DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
University of Waterloo, 200 University Avenue West Waterloo, Ontario, Canada, N2L 3G1
519-888-4567, ext. 00000 | Fax: 519-746-1875 | www.stats.uwaterloo.ca
UNIVERSITY OF WATERLOO
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
WORKING PAPER SERIES
2009-04
Atmospheric Concentration of
Chlorofluorocarbons: Addressing the
Global Concern with the Longitudinal
Bent-Cable Model
Shahedul Ahsan Khan,
University of Waterloo, e-mail:
[email protected]
Grace Chiu,
University of Waterloo, e-mail:
[email protected]
Joel A. Dubin,
University of Waterloo, e-mail:
[email protected]
Abstract
The recent steady decline in atmospheric chlorofluorocarbon (CFC) concentrations could be a direct result of the Montréal Protocol on Substances
That Deplete the Ozone Layer, in effect since 1987. To study the extent of the
decline, we introduce the longitudinal bent-cable model to describe CFC concentrations observed over a global detection network. The bent cable is a parametric regression model to study data that exhibit a trend change. It comprises
two linear segments to describe the incoming and outgoing phases, joined by
a quadratic bend to model the transition period. For longitudinal data, measurements taken over time are nested within observational units drawn from
some population of interest. Here, it is useful to develop a mixed-effects model
extension of existing (frequentist) bent-cable methodology for a single time series. We do so in a hierarchical Bayesian framework, where each observational
unit is associated with a random bent cable and within-unit serial correlation.
Our analysis using the longitudinal bent-cable model reveals a global decrease
in atmospheric levels of CFC-11. The global drop took place between January,
1989 and September, 1994 approximately, with November, 1993 being the estimated time point at which CFC-11 levels went from increasing to decreasing.
keywords: Longitudinal data; Bent-cable regression; Changepoint model; Bayesian
hierarchical model; Random effects.
1
Introduction
Many biological consequences such as skin cancer and cataracts, irreversible damage to plants, and reduction of drifting organisms (animals, plants, archaea, bacteria) in the ocean’s photic zone may result from the increased ultraviolet (UV) exposure due to ozone depletion. According to “Ozone Science: The Facts Behind the
Phaseout” by the U.S. Environmental Protection Agency (U.S. EPA, “http://www.epa.
gov/ozone/science/sc fact.html”), each natural reduction in ozone levels has been
followed by a recovery, though there is convincing scientific evidence that the
ozone shield is being depleted well beyond changes due to natural processes. In
particular, ozone depletion due to human activities is a major concern, and may be
controlled. One such human activity is the use of chloroflurocarbons (CFCs). As
cited in “The Ozone Hole Tour” by the University of Cambridge (“http://www.atm.
ch.cam.ac.uk/tour/part3.html”), the catalytic destruction of ozone by atomic chlorine and bromine is the major cause of the forming of polar ozone holes, and photodissociation of CFC compounds is the main reason for these atoms to be in the
stratosphere.
CFCs are nontoxic, nonflammable chemicals containing atoms of carbon, chlorine and fluorine. CFC-11 is one such compound. CFCs were extensively used
in air conditioning/cooling units, and as aerosol propellants prior to the 1980’s.
While CFCs are safe to use in most applications and are inert in the lower atmosphere, they do undergo significant reaction in the upper atmosphere. As cited in
The Columbia Encyclopedia, CFC molecules take an average of fifteen years to travel
1
270
265
260
250
255
CFC-11 (in ppt)
275
from the ground to the upper atmosphere, and can stay there for about a century.
Chlorine inside the CFCs is one of the most important free radical catalysts to destroy ozone. The destruction process continues over the atmospheric lifetime of
the chlorine atom (one or two years), during which an average of 100,000 ozone
molecules are broken down. Because of this, CFCs were banned globally by the
1987 Montréal Protocol on Substances That Deplete the Ozone Layer (“Montreal
Protocol” in The Columbia Encyclopedia). Since this protocol came into effect, the
atmospheric concentration of CFCs has either leveled off or decreased. For example, Figure 1 shows the monthly average concentrations of CFC-11 monitored
from a station in Mauna Loa, Hawaii (Global Monitoring Division of the National
Oceanic and Atmospheric Administration (NOAA)/Earth System Research Laboratory (ESRL)). We see roughly three phases: an initial increasing trend (incoming
phase), a gradual transition period, and a decreasing trend after the transition period (outgoing phase). This trend is representative of CFC-11 measurements taken
by stations across the world and illustrated in this paper.
