model selection, parameter and state estimation with uncertain...

Model selection, parameter and state estimation withuncertain chronology

Jake Carson, Michel Crucifix, Richard Wilkinson, Simon Preston

Imperial College, UCLouvain, U. Sheffield, U. [email protected]

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


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.

http://dx.doi.org/10.1175/1520-0469(1978)035<2362:LTVODI>2.0.CO;2

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


Bayesian Model Selection

Bayes Factor =P(Y1:M |Model1)P(Y1:M |Model2)

(1)


Plan

Introduction

Methodology



Discussion


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)


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


Testing

I Not shown here for lack of timeI But of course all methods need to be tested with simulation studies


Plan

Introduction

Methodology



Discussion


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


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


Plan

Introduction

Methodology



Discussion


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)


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


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


http://dx.doi.org/10.2973/odp.proc.sr.138.160.1995

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


Plan

Introduction

Methodology



Discussion


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


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.


http://dx.doi.org/10.1111/j.1467-9868.2012.01046.x

http://dx.doi.org/10.1111/rssc.12222

http://perso.uclouvain.be/michel.crucifix

model selection, parameter and state estimation with uncertain...

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