This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
• 115 samples at Sluggan MossFor 115 : core depths di
For 32 : reported 14C ages yi ± σi
• Seek θi true calendar age θi all di
“chronology model”– r(θ) 14C calibration curve
• yi ~ N( r(θi), σi2) (outliers, so long tails)
• r(θ)~ N( μ(θ), σ2(θ)) prior
– Piecewise constant sedimentation rate• Gaussian random walk
ChronologyGiven complete knowledge of sedimentation history, age may be determined from depth
Calendar age θ
d = depth of accumulated sediment
ButOnly know 14C age at some depthsSeek realisations of sediment history,conditional on dataPrior: Gaussian random walk with driftconstrained to be monotonePiecewise const iid sedimentation rate
Temporal uncertainty:single dated sample
Lab report3180 ± 30
Implied post dist
Schematic of Bayesian 14C calibration curveBuck
Temporal uncertainty:all dated samples
Prior:Discrete time (20 year intervals)Random Walk with drift (monotone)
• Draw random θi | yi σi each of 32 di
– Order constraint θi > θk if di > dk
• Stoch. interpolation to undated samples– Sample θm (undated)| θi (dated), all depths
Draw set of random dates for 14C dated samples
x
x
x
x
Calendar age θ
Depth d
Realisations of order constrained radio-carbon dates
drift
drift
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
Climate Smoothness
• Climate changes δi = c(timei) - c(timei-1)
– Mostly small/sometimes large
– Depends on increments |timei - timei-1|
• Prior for smoothness rejection of histories with large |δi |
implicit smoothing / borrowing strength
• Issues– Prior for δi long tail random walk
Climate over 100,000 yearsGreenland Ice Core
Temporal structure for climate (20 yr. resolution)Frequent small changes, occasional large changes
Ice Core data time series Greenland Ice Core Data10,000 year intervals
Irish study periodOxygen isotope – proxy for Greenland temp
Ice Core data time series Greenland Ice Core Data10,000 year intervals
Long-tailed Random Walk Prior
Model δ = c(t) - c(t-20) as iid NIGNormal Inverse Gamma Random Walk
Sampling Climate Histories
Given – Realisation of pollen response surfaces– Sample pollen at each of 115 depths– Realisation of complete chronology
• 115 dates given 14C dates for 32 samples
– Model for climate smoothness
Sample realisations of climate at 115 dates
Sample climate history every 20 years
Modelled Climate Histories
ClimateSmoothmostly
Modelled Climate Histories
ClimateSmoothmostly
Modelled Climate Histories
ClimateSmoothmostly
Modelled Climate Histories
ClimateSmoothmostly
Modelled Climate Histories
ClimateSmoothmostly
Better Reconstruction of GDD5
Reconstruction of GDD5
Note dates in Radio-carbon YBP
Monte Carlo ModulesResp Surface
Randomset of surfaces
Modern dataClimate and pollen
Random set of 115 dates
Depths andradiocarbon
dates
Dating
RandomClimate History
length 115
Temporal Stochastic
Smoothness
Stochastic Interpolation
Random Climate History 12,600y by
20y step
Summaries
Fossil Pollen
Point wise Recon-
struction
Randompoint-wisehistories
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
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