1988
1990
1992
1994
1996
1998
2000
Time
Figure 1: Trend of the monthly average concentrations of CFC-11 in Mauna Loa,
Hawaii. (Data source: NOAA/ESRL global monitoring division)
The effects of CFCs in ozone depletion is a global concern. Although exploratory
data analyses reveal a decrease of CFCs in the earth’s atmosphere since the early
1990’s, so far no sophisticated statistical analysis has been conducted to evaluate the global trend. In addition, there are several other important questions regarding the CFC concentration in the atmosphere that could be useful not only to
policy makers, but also for human awareness. For example, (1) How long did it
take for the CFC concentration to show an obvious decline? (2) What were the
rates of change (increase/decrease) in CFCs before and after the transition period? (3) What was the critical time point (CTP) at which the CFC trend went
from increasing to decreasing? In this article, we will address these questions
2
statistically by fitting a special changepoint model for CFC-11 data. We focus
on CFC-11, because it is considered one of the most dangerous CFCs to reduce
the ozone layer in the atmosphere. In fact, it has the shortest lifetime of common CFCs, and is regarded as a reference substance in the definition of the ozone
depletion potential (ODP). The ODP of a chemical is the ratio of its impact on
ozone compared to the impact of a similar mass of CFC-11. Thus, the ODP is 1
for CFC-11 and ranges from 0.6 to 1 for other CFCs. (These facts about CFCs are
taken from the U.S. EPA websites, “http://www.epa.gov/ozone/defns.html” and
“http://www.epa.gov/ozone/science/ods/classone.html”.)
In a broader sense, we will comment in this article on (i) the global trend of
CFC-11, and (ii) the effectiveness of the Montréal Protocol on preserving the ozone
level by reducing the use of CFC-11. Our findings will also provide a rough idea
of how long it may take to diminish CFC-11 from the earth’s atmosphere. Note
that this article is associated with that of Khan, Chiu and Dubin (2009, to appear in
CHANCE) with more details on our proposed methodology.
2
Data
CFCs are monitored from different stations all over the globe by the NOAA/ESRL
global monitoring division (“ftp://ftp.cmdl.noaa.gov/hats/cfcs/cfc11/insituGCs/”),
and by the Atmospheric Lifetime Experiment/Global Atmospheric Gases Experiment/Advanced Global Atmospheric Gases Experiment (ALE/GAGE/AGAGE)
global network program (“http://cdiac.esd.ornl.gov/ndps/alegage.html”). Henceforth, we will refer to these two programs simply as NOAA and AGAGE.
Under the Radiatively Important Trace Species (RITS) program, NOAA began
measuring CFCs using in situ gas chromatographs at their four baseline observatories — Pt. Barrow (Alaska), Cape Matatula (American Samoa), Mauna Loa
(Hawaii) and South Pole (Antarctica) — and at Niwot Ridge (Colorado) in collaboration with the University of Colorado. We will label these five stations from 1 to 5
respectively. During the period of 1988-1999, a new generation of gas chromatography called Chromatograph for Atmospheric Trace Species (CATS) was developed
and has been used to measure CFC concentrations ever since.
The AGAGE program consists of three stages corresponding to advances and
upgrades in instrumentation. The first stage (ALE) began in 1978, the second
(GAGE) began during 1981-1985, and the third (AGAGE) began during 1993-1996.
The current AGAGE stations are located in Mace Head (Ireland), Cape Grim (Tasmania), Ragged Point (Barbados), Cape Matatula (American Samoa), and Trinidad
Head (California). These five stations will be labeled from 6 to 10 respectively.
