old,site2 C)! Pr( pollen data old,site3 C)!...

6 Prior Pr(C) Descriptive {C(x,t)} stochastically smooth eg Gaussian process eg Heavy tail Random Walk Physical {C(x,t)} satisfies GCM equations

7 Sampling the Palaeoclimate Samples of C(x,t) (sets of random nums) = plausible equally likely stories of what happened consistent with data & theory from which can (eg) Construct space-time averages (eg W Europe, 500y) Time series at one location Research Dynamics, Extremes, Comparisons

8 Modelled Climate Histories; eg at one site summary MTCO t t = 20,40, marginal summaries Multi-modal Highest Posterior Density Regions

9 Sampled climate histories

10 Sampled climate histories

11 Sampled climate histories

12 Sampled climate histories

13 Sampled climate histories smooth

14 Modelled Climate Histories at one site MTCO t t = 20,40, marginal summaries Highest Posterior Density Regions Multi-modal Other summaries Eg Max change in 20 years

15 Alder response Alder percentage GDD5 MTCO Ireland, currently

16 Alder mean response parameter 20 0 MTCO GDD5 High alder count! about (1600,6) But large noise parameters!

17 Multivar non para regression Response surfaces x 1 (c), x 2 (c),... Modern data, Zero-inf. Poisson Gaussian prior 2D climate

18 Zero Inflated Poisson 1D climate Latent x j (c);poisson! = e x(c) ; Pr(0)= " $ $ # ex(c) 1+e x(c) % ' ' & (

19 Climate inf, given counts y and x(c) count=hi count=lo Bimodal Likelihood of obs count, for every possible c

20 Climate inf, given counts y only count=hi count=lo Likelihood of obs count, for every possible c

21 Climate likelihoods, given counts y Implied climate likelihoods given data and climate resp surfaces Taxon A Taxon B Taxon C All taxa marginal joint 1D climate

22 Climate history; joint inference Implied joint climate likelihoods given data + Joint prior reflecting smooth climate 28 taxa at Depth 1 Depth 2 Depth 3 1D climate Regular depths! Irregular uncertain times

23 Why Bayes? Why? Need joint statements of uncertainty Multiple sources of uncertainty Weak priors eg MTCO at 10000BP Strong Priors eg stochastically smooth Flexible

24 Why Flexible? Non-normal Multi-modal Zero-Inflation Presence/Absence Hierarchical Missing data Constraints monotonicity in chronologies Stochastically smooth in space time

25 Why Bayes? Problems New, non-standard, software! Display and publication of data Solutions Use Monte Carlo, modularise, software " Bchron R software, Parnell 2008

26 Monte Carlo Generate multiple random copies { } at one site of (eg) C = c 1,c 2,...c t,... each probabilistically consistent jointly with data, information Hence form multiple random copies of c t! est marginal dist of c t " c t"20! est marginal dist of diff of max(c t,c t"20 )! est marginal dist of max

27 Monte Carlo Rejection Sampling 1 Generate multiple random copies of { c 1,c 2,...c t,...}from prior 2 Compute likelihood L(data c t ) 3 Reject c t with high prob if L(data c t ) is low low prob if L(data c t ) is high 4 Hence copies probabilistically consistent jointly with data and prior

28 Using joint prior information Chronology example: Age at depth ± 500 (Normal model, SD = 250) Age at depth ± 600 (SD = 150) Info : (Depth 2 > Depth 1)! (Age 1 > Age 2) Algorithm: rejection sampling reject if inverted

29 Using joint prior information Monte Carlo Samples Accept? Depth 1 Depth Y Y Y Y Y Y Y With constraint: Age1 Age2 Without Age1 Age2 Mean SD

30 Sampling, using joint information Post Dist [ c proxies]! = Model Prob[ proxies c] " Prior[c] Prior[c] - Joint prior for c = { c, c,... c,..} Eg if c, c,... c denote climate at times 1,2,... t,... t then c will usually be more like c than 2 3 { c, c,... c,..} is stochastically smooth 1 2 t Prior: time series model eg Random Walk Prior ties things together t c 20

31 Posterior Dists Probabilistically consistent with data & info Random samples {,,...,..}from c = c1 c2 c t Post Dist [ c proxies]! Likelihood [ proxies c] " Prior[c] = Model Prob[ proxies c] " Prior[c] Inverse model! Forward model " Prior[c] Modules: Decomposing and Integrating

32 Modules Decomposing the Likelihood via Conditional Independence Typically : at least as an approximation Model Prob[ all proxies c] = Product of Probs[ each proxy type c] = Probs[ pollen c]! Probs[ diatoms c]... " separate modules!!!!

33 Modules One sample at a time With multivar count data y Compute Prob[ c counts] for all climates for each sample separately Fast approximations, no Monte carlo

34 Rejecting Climate Histories Algorithm in principle MCMC just efficiency Generate entirely random histories Reject with hi prob those that are improbable, given data&info Reject with lo prob those that are quite probable Accept the remainder

35 Temporal Smoothing Module Temporal smoothing module Generate random histories for each sample separately Reject with hi prob those that are not smooth Reject with lo prob those that are smooth Accept the remainder MCMC just efficiency

36 Multiple Cores in Space Sample space-time histories Random movies Consistent with data and models Reject movies with hi prob if with hi prob if not spatio-temporally smooth with lo prob if spatio-temporally smooth But Different and irregular depths Different, irregular and uncertain times

37 Chronology Module Known depths, uncertain dates Randomly generate dates for each sample consistent with depths & 14 C via rejection sampling (Round to nearest 20 years) info consistent with monotone order

38 Vision Multiple-proxy types Space-time reconstructions movies arbitrary resolution Noisy if weak signal One model Many modules GCM comparisons

39 GCM comparisons Different spatio-temporal scales Modelling dynamics Uncertainties