model selection, parameter and state estimation with uncertain...
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Model selection, parameter and state estimation withuncertain chronology
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston
Imperial College, UCLouvain, U. Sheffield, U. Nottinghammichel.crucifix@uclouvain.be
PAGES DAPS Louvain-la-Neuve 30 May 2017
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 1.
Plan
Introduction
Methodology
Application 1 : selection with known chronology
Application 2 : uncertain chronology
Discussion
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 2.
Ice ages
-800 -600 -400 -200 0
44
04
80
52
05
60
Mid
-Ju
ne
In
so
latio
n 6
5N
[
W/m
2]
-800 -600 -400 -200 0
2.0
3.0
4.0
Time [ka]
DS
DP
60
7
∂ 1
8-O
(p
er
mil)
A. L. Berger. In: Journal of the Atmospheric Sciences (1978). DOI:10.1175/1520-0469(1978)035<2362:LTVODI>2.0.CO;2, M. E. Raymo et al. In:Paleoceanography (1989)Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 3.
Ice ages as the output of a (forced, random) dynamicalsystem
Forcing (mid−June insolation at 65N)
time [ka]
A.U
.−800 −600 −400 −200 0
−2
02
Saltzman and Maasch 1990
time [ka]
Ice
volu
me
(A. U
.)
−800 −600 −400 −200 0
−1
1
Tziperman et al., 2006
time [ka]
Ice
volu
me
(1E
15 m
3)
−800 −600 −400 −200 0
020
40
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 4.
Bayesian Model Selection
Bayes Factor =P(Y1:M |Model1)P(Y1:M |Model2)
(1)
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 5.
Plan
Introduction
Methodology
Application 1 : selection with known chronology
Application 2 : uncertain chronology
Discussion
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 6.
Directed Acyclic Graph (DAG)
∫ψ
��
H1:m
��
{∫χ,
∫ε1:m}
��∫X0 // X (T ) //
$$
X1:m // Y1:m
F (T )
;;
∫ωp(T )
OO
H(T )
OO
��
{∫ξ,∫ωc(T )}oo
T (Hj)
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 7.
Sequential Monte-Carlo Square (SMC2) [Chopin et al. 2012]
X(1)0 , θ(1)
X(2)0 , θ(2)
X(3)0 , θ(3)
X(4)0 , θ(4)
X(5)0 , θ(5)
ω(1)1
X(1)1
observations Y1
ω(2)1
Y2 Y3
ω(2)1 X
(2)1 ω
(2)2
ω(3)1 X
(3)1 ω
(3)2
ω(4)1 X
(4)1
ω(4)2
ω(5)1 X
(5)1 ω
(5)2
w(i)3
P(Y1...3|M) ∝ ∑w
(i)3
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 8.
Testing
I Not shown here for lack of timeI But of course all methods need to be tested with simulation studies
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 9.
Plan
Introduction
Methodology
Application 1 : selection with known chronology
Application 2 : uncertain chronology
Discussion
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 10.
Three ice age models
Selecting between competing models
CR12
dX1 = ���0 + �1X1 + �2
�X1 � X 3
1
�+ �X2 + F (�p , �e)
�dt + �1dW1
dX2 = ↵�
✓X1 + X2 � X 3
2
3
◆dt + �2dW2
SM91
dX1 = � (X1 + X2 + vX3 + F (�p , �e)) dt + �1dW1
dX2 =�rX2 � pX3 � sX 2
2 � X 32
�dt + �2dW2
dX3 = �q (X1 + X3) dt + �3dW3
T06
dX1 = ((p0 � KX1) (1 � ↵X2) � (s + F (�p , �e))) dt + �1dW1
X2 : 0 ! 1 when X1 exceeds some threshold Xu
X2 : 1 ! 0 when X1 decreases below Xl
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 11.
Selection Table: The selected models depends on datingassumpions}
Results: ODP677
Model EvidenceODP677: H07(unforced) ODP677: LR04(forced)
SM91 Forced 4.0 ⇥ 1024 1.1 ⇥ 1028
Unforced 3.5 ⇥ 1026 1.6 ⇥ 1018
T06 Forced 3.3 ⇥ 1025 4.5 ⇥ 1029
Unforced 1.7 ⇥ 1028 3.3 ⇥ 1021
PP12 Forced 1.5 ⇥ 1022 1.8 ⇥ 1034
The dating method applied changes the answer
Using Huybers’ non-orbitally tuned data, we find evidence in favour of theunforced T06 model.
