Bayesian PalaeoClimate Reconstruction from proxies: Framework Bayesian modelling of space-time processes General Circulation Models
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) 14000 years at 20 year intervals on 50 x 50 grid = 5250000 dim random variable
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) = 5250000 Sample from Pr(C proxy data)
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)...
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)!...
Prior Pr(C) Descriptive {C(x,t)} stochastically smooth eg Gaussian process eg Heavy tail Random Walk Physical {C(x,t)} satisfies GCM equations
Sampling the Palaeoclimate Samples of C(x,t) (sets of 5250000 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
Modelled Climate Histories; eg at one site summary MTCO t t = 20,40, 14000 700 marginal summaries Multi-modal Highest Posterior Density Regions
Sampled climate histories
Sampled climate histories
Sampled climate histories
Sampled climate histories
Sampled climate histories smooth
Modelled Climate Histories at one site MTCO t t = 20,40, 14000 700 marginal summaries Highest Posterior Density Regions Multi-modal Other summaries Eg Max change in 20 years
Alder response Alder percentage 100 100 50 50 0 0 0 2500 5000-40 -20 0 20 GDD5 MTCO Ireland, currently
Alder mean response parameter 20 0 MTCO -20-40 0 2500 5000 GDD5 High alder count! about (1600,6) But large noise parameters!
Multivar non para regression Response surfaces x 1 (c), x 2 (c),... Modern data, Zero-inf. Poisson Gaussian prior 2D climate
Zero Inflated Poisson 1D climate Latent x j (c);poisson! = e x(c) ; Pr(0)= " $ $ # ex(c) 1+e x(c) % ' ' & (
Climate inf, given counts y and x(c) count=hi count=lo Bimodal Likelihood of obs count, for every possible c
Climate inf, given counts y only count=hi count=lo Likelihood of obs count, for every possible c
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
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
Why Bayes? Why? Need joint statements of uncertainty Multiple sources of uncertainty Weak priors eg MTCO at 10000BP Strong Priors eg stochastically smooth Flexible
Why Flexible? Non-normal Multi-modal Zero-Inflation Presence/Absence Hierarchical Missing data Constraints monotonicity in chronologies Stochastically smooth in space time
Why Bayes? Problems New, non-standard, software! Display and publication of data Solutions Use Monte Carlo, modularise, software " Bchron R software, Parnell 2008
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
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
Using joint prior information Chronology example: Age at depth 1 5000 ± 500 (Normal model, SD = 250) Age at depth 2 5300 ± 600 (SD = 150) Info : (Depth 2 > Depth 1)! (Age 1 > Age 2) Algorithm: rejection sampling reject if inverted
Using joint prior information Monte Carlo Samples Accept? Depth 1 Depth 2 1 4784 5565 Y 2 5050 5083 Y 3 5092 5297 Y 4 4690 5172 Y 5 4926 5260 Y 6 4924 5118 Y 7 5211 5438 Y With constraint: Age1 Age2 Without Age1 Age2 Mean 4901 5213 4901 5213 SD 196 97 250 150
Sampling, using joint information Post Dist [ c proxies]! = Model Prob[ proxies c] " Prior[c] Prior[c] - Joint prior for c = { c, c,... c,..} 1 2 1 2 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
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
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!!!!
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
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
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
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
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
Vision Multiple-proxy types Space-time reconstructions movies arbitrary resolution Noisy if weak signal One model Many modules GCM comparisons
GCM comparisons Different spatio-temporal scales Modelling dynamics Uncertainties