1. Palaeoclimate Reconstruction: Modelling Temporal Uncertainty Many collaborators: Stats: Bhattacharya, Gelfand, Salter-Townshend, Parnell, Whiley, Wilson, others Botany: Allen, Huntley, Mitchell, Support: SFI and previously by Enterprise Ireland and PRTLI Glendalough Co Wicklow

2. Trinity College Dublin Bay

3. Sunday Times “Gobsmacking” says Haslett “Not what I said” says Haslett

4. Courtesy of Sunday Times graphics dept

5. Reconstruction of GDD5

7. core samples mult. counts by taxa

8. Changing pollen composition 10,000 BP Use Matts pollen diagram Observed pollen proportions vs 14 C y BP Pollen composition changes  Climate changes Recent Shallow Ancient Deep

9. Science Changing pollen composition in carefully selected sites Reflects changing vegetation, which Reflects changing climate –GDD5 Growing Deg Days > 5 o –MTCOMean Temp Coldest Month whence Can reconstruct climate quantitatively Can reduce uncertainty about past climate GDD5 eg Avg temp on four successive days 58410 Excess over 50305 ThusGDD5 = 8

10. Data Pollen Data – multivariate counts –14 distinguishable taxa –115 samples at Sluggan Moss, Lough Neagh –115 depths of which 32 radiocarbon dated –Climate unknown (2D GDD5, MTCO) Modern data –7815 modern sites –Counts known – surface pollen (depth = 0) –Climate known

11. Statistical Tasks Aspects Pollen response to climate –Use modern data –Transfer functions/response surfaces –Climate one sample at a time Climate smoothness in time –Climate history –Greenland ice cores –Dating uncertainties

12. Statistical Tasks Given Modern data (p m,c m ) (7815 records); and Fossil data (p f,?) at one depth seek post dist π(c f | p m,c m,p f ) at that depth Additionally, given ‘Climate smoothness’; and Fossil data (p f,?, d) at 115 depths Radiocarbon dating info at 32 depths seek post dist π(c f | p m,c m,p f ) entire climate history

13. Glendalough Modern Training Data Data on modern pollen compositions p m 7815 sites in Eur/ N. America Modern climate c m known. Hence relationship π(p m | c m ) Adopt for fossil data π(p f | c f ) Sluggan Moss

14. Physical and (2D) climate spaces Climate space grid Glendalough An extreme climate? Impossible climates? Unknown climates? Growing Deg Day > 5 o Mean Temp Coldest Month 7815 data locations - grey points. Computational grid - black points 7815 data locations

15. Pollen response to (2D) climate π(p | c) pdf p14 dim comp vect c2 dim climate (here) Latent Gaussian proc mixture of multinomials zero inflation MCMC MTCO GDD5 Small change in climate c  Small change in vegetation  p = p(c) smooth multivariate function 7815 data locations - grey points. Computational grid - black points Two stage implementation 1MCMC creates/stores many realisations of p(c) 2(a) Draw one at random 2(b)MCMC Climate recon 2(c)Repeat (a,b)

16. Reconstruction of GDD5 Here depth to radiocarbon age presumed known Later address dating uncertainty Note dates in Radio-carbon YBP One sample at a time

17. Post Dist c f given p f Given vector of counts at given depth, whence p f Find π( GDD5 f, MTCO f | p f ) by MCMC for each depth Here concentrate on GDD5 eg depth 10 m

18. Differential Response to 1D Climate 05001000150020002500 C being 1D Climate A B Inverse relationship Model taxon productivity response to ‘climate’ Multi-modal climate posteriors natural Toy example; two taxa one climate dimension Prop of A high Post prob C given A high 05001000150020002500 Prop of A low 05001000150020002500 Post prob C given A low

19. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

20. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

21. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

22. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

23. Implied Climate Histories Pointwise realisations of climate Climate apparently volatile

24. Climate Smoothness Climate changes δ i = c(time i ) - c(time i-1 ) –Mostly small/sometimes large “smooth” –Depends on increments |time i - time i-1 | –Reject (most) volatile climates Issues –How smooth? Greenland ice cores –Uncertainty in 14 C dating? Random chronology

25. Temporal uncertainty 115 samples at Sluggan Moss For 115: core depths d i For 32: reported 14 C ages y i ± σ i Seekθ i true calendar age θ i all d i “ chronology model” –r(θ) 14 C calibration curve y i ~ N( r(θ i ), σ i 2 )(outliers, so long tails) r(θ)~ N( μ(θ), σ 2 (θ)) prior –Piecewise constant sedimentation rate Gaussian random walk