Using Lisiecki’s orbitally tuned data, we find strong evidence for PP12 atuned model (PP12)
Moreover, orbitally tuned data leads us to strongly prefer the orbitally tunedversion of each model (and vice versa)
The age model used to date the stack (often taken as a given) has a strong
e↵ect on model selection conclusions
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 12.
Plan
Introduction
Methodology
Application 1 : selection with known chronology
Application 2 : uncertain chronology
Discussion
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 13.
Model
I Mean sedimentation rate + random component
dH = −µsdT + σdW (2)
I Accounts for sediment compaction similar to Huybers (2007)I Acconts for dissolution events (age grows monotonously)
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 14.
Account for dissolution events
Age Model
Begin by considering a simple model for sediment (S) accumulation:
dS = µsdT + �sdW
mean sediment rate is constant, stochastic contributions account for periods of errosionetc. We assume that a sediment core is constructed from this model. To model timevariation according to core depth we also need to consider how a core is sampled:
200 202 204 206 208 210
6.8
6.7
6.6
6.5
Time (kyr BP)
Dep
th (m
)
Figure 1: Demonstration of core sampling. Line represesents change in sediment overtime, horizontal lines are sampling depths, and vertical lines are the sampled times.
Firstly, we know that the top of the core (which we will assume is the final observation)is sampled in the present, such that TM = 0 at HM = 0 (where depth is represented byH). The change in depth is the negative of the change in sediment accumulation:
dH = �µsdT + �sdW
Slices are then taken through the core at specific depths. According to our depth modelthere may have been multiple times in the past at which this depth was reached. We areinterested in the most recent time; if sediment accumulated beyond this point before thattime then the information has been eroded away. In other words we have a first passagetime problem, which is solvable for our linear model. Given depth Hm the distributionof the first passage time of Hm�1 is inverse Gaussian:
Tm�1 ⇠ IG
✓Tm � Hm�1 � Hm
µs
,(Hm�1 � Hm)2
�2s
◆
At this point we should also consider accounting for compaction. Compaction is oftenmodelled as the expulsion of water due to the load of the above sediment. This can
2
There may have been multiple times when a certain depth was reached: themost recent time is the age of that slice, i.e., it is a first passage problem.Given (Tm, Hm), then Tm�1 is the first passage time of Hm�1 with
Tm�1|Tm ⇠ IG
✓Tm � Hm�1 � Hm
µs,(Hm�1 � Hm)2
�2s
◆
We then add a model to account for compaction in the core, and apply Bayes
theorem to find ⇡(Tm|Tm�1) so that we can run the model forward in time
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 15.
Age Estimates
ODP677
δ18O
34
5−
400
Age
ano
mal
y
32 28 24 20 16 12 8 4 0
Depth (m)
ODP846
δ18O
34
5−
400
Age
ano
mal
y
28 24 20 16 12 8 4 0
Depth (m)
— : as used for Lisiecki & Raymo 2005— : as used for Huybers 2007
N. J. Shackleton, A. Berger, and W. R. Peltier. In: Transactions of the Royal Society ofEdinburgh-Earth Sciences (1990), A. Mix, J. Le, and N. Shackleton. In: Proceedings of theOcean Drilling Program (1995). DOI: 10.2973/odp.proc.sr.138.160.1995
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 16.
State Estimate (consistent for two different cores)
ODP677
800 600 400 200 0
−4
−2
01
23
Time (kya)
Ice
Vol
ume
Ano
mal
y
ODP846
800 600 400 200 0
−4
−2
01
23
Time (kya)
Ice
Vol
ume
Ano
mal
y
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 17.
Plan
Introduction
Methodology
Application 1 : selection with known chronology
Application 2 : uncertain chronology
Discussion
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 18.
Discussion
I Elementary example
I only one type of dataI one coreI about one thousand data pointsI tested for robustness to prior and implementation details
I Yet highly computationaly expensive already (Bayesian framework)
I 1 week for one model
I Joint age-state-parameter necessary for model selection
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 19.
References:
I N. Chopin, P. E. Jacob, and O. Papaspiliopoulos. In: Journal of theRoyal Statistical Society: Series B (Statistical Methodology)(2013). DOI: 10.1111/j.1467-9868.2012.01046.x
I J. Carson et al. In: Journal of the Royal Statistical Society:Series C (Applied Statistics) (2017). DOI: 10.1111/rssc.12222
see http://perso.uclouvain.be/michel.crucifix for more ondynamical systems of ice ages.
Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston, 30 mai 2017, p. 20.
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