1
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)
6
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
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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
13
Inference (cont’d)(1) i|. ~ Normal
(2) i|. ~ No closed-form expression
(3) ~ Gamma
(4) ~ Wishart
(5) ~ Wishart
(6) |. ~ Normal
(7) *|. ~ Normal
(8) |. ~ Normal.|1
1D
.|12D
.|σ 2ui
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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
19
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
21
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
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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)
24
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)?
25
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?
26
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)
27