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