1. 1 Monitoring Atmospheric Chlorofluorocarbons by the Longitudinal Bent-Cable Model S.A. Khan, G. Chiu * and J.A. Dubin TIES 2009 * presenter

2. 2 Outline Introduction CFC-11 Data Model Inference Results Further Extension of the Methodology Limitations

3. 3 Introduction Incoming Phase - + 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 ) t i of shock-though data by the bent-cable function Concentration of CFC-11 in response to the Montreal Protocols 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. 4 Introduction (contd) Bent-cable function (Chiu, Lockhart & Routledge, 2006) f(x i,, ) = 0 + 1 t i + 2 q(t i, ), where = ( 0, 1, 2 ), = (, ), q(t i, ) =, Bent-cable Regression: y i = f(t i,, ) + i i iid (Chiu, Lockhart & Routledge, 2006, JASA) i AR(p) (Chiu and Lockhart, revisions submitted) R Package bentcableAR handles both

5. 5 Introduction (contd) 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. 6 Skin Cancer and Cataracts Damage to Plants Reduction of Organisms in the Oceans 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. 7 Banned globally by the 1987 Montreal Protocol CFC-11 Data (contd) 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 propellants prior to the 1980s Destroy Ozone

8. 8 Monitoring 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 (contd)

9. 9 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 (contd)

10. 10 Model Level 1 y ij = f ij + ij, y ij = ij + u ij, j = p+1, …, n i Y ij | y i1, …, y ip, i, i,, Y i (2) | y i (1), i, i,, ~ MVN( i, I i ), where, i = ( i,p+1, …, )' f ij = f(t ij, i, i ), q ij =q(t ij, i ) i = ( 0i, 1i, 2i )', i = ( i, i )' = ( 1, …, p )' y i (1) = (y i1, …, y ip )' y i (2) = (y i,p+1, …, )'

11. 11 Model (contd) Level 2 i and i are independent i |, D 1 ~ MVN(, D 1 ), i | *, D 2 ~ BVLN( *, D 2 ) Level 3, ~ MVN(h, H ) ~ MVN(h 1, H 1 ), * ~ BVN(h 2, H 2 ),,

12. 12 Inference Bayesian inference for longitudinal bent-cable regression MCMC (Metropolis Within Gibbs) Full conditionals (1) i |. (2) i |. (3) (4) (5) (6) |. (7) * |. (8) |. Implementation Drawing MCMC samples – C MCMC output Analysis – R (coda package) Computation

13. 13 Inference (contd) (1) i |. ~ Normal (2) i |. ~ No closed-form expression (3) ~ Gamma (4) ~ Wishart (5) ~ Wishart (6) |. ~ Normal (7) * |. ~ Normal (8) |. ~ Normal

14. 14 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. 15 Results (contd) Black: Observed data Red: Station- specific fit Green: Population/ global fit Estimated transition is marked by the vertical lines

16. 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 1 2 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 2 2 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 3 2 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 4 2 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 (contd)

17. 17 Incoming slope (95% C.I.) Outgoing slope (95% C.I.) Transition period (Duration) CTP (99% C.I.) Ragged Point 5 2 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 6 2 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 7 2 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 8 2 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 (contd)

18. 18 Results (contd) 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. 19 Results (contd) 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. 20 Results (contd) 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. 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 (contd) Station-Specific (South Pole)

22. 22 Results (contd) 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. 23 Further Extension of the Methodology Gradual change ( > 0) Abrupt change ( = 0)

24. 24 Further Extension of the Methodology (contd) Gradual ( > 0)? Abrupt ( = 0)?

25. 25 Further Extension of the Methodology (contd) Longitudinal bent-cable Methodology for smooth/gradual 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 What if the sample comes from two potential populations: one with a gradual transition period, and the other with an abrupt transition?

26. 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. 27