We consider monthly mean data for our statistical analysis. Ideally, we wish
to have (a) full data for all stations, (b) a long enough period to capture all three
phases of the CFC trend, and (c) no change in instrumentation to avoid the elements of non-stationarity and biased measurement, if any. However, we do not
have the same duration of consecutive observations for all stations. Moreover, data
were recorded by instrumentation that switched from one type to another. Table 1
3
summarizes the availability of the consecutive observations, and the instrumentations used to record data.
Table 1: CFC-11 data summary
Station
Available consecutive observations
Instrumentation
Barrow (Station 1)
Nov, 1987 - Feb, 1999
RITS
Jun, 1998 - Aug, 2008
CATS
May, 1989 - Apr, 2000
RITS
Dec, 1998 - Aug, 2008
CATS
Jul, 1987 - Aug, 2000
RITS
Jun, 1999 - Aug, 2008
CATS
Jun, 1990 - Nov, 2000
RITS
Feb, 1998 - Aug, 2008
CATS
Feb, 1990 - Apr, 2000
RITS
May, 2001 - Jul, 2006
CATS
Feb, 1987 - Jun, 1994
GAGE
Mar, 1994 - Sep, 2007
AGAGE
Dec, 1981 - Dec, 1994
GAGE
Aug, 1993 - Sep, 2007
AGAGE
Aug, 1985 - Jun, 1996
GAGE
Jun, 1996 - Sep, 2007
AGAGE
Jun, 1991 - Sep, 1996
GAGE
Aug, 1996 - Sep, 2007
AGAGE
–
GAGE
Oct, 1995 - Sep, 2007
AGAGE
Cape Matatula (Station 2)
Mauna Loa (Station 3)
South Pole (Station 4)
Niwot Ridge (Station 5)
Mace Head (Station 6)
Cape Grim (Station 7)
Ragged Point (Station 8)
Cap Matatula (Station 9)
Trinidad Head (Station 10)
Thus, ideal statistical conditions are not achievable in this case. As a compromise, we remove Stations 9 and 10 from our analysis due to insufficient data, and
choose a study period in such a way that it can reflect the changing behavior of the
CFC-11 concentration in the atmosphere. The Montréal Protocol came into force on
January 1, 1989. So, we expect an increasing trend in CFC-11 prior to 1989 because
of its extensive use during that period. After the implementation of the protocol, we expect a change (either decreasing or leveling off) in the CFC-11 trend. To
characterize this change, we wish to have a study period starting from some point
before the implementation of the protocol. Moreover, we must have sufficient data
to observe the CFC-11 trend, if any. Thus, we settle for a relatively long study period of 152 months from January, 1988 to August, 2000, which is perhaps the most
reasonable to satisfy (a)-(c) as much as possible. In particular, it covers Stations 3-4
with a single measuring device, RITS. Stations 2 and 5 have RITS data until April,
2000, at which point we truncate their data so that only RITS is present for all of Sta4
260
250
240
Barrow (Station 1)
Cape Matatula (Station 2)
Mauna Loa (Station 3)
South Pole (Station 4)
Niwot Ridge (Station 5)
Mace Head (Station 6)
Cape Grim (Station 7)
Ragged Point (Station 8)
230
CFC-11 in parts-per-trillion (ppt)
270
280
tions 2-5 throughout the study period. Data for the remaining four stations during
this period were recorded by two measuring devices – RITS and CATS for Station
1, and GAGE and AGAGE for Stations 6-8 – each device occupying a substantial
range of the 152 months. Figure 2 shows the eight profiles of the corresponding
CFC-11 data. Specifically, each station constitutes an individual curve, which is
different from the others due to actual CFC-11 levels during measurement, exposure to wind and other environmental variables, sampling techniques, and so on.
Our objective is to assess the global CFC-11 concentration in the atmosphere, as
well as station-specific characterization of the trends.
1988
1990
1992
1994
1996
1998
2000
Time
Figure 2: CFC-11 profiles of eight stations (monthly mean data)
Note that in this paper, we do not take into consideration the effects of change
in instrumentation (RITS/CATS and GAGE/AGAGE). Modeling these effects is
currently in progress (see the Conclusion section), and a preliminary analysis reveals statistically insignificant results. However, between-station differences are
accounted for, at least partially, by the random components introduced in our modeling (see the Methods section).