26. Chronology Given complete knowledge of sedimentation history, age may be determined from depth Calendar age θ d = depth of accumulated sediment But Only know 14 C age at some depths Seek realisations of sediment history, conditional on data Prior: Gaussian random walk with drift constrained to be monotone Piecewise const iid sedimentation rate

27. Temporal uncertainty: single dated sample Lab report 3180 ± 30 Implied post dist Schematic of Bayesian 14 C calibration curve Buck

28. Modelling chronology

29. Temporal uncertainty: all dated samples Prior: Discrete time (20 year intervals) Random Walk with drift (monotone) Draw random θ i | y i σ i each of 32 d i –Order constraint θ i > θ k if d i > d k Stoch. interpolation to undated samples –Sample θ m (undated)| θ i (dated), all depths

30. Sample from joint post dist (θ 1, θ 2,… θ 115 ) –know marginal post dist for each of 32 from radiocarbon dating –date ordering must follow depth ordering Model depth/date relationship θ(d) –prior – monotone random walk with drift Temporal uncertainty: all samples jointly Age θ d high low Piece-wise constant sedimentation rate

31. Draw set of random dates for 14 C dated samples x x x x Calendar age θ Depth d Realisations of order constrained radio-carbon dates drift

32. Complete random chronology x x x x Realisations of order constrained stochastic chronology, conditional on radio-carbon derived dates Monotone random walk with drift Depth d Calendar age θ Given set of depths Realisation of a set of calendar dates

33. Climate Smoothness Climate changes δ i = c(time i ) - c(time i-1 ) –Mostly small/sometimes large –Depends on increments |time i - time i-1 | Prior for smoothness  rejection of histories with large |δ i |  implicit smoothing / borrowing strength Issues –Prior for δ i long tail random walk

34. Climate over 100,000 years Greenland Ice Core Temporal structure for climate (20 yr. resolution) Frequent small changes, occasional large changes Ice Core data time series Greenland Ice Core Data 10,000 year intervals Irish study period Oxygen isotope – proxy for Greenland temp

35. Climate over 100,000 years Greenland Ice Core Normal prob plot First diffs

36. Climate Smoothness Ice Core data time series Greenland Ice Core Data 10,000 year intervals Long-tailed Random Walk Prior Model δ = c(t) - c(t-20) as iid NIG Normal Inverse Gamma Random Walk

37. Sampling Climate Histories Given –Realisation of pollen response surfaces –Sample pollen at each of 115 depths –Realisation of complete chronology 115 dates given 14 C dates for 32 samples –Model for climate smoothness Sample realisations of climate at 115 dates  Sample climate history every 20 years

38. Modelled Climate Histories Climate Smooth mostly

39. Modelled Climate Histories Climate Smooth mostly

40. Modelled Climate Histories Climate Smooth mostly

41. Modelled Climate Histories Climate Smooth mostly

42. Modelled Climate Histories Climate Smooth mostly

43. Better Reconstruction of GDD5

44. Reconstruction of GDD5 Note dates in Radio-carbon YBP

45. Irregular time intervals –Time-incs k steps of 20 yrs approx ~ t ν (k) –ν (k) chosen wrt kurtosis –Irreg time-inc approx likelihood П t ν (k) –Stochastic interpolation Stochastic Smoothness Regular time intervals –(Coupled) random walk 20 year incs ~ t 8 –Regular time-inc likelihood П t 8

46. Monte Carlo Modules Resp Surface Random set of surfaces Modern data Climate and pollen Random set of 115 dates Depths and radiocarbon dates DatingRandom Climate History length 115 Temporal Stochastic Smoothness Stochastic Interpolation Random Climate History 12,600y by 20y step Summaries Fossil Pollen Point wise Recon- struction Random point-wise histories

47. Rapidity of Climate Change Prob (GDD5 change in 20 years > 500) alt summary of climate histories

48. Next Stages Multiple sites –Joint reconstruction of two sites –Probable synchronicity of climate change Borrow more strength –for dates, for climate smoothness –Joint reconstruction of many sites in space More climate dimensions and taxa –Many high dim response surfaces Other proxies, covariates Confront General Circ. Models

49. Methodological Issues MCMC - the way forward? –Speed and convergence –Approximations esp for response surfaces –Model checking and model choice Technical issues –Zero inflation –Fast high-dim non-parametric smoothing –Long tailed space-time prior for climate –Latent (mixtures of) Gaussian processes