Post on 26-Sep-2019
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
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)
Inference on C
Decompose
Pr( proxy data | C)
= Pr( proxy dataold | C)Pr( proxy datanew | C)
Pr( proxy dataold | C)
= Pr( pollen dataold | C)Pr(diatom dataold | C)...
Modules for Inference on C
Decompose
Pr( pollen dataold | C)
= Pr( pollen dataold ,site1 | C)!
Pr( pollen dataold ,site2 | C)!
Pr( pollen dataold ,site3 | C)!
.........
Modules for Inference on C
Descriptive
{C(x,t)} stochastically smooth
eg Gaussian process
eg Heavy tail Random Walk
Physical
{C(x,t)} satisfies GCM equations
Prior Pr(C)
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
MTCOt t = 20,40,…14000
700 marginal summaries
Multi-modal
Highest Posterior Density Regions
summary
Sampled climate histories
Sampled climate histories
Sampled climate histories
Sampled climate histories
Sampled climate histories
smooth
Modelled Climate Histories
at one site
MTCOt 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
0
50
100
0 2500 5000
GDD5
Ireland, currently
0 20-20-40
MTCO
0
50
100
Alder mean response parameter
GDD5
0 2500 5000
0
20
-20
-40
MTCO
But large noise parameters!
High alder count ! ‘about’ (1600,6)
Multivar non para regression
Response surfaces
x1(c), x2(c),........
Modern data,
Zero-inf. Poisson
Gaussian prior
2D climate
Zero Inflated Poisson
Latent x j(c);Poisson! = ex(c); Pr(0)= ex(c)
1+ex(c)
"
#$$
%
&''
(
1D climate
Climate inf, given counts y and x(c)
Likelihood of obs count, for every possible c
count=lo
count=hi
Bimodal
Climate inf, given counts y only
Likelihood of obs count, for every possible c
count=lo
count=hi
Climate likelihoods, given counts y
Implied
climate
likelihoods
given
data
and
climate
resp
surfaces
marginal
joint
Taxon A
Taxon B
Taxon C
All taxa
1D climate
28 taxa at
Depth 1
Climate history; joint inference
Implied
joint
climate
likelihoods
given
dataDepth 2
Depth 3
Joint prior
reflecting
“smooth”
climate
+
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
Generate multiple random copies
of (eg) C = c1,c
2,...c
t,..............{ } at one site
each probabilistically consistent jointly
with data, information
Hence form multiple random copies
of ct
! est marginal dist
of ct" c
t"20! est marginal dist of diff
of max(ct,c
t"20) ! est marginal dist of max
Monte Carlo
1 Generate multiple random copies
of c1,c
2,...c
t,..............{ }from prior
2 Compute likelihood L(data | ct)
3 Reject ct with high prob if L(data | c
t) is low
low prob if L(data | ct) is high
4 Hence copies probabilistically consistent
jointly with data and prior
Monte Carlo Rejection Sampling
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)
Using joint prior information
Algorithm: rejection sampling – reject if ‘inverted’
Using joint prior information
With constraint: Without
Age1 Age2 Age1 Age2
Mean 4901 5213 4901 5213
SD 196 97 250 150
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
Monte Carlo Samples
1 2
1 2
2 3
Post Dist [ | ]
= Model Prob[ | ] Prior[c]
Prior[c] - prior for { , ,... ,..}
Eg if , ,... denote climate at times 1,2,.... ,...
then will than
t
t
c proxies
proxies c
c c c c
c c c t
c usually be more like
J
c c
oint
!
"
=
20
1 2{ , ,... ,..} is
Prior: time series model eg Random Walk
tc c c stochastically smooth
Prior ties things together
Sampling, using joint information
1 2
with data & info
Random samples { , ,... ,..}from
Post Dist [ | ]
Likelihood [ | ] Prior[c]
= Model Prob[ | ] Prior[c]
Inverse model Forward model
tc c c c
c pro
Probabilistically consisten
xies
proxies c
proxies c
t
=
!
"
"
! Prior[c]"
Posterior Dists
Modules: Decomposing and Integrating
Typically :
Model Prob[ | ]
Probs[ | ]
Probs[ | ] Probs[ | ].
..
separate modules!!!!
at least as an approximation
all proxies c
each proxy type c
pollen c diatoms
Product
c
of=
= !
"
ModulesDecomposing the Likelihood
via Conditional Independence
With multivar count data
Compute Prob[ | ] for all climates
for each sample separately
Fast approximations, no Monte carlo
y
c counts
ModulesOne sample at a time
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 MCMC just efficiency
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
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
via rejection sampling
Known depths, uncertain dates
Randomly generate dates for each sample
C
(Round to nearest 20 years)
14consistent with depths & info
consistent with monotoneorder
Chronology Module
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