3
Methods
We wish to unify information from each station to aid the understanding of the
global as well as station-specific behavior of CFC-11 in the atmosphere. We will
index the stations by i = 1, 2, . . . , 8, and the months from January, 1988 to August,
2000 by j = 1, 2, . . . , 152. Let tij denote the j th measurement occasion of the ith
station. We model the CFC-11 measurement for the ith station at time tij , denoted
5
by yij , by the relationship
yij = f (tij , θ i ) + ij
(1)
where θ i is a vector of regression coefficients for the ith station, f (·) is a function
of tij and θ i characterizing the trend of the station-specific data over time, and ij
represents the random error component.
Some remarks are required to explain tij . Recall that some stations do not have
data for all 152 months. We employ the following system for defining tij . For
example, the first and last months with recorded data by Station 3 are January, 1988
and August, 2000, respectively; thus, t3,1 = 1, t3,2 = 2, . . . , t3,152 = 152. In contrast,
Station 2 had its first and last recordings in May, 1989 and April, 2000, respectively;
hence, t2,1 = 17, t2,2 = 18, . . . , t2,132 = 148. The same approach is used to define tij ’s
for other i’s. Note that a few yij ’s (from 1 to 5) were missing between the first and
last months for a given station. We replace them by observations from another
data set (e.g. CATS or AGAGE) or by mean imputation based on neighboring time
points if not available from another data set. Though mean imputation can be
problematic resulting in biased estimates, if just a few missing values are replaced
by the mean, the deleterious effect of mean substitution is reduced (McKnight,
Figueredo and Sidani, 2007). So, we expect our findings to be minimally affected
by this replacement for so few time points.
Next, we will describe the complete formulation of the model given by (1). We
would like an expression for f that not only describes the CFC-11 profiles as shown
in Figure 2, but also gives useful information regarding the rates of change and the
transition. Although a simple quadratic model such as f (tij , θ i ) = β0i +β1i tij +β2i t2ij ,
where θ i = (β0i , β1i , β2i )0 , might be appealing to characterize the overall convexity
of the trend, it would not be expected to fit the observed data all that well, in
light of the apparent three phases: incoming and outgoing, joined by the curved
transition. A characterization of such a trend can be well-accomplished by the
so-called bent-cable function (Chiu, Lockhart and Routledge, 2006), given by
f (tij , θ i ) = β0i + β1i tij + β2i q(tij , γi , τi )
(2)
(tij − τi + γi )2
1{|tij − τi | ≤ γi } + (tij − τi )1{tij − τi > γi }.
q(tij , γi , τi ) =
4γi
(3)
where
A graphical description of this function is given in Figure 3. The model is parsimonious in that it has only five regression coefficients, and is appealing due to the
greatly interpretable regression coefficients. We will extend their model to account
for the between-station heterogeneity as suggested by the different profiles in Figure 2. Henceforth, we will denote f (tij , θ i ) and q(tij , γi , τi ) by simply fij and qij ,
respectively.
Overall, the bent-cable function, f , represents a smooth trend over time. The
random error components, ij ’s in (1), account for within-station variability in the
measurements around the regression f . Many standard assumptions for regression analyses do not hold for longitudinal data, including independence between
6
Bent-cable function, f
Incoming
phase
Outgoing phase
𝛾𝛾𝑖𝑖
𝜏𝜏𝑖𝑖 − 𝛾𝛾𝑖𝑖
𝛾𝛾𝑖𝑖
𝜏𝜏𝑖𝑖
𝜏𝜏𝑖𝑖 + 𝛾𝛾𝑖𝑖
Time
Figure 3: The bent-cable function, comprising two linear segments (incoming and
outgoing) joined by a quadratic bend. There are three linear parameters β0i , β1i
and β2i , and two transition parameters γi and τi , with θ i = (β0i , β1i , β2i , γi , τi )0 . The
intercept and slope for the incoming phase are β0i and β1i , respectively; the slope
for the outgoing phase is β1i + β2i ; and the center and half-width of the bend are τi
and γi , respectively. The critical time point (at the green line) is the point at which
the response takes either an upturn from a decreasing trend or a downturn from
an increasing trend.
all measurements (Fitzmaurice, Laird and Ware, 2004). Instead, it is necessary
to account for correlation between repeated measurements within the same unit
(here, station) over time. Though, for some data, the unit-specific coefficients θ i
may adequately account for this correlation, quite often there is additional serial
correlation remaining that can be accounted for by these ij ’s. An order-p (p > 0)
autoregressive (AR(p)) serial correlation structure (Box and Jenkins, 1994) may be
well-suited for many types of longitudinal data. Thus, we consider here an AR(p),
p > 0, model for the ij ’s, that is,
ij = φ1 i,j−1 + φ2 i,j−2 + . . . + φp i,j−p + uij
(4)
where φ1 , φ2 , . . . , φp are the AR(p) parameters, which we denote by the vector φ,
that is, φ = (φ1 , φ2 , . . . , φp )0 . We assume that uij ’s in (4) are independently and
2
identically normally distributed with mean 0 and variance σui
. Furthermore, we
assume that the initial p observations for each i are known. We develop a Bayesian
methodology conditional on these initial observations, that is, the posterior (Section
3.1) is conditioned on these initial observations. This assumption is customary in
the frequentist’s estimation procedures, and was considered by Chib (1993) in a
Bayesian approach. To analyze a single profile (measurements from a single station
over time) using a bent-cable model with AR(p) noise, the reader may refer to Chiu
and Lockhart ([3] and [4]).
7
In particular, an AR(1) model, being very parsimonious, is characterized by
an exponential decay of correlations as the separation between a pair of measurements increases, a feature that is common in many longitudinal data sets. For
our CFC-11 analysis, we assume an AR(1) structure with correlation parameter
ρ = corr(i,j , i,j±1 ), and variance parameters σi2 = var(ij ). To define a population
level of CFC-11, we wish to relate the station-specific coefficients θ i to the population coefficients in some meaningful way. Before describing such a relation, we
mention here one more assumption we make in developing the model: the linear
coefficients β i = (β0i , β1i , β2i )0 and transition coefficients αi = (γi , τi )0 are independent a priori (see Section 3.1). Practically, β i characterizes the intercepts and
the rates of change, while αi characterizes the structure of the transition. There
is generally no way to infer about αi from prior knowledge of β i , without first
seeing the data. For example, suppose the two linear phases are very steep. This
information alone is not sufficient to answer the question “what is the chance of αi
being, say, (30, 40)0 versus (35, 50)0 ?” Hence, it is reasonable to assume that β i and
αi are a priori independent. Now, we relate β i to the population coefficient vector
β = (β0 , β1 , β2 )0 plus a random component η 1i , that is, β i = β + η 1i , where η 1i is
a 3 × 1 column vector. Under the assumption that η 1i has mean 0 and a 3 × 3 covariance matrix D1 , conceptually we can regard stations as having their own linear
coefficients, and the population coefficients as the average across stations. Then,
the covariance matrix D1 provides information on the variability of the deviation of
the station-specific coefficient β i from the population coefficient β. As an extreme
example, a zero variation of the deviation between β1i and β1 indicates that the
station-specific and global incoming slopes are identical. In other words, rates of
change in the incoming phase are identical across stations. To complete the model
formulation for the linear coefficients, an assumption on the distribution of η 1i , or
equivalently β i , is necessary. Since (i) the dependence structure can be fully specified by the covariance matrix through an assumption of the multivariate normal
distribution, and (ii) it is convenient theoretically and computationally, we assume
that η 1i ∼ MVN (0, D1 ), or equivalently, β i ∼ MVN (β, D1 ).
Let us turn now to a model for the transition coefficients. Note that γi and τi
are positive. We assume that α∗i = (log (γi ), log (τi ))0 = α∗ + η 2i , α∗ = (γ ∗ , τ ∗ )0 ,
and η 2i is a 2 × 1 column vector with η 2i ∼ MVN (0, