Monitoring Atmospheric
Chlorofluorocarbons by the
Longitudinal Bent-Cable Model
S.A. Khan, G. Chiu* and J.A. Dubin
TIES 2009
* presenter
1
Outline
Introduction
 CFC-11 Data
 Model
 Inference
 Results
 Further Extension of the Methodology
 Limitations

2

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)
265
255
260
Transition
period

250
CFC-11 (in ppt)
270
275
Introduction
-
1988
1990
1992


CTP: the point
at which it took
a downturn from
an increasing
trend
+
1994
1996
1998
2000
Time
Figure 1: Characterizing a trend
of shock-though data by the bent-cable function
3
Introduction (cont’d)

Bent-cable function (Chiu, Lockhart & Routledge, 2006)
f(xi, , ) = 0 + 1 ti + 2 q(ti, ),
where  = (0 , 1, 2),  = (, ),
2
(
t

τ

γ
)
q(ti, ) = i
I{| t i  τ | γ}  (t i  τ)I{t i  τ  γ} ,
4γ

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
4
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)
5
CFC-11 Data
Natural
(followed by a
natural recovery)
Skin Cancer and
Cataracts
Reduction of Ozone
Layer in the Upper
Atmosphere
Increased UV
Exposure
Human Activities
(e.g. use of
CFCs)
Reduction of
Organisms in the
Ocean’s Photic
Zone
Damage to
Plants
6
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
Destroy
Ozone
Used in air
conditioning/cooling units,
and aerosol propellants
prior to the 1980’s
Banned globally by the
1987 Montreal Protocol
7
CFC-11 Data (cont’d)
Pt. Barrow,
Alaska
Ragged Point,
Barbados
Mace Head,
Ireland
South Pole,
Antarctica
Cape Matatula,
American Samoa
Niwot Ridge,
Colorado
Mauna Loa,
Hawaii
Cape Grim,
Tasmania
Monitoring stations of CFCs all over the globe (Data collected by NOAA/ESRL
global monitoring division and ALE/GAGE/AGAGE global network program) 8
280
CFC-11 Data (cont’d)
260
240
250
Barrow
Cape Matatula
Mauna Loa
South Pole
Niwot Ridge
Mace Head
Cape Grim
Ragged Point
230
CFC-11(in ppt)
270
 What were the rates
of change before and
after the transition
period?
1988
1990
1992 1994
1996 1998
 How long did it take
to show an obvious
decline?
 What was the CTP at
which the trend went
from increasing to
decreasing?
2000
Time
CFC-11 profiles of eight stations (monthly mean data)
9
Model
Level 1

• fij = f(tij, i, i), qij =q(tij, i)
p
yij = fij + ij, ij   k i, jk  u ij , u ij ~ N(0, 2ui )
• i = (0i, 1i, 2i)', i = (i, i)'
yij = ij + uij, j = p+1, …, ni
•  = (1, … , p)'
k 1

p
p


μ ij  1   φ k  β 0i  β1i x ij  β 2i rij   φ k yi , jk
k 1
 k 1 


Yij| yi1, …, yip, i, i, , σ
2 i.i.d.
ui
~ N(μ ij , σ 2ui )
Yi(2)| yi(1), i, i, , σ 2ui ~ MVN(i, σ 2uiIi),
where, i = (i,p+1, … , μ ini )'
• yi(1) = (yi1, …, yip)'
• yi(2) = (yi,p+1, …, y i , n )'
i
• x t 
ij
ij
• r q 
ij
ij
p
φ t
k 1
p
k i , j k
φ q
k 1
k
i , j k
10
Model (cont’d)
Level 2

i and i are independent

i| , D1 ~ MVN(, D1), i| *, D2 ~ BVLN(*, D2)
Level 3
2
ui
 a 0 a1 
~ G ,  ,
 2 2
 ~ MVN(h, H)



 ~ MVN(h1, H1) , * ~ BVN(h2, H2),

D11 ~ W( ν1 , ( ν1A1 ) 1 ) , D21 ~ W( ν 2 , ( ν 2 A 2 ) 1 )
11
Inference
Bayesian inference
for longitudinal
bent-cable regression
Implementation
MCMC
(Metropolis
Within Gibbs)
Full conditionals
(1) i|.
(2) i|.
(4) D11 | . (5) D21 | .
(6) |.
(8) |.
(7) *|.
(3) σ ui2 | .
Computation
• Drawing MCMC
samples
–C
• MCMC output
Analysis
– R (coda package)
12
Inference (cont’d)
(1)
i|. ~ Normal
(5)
D21 | .
(2)
i|. ~ No closed-form expression
(6)
|. ~ Normal
(3)
σ ui2 | . ~ Gamma
(7)
*|. ~ Normal
(8)
|. ~ Normal
(4) D11 | .
~ Wishart
~ Wishart
13
Results assuming AR(1) within-station noise
1988
1992
1996
 Black: Observed
230 250 270
CFC-11 (in ppt)
Cap Matatula
230 250 270
CFC-11 (in ppt)
Barrow
2000
data
 Red: Station1988
Time
1992
1996
2000
specific fit
 Green: Population/
Time
global fit
1988
1992
1996
Time
2000
 Estimated
230 250 270
CFC-11 (in ppt)
South Pole
230 250 270
CFC-11 (in ppt)
Mauna Loa
transition is marked
by the vertical lines
1988
1992
1996
Time
2000
14
Results (cont’d)
Mace Head
1988
1992
1996
 Black: Observed
230 250 270
CFC-11 (in ppt)
230 250 270
CFC-11 (in ppt)
Niwot Ridge
2000
data
 Red: Station1988
Time
1992
1996
2000
specific fit
 Green: Population/
Time
global fit
1988
1992
1996
Time
2000
230 250 270
CFC-11 (in ppt)
Ragged Point
230 250 270
CFC-11 (in ppt)
Cape Grim
 Estimated
transition is marked
by the vertical lines
1988
1992
1996
Time
2000
15
Results (cont’d)
Incoming
slope
(95% C.I.)
Outgoing
slope
(95% C.I.)
Transition
period
(Duration)
CTP
(99% C.I.)
0.65
-0.12
Jan, 89 – Sep, 94
Nov, 93
(0.50, 0.80)
(-0.22, -0.01)
(69 months)
(Aug, 92 to May, 95)
0.74
-0.10
May, 89 – Jan, 95
May, 94
12  1.01
(0.56, 0.94)
(-0.13, -0.07)
(69 months)
(Oct, 93 to Feb, 95)
Mauna Loa
0.67
-0.12
Mar, 89 – Jun, 94
Aug, 93
22  1.81
(0.52, 0.83)
(-0.16, -0.09)
(64 months)
(Dec, 92 to May, 94)
0.56
-0.11
Nov, 88 – Jul, 94
Aug, 93
32  0.82
(0.34, 0.79)
(-0.13, -0.08)
(69 months)
(Dec, 92 to May, 94)
Mace Head
0.59
-0.11
Sep, 88 – Jan, 94
Mar, 93
42  1.20
(0.44, 0.74)
(-0.13, -0.08)
(65 months)
(Jul, 92 to Dec, 93)
Global
Cap Matatula
Niwot Ridge
16
Results (cont’d)
Incoming
slope
(95% C.I.)
Outgoing
slope
(95% C.I.)
Transition
period
(Duration)
CTP
(99% C.I.)
0.70
-0.10
Jan, 89 – Apr, 94
Aug, 93
(0.55, 0.86)
(-0.14, -0.07)
(64 months)
(Nov, 92 to Jun, 94)
0.55
-0.19
Jan, 89 – Aug, 94
Mar, 93
62  2.97
(0.39, 0.72)
(-0.24, -0.15)
(68 months)
(Jul, 92 to Nov, 93)
Cape Grim
0.78
-0.07
Mar, 89 – Nov, 94
Jun, 94
72  0.29
(0.68, 0.93)
(-0.09, -0.06)
(69 months)
(Jan, 94 to Oct, 94)
South Pole
0.60
-0.12
Dec, 88 – Nov, 95
Sep, 94
82  0.30
(0.42, 0.77)
(-0.15, -0.10)
(84 months)
(Apr, 94 to Mar, 95)
Ragged Point
52  2.25
Barrow
17
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
18
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
19
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
20
Results (cont’d)

Station-Specific (South Pole)
 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
Outlier
CFC-11 measurements showed little
variation over time
21
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
22
36.0

35.0
37.8

35.5
38.0
38.2
36.5
38.4
37.0
Further Extension of the
Methodology
0
20
40
60
80
Time
100
120
140
0
50
100
150
200
250
Time
Gradual change
Abrupt change
( > 0)
( = 0)
23
37.5
37.5
37.6
37.6
37.7
37.7
37.8
37.8
Further Extension of the
Methodology (cont’d)
0
50
100
150
200
0
50
100
Time
Time
Gradual
Abrupt
( > 0)?
( = 0)?
150
200
24
Further Extension of the
Methodology (cont’d)
Longitudinal bent-cable
Methodology for
smooth/gradual transition
√
What if the sample comes from two potential populations:
one with a gradual transition period, and
the other with an abrupt transition?
Longitudinal bent cable to account for
either 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
25
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 nonstationary, a restrictive prior for  could be
required

in progress: alternative modeling approach and/or
prior specification for  (e.g. Fisher transformation)
26
27