1 Monitoring Atmospheric Chlorofluorocarbons by the Longitudinal Bent-Cable Model S.A. Khan, G. Chiu...

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Monitoring Atmospheric Chlorofluorocarbons by the Longitudinal Bent-

Cable Model

S.A. Khan, G. Chiu* and J.A. Dubin

TIES 2009* presenter

2

Outline Introduction CFC-11 Data Model Inference Results Further Extension of the Methodology Limitations

3

Introduction

Time

CF

C-1

1 (

in p

pt)

1988 1990 1992 1994 1996 1998 2000

25

02

55

26

02

65

27

02

75

Inco

min

g P

has

e

- +

CTP: the point at which it took a downturn from an increasing trend

Transition period

Outgoing Phase

Figure 1: Characterizing a trend

0 +

1 t i

(0 -

2 ) +

(1 +

2 ) ti

of shock-though data by the bent-cable function

Concentration of CFC-11 in response to the Montreal Protocol’s ban on CFC products (monitored from Mauna Loa)

Shock-through data – a trend characterized by a change due to a shock (the Montreal Protocol)

4

Introduction (cont’d) Bent-cable function (Chiu, Lockhart & Routledge, 2006)

f(xi, , ) = 0 + 1 ti + 2 q(ti, ),

where = (0 , 1, 2), = (, ),

q(ti, ) = ,

Bent-cable Regression: yi = f(ti, , ) + i

i iid (Chiu, Lockhart & Routledge, 2006, JASA) i AR(p) (Chiu and Lockhart, revisions submitted)

• R Package ‘bentcableAR’ handles both

}γτt{I)τt(}γ|τt{|Iγ4

)γτt(iii

2i

5

Introduction (cont’d) We have extended the bent-cable

regression for longitudinal data using

random coefficients and within-individual

noise that is AR(p), p 0

We have applied our methodology to

CFC-11 data monitored from different

stations all over the globe (Khan, Chiu &

Dubin, to appear in CHANCE, 2009)

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Skin Cancer and Cataracts

Damage to Plants

Reduction of Organisms in the Ocean’s Photic

Zone

Natural (followed by a

natural recovery)

Human Activities(e.g. use of

CFCs)

Reduction of Ozone Layer in the Upper

Atmosphere

Increased UV Exposure

CFC-11 Data

7

Banned globally by the 1987 Montreal Protocol

CFC-11 Data (cont’d)CFCs (11, 12, 113, 114, 115)

CFC-11: One of the most dangerous CFCs to reduce the ozone layer in the atmosphere (ODP = 1)

Nontoxic, nonflammable chemicals containing

atoms of carbon, chlorine and fluorine

Used in air conditioning/cooling units,

and aerosol propellantsprior to the 1980’s

DestroyOzone

8Monitoring stations of CFCs all over the globe (Data collected by NOAA/ESRL global monitoring division and ALE/GAGE/AGAGE global network program)

Cape Grim,

Tasmania

Mauna Loa,

Hawaii

Cape Matatula, American

Samoa

Niwot Ridge,

Colorado

Pt. Barrow, Alaska

South Pole,

Antarctica

Mace Head,

Ireland

Ragged Point,

Barbados

CFC-11 Data (cont’d)

9Time

CF

C-1

1(i

n p

pt)

1988 1990 1992 1994 1996 1998 2000

23

02

40

25

02

60

27

02

80

BarrowCape MatatulaMauna LoaSouth PoleNiwot RidgeMace HeadCape GrimRagged Point

CFC-11 profiles of eight stations (monthly mean data)

What were the rates of change before and after the transition period?

How long did it take to show an obvious decline?

What was the CTP at which the trend went from increasing to decreasing?

CFC-11 Data (cont’d)

10

ModelLevel 1

yij = fij + ij,

yij = ij + uij, j = p+1, …, ni

Yij| yi1, …, yip, i, i, ,

Yi(2)| yi

(1), i, i, , ~ MVN(i, Ii),

where, i = (i,p+1, … , )'

),0(N~u ,u 2uiijij

p

1kkj,ikij

• fij = f(tij, i, i), qij =q(tij, i)

i = (0i, 1i, 2i)', i = (i, i)'

= (1, … , p)'

• yi(1) = (yi1, …, yip)'

• yi(2) = (yi,p+1, …, )'

•

•

in,iy

p

1kkj,ikijij tφtx

p

1kkj,ikijij qφqr

p

1kkj,ikiji2iji1i0

p

1kkij yφrβxββ φ1μ

2uiσ )σ,μ(N ~ 2

uiij

.d.i.i

2uiσ 2

uiσ

iinμ

11

Model (cont’d)Level 2

i and i are independent

i| , D1 ~ MVN(, D1), i| *, D2 ~ BVLN(*, D2)

Level 3

, ~ MVN(h, H)

~ MVN(h1, H1) , * ~ BVN(h2, H2),

,

2

a,

2

aG~ 102

ui

))ν(,ν(W~ 1222

12

AD))ν(,ν(W~ 1111

11

AD

12

InferenceBayesian inference

for longitudinalbent-cable regression

MCMC(Metropolis

Within Gibbs)

Full conditionals

(1) i|. (2) i|. (3)

(4) (5)

(6) |. (7) *|.

(8) |.

Implementation

.|12D.|1

1D

.|σ 2ui • Drawing MCMC

samples – C• MCMC output Analysis – R (coda package)

Computation

14

Barrow

Time

CF

C-1

1 (

in p

pt)

1988 1992 1996 2000

230

250

270

Cap Matatula

Time

CF

C-1

1 (

in p

pt)

1988 1992 1996 2000

230

250

270

Mauna Loa

Time

CF

C-1

1 (

in p

pt)

1988 1992 1996 2000

230

250

270

South Pole

Time

CF

C-1

1 (

in p

pt)

1988 1992 1996 2000

230

250

270

Black: Observed

data

Red: Station-

specific fit

Green:

Population/

global fit

Estimated

transition is

marked by the

vertical lines

Results assuming AR(1) within-station noise

15

Niwot Ridge

Time

CF

C-1

1 (

in p

pt)

1988 1992 1996 2000

230

250

270

Mace Head

Time

CF

C-1

1 (

in p

pt)

1988 1992 1996 2000

230

250

270

Cape Grim

Time

CF

C-1

1 (

in p

pt)

1988 1992 1996 2000

230

250

270

Ragged Point

Time

CF

C-1

1 (

in p

pt)

1988 1992 1996 2000

230

250

270

Results (cont’d)

Black: Observed

data

Red: Station-

specific fit

Green:

Population/

global fit

Estimated

transition is

marked by the

vertical lines

16

Incoming slope

(95% C.I.)

Outgoing slope

(95% C.I.)

Transition period

(Duration)

CTP(99% C.I.)

Global 0.65

(0.50, 0.80)

-0.12

(-0.22, -0.01)

Jan, 89 – Sep, 94

(69 months)

Nov, 93

(Aug, 92 to May, 95)

Cap Matatula

12 1.01

0.74

(0.56, 0.94)

-0.10

(-0.13, -0.07)

May, 89 – Jan, 95

(69 months)

May, 94

(Oct, 93 to Feb, 95)

Mauna Loa

22 1.81

0.67

(0.52, 0.83)

-0.12

(-0.16, -0.09)

Mar, 89 – Jun, 94

(64 months)

Aug, 93

(Dec, 92 to May, 94)

Niwot Ridge

32 0.82

0.56

(0.34, 0.79)

-0.11

(-0.13, -0.08)

Nov, 88 – Jul, 94

(69 months)

Aug, 93

(Dec, 92 to May, 94)

Mace Head

42 1.20

0.59

(0.44, 0.74)

-0.11

(-0.13, -0.08)

Sep, 88 – Jan, 94

(65 months)

Mar, 93

(Jul, 92 to Dec, 93)

Results (cont’d)

17

Incoming slope

(95% C.I.)

Outgoing slope

(95% C.I.)

Transition period

(Duration)

CTP(99% C.I.)

Ragged Point

52 2.25

0.70

(0.55, 0.86)

-0.10

(-0.14, -0.07)

Jan, 89 – Apr, 94

(64 months)

Aug, 93

(Nov, 92 to Jun, 94)

Barrow

62 2.97

0.55

(0.39, 0.72)

-0.19

(-0.24, -0.15)

Jan, 89 – Aug, 94

(68 months)

Mar, 93

(Jul, 92 to Nov, 93)

Cape Grim

72 0.29

0.78

(0.68, 0.93)

-0.07

(-0.09, -0.06)

Mar, 89 – Nov, 94

(69 months)

Jun, 94

(Jan, 94 to Oct, 94)

South Pole

82 0.30

0.60

(0.42, 0.77)

-0.12

(-0.15, -0.10)

Dec, 88 – Nov, 95

(84 months)

Sep, 94

(Apr, 94 to Mar, 95)

Results (cont’d)

18

Results (cont’d) Global

Significant increase/decrease in CFC-11 in the incoming/outgoing phases

incoming phase: average increase in CFC-11 was about 0.65 ppt/month during the

outgoing phase: average decrease was about 0.12 ppt/month

Transition: Global drop in CFC-11 took place between Jan ’89 and Sep ’94, approximately

Estimated CTP was Nov ’93 CFC-11 went from increasing to decreasing in around

Nov ’93

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Results (cont’d) Station-Specific

Significant increase/decrease of CFC-11 in

the incoming/outgoing phases for all

stations individually Rates at which these changes occurred agree

closely Approximately constant rates of change before and

after the enforcement of the Montreal Protocol,

observable despite a geographically spread-out

detection network

20

Results (cont’d) Station-Specific

Transition periods and CTPs varied somewhat across stations This may be due to the extended phase-out

schedules contained in the Montreal Protocol – 1996 for developed countries and 2010 for developing countries

Durations of the transition periods are very similar among stations except for South Pole

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Highly unusual weather conditions CFCs are not disassociated during

the long winter nights It may be expected for CFCs to

remain in the atmosphere for a long period of time, and hence, an extended transition period

CFC-11 measurements showed little variation over time

Outlier

Results (cont’d) Station-Specific (South Pole)

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Results (cont’d) Key Findings

Substantial decrease in global CFC-11 levels after the gradual transition suggest

The Montreal Protocol, which came into force in Jan ’89, can be regarded as a successful international agreement to reduce the atmospheric concentration of CFCs globally

The rate by which CFC-11 has been decreasing suggests that it will remain in the atmosphere throughout the 21st century, should current conditions prevail

23

Further Extension of the Methodology

0 20 40 60 80 100 120 140

37

.83

8.0

38

.23

8.4

Time

Gradual change

( > 0)

0 50 100 150 200 2503

5.0

35

.53

6.0

36

.53

7.0

Time

Abrupt change

( = 0)

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Further Extension of the Methodology (cont’d)

Gradual

( > 0)?

0 50 100 150 200

37

.53

7.6

37

.73

7.8

Time

0 50 100 150 200

37

.53

7.6

37

.73

7.8

Time

Abrupt

( = 0)?

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Further Extension of the Methodology (cont’d)

Longitudinal bent-cable Methodology for

smooth/gradual transition

Longitudinal bent cable to account foreither type of transition

– gradual or abrupt –driven by the data rather than

assuming that only one type is possible

√

Flexible methodology

for longitudinal changepoint

data

What if the sample comes from two potential populations:one with a gradual transition period, and

the other with an abrupt transition?

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Limitations Assumes stationarity of the AR process

Can be sensitive to the values of the

hyper-prior parameters Example: If the AR process is close to non-

stationary, a restrictive prior for could be

required in progress: alternative modeling approach and/or

prior specification for (e.g. Fisher transformation)

1 Monitoring Atmospheric Chlorofluorocarbons by the Longitudinal Bent-Cable Model S.A. Khan, G. Chiu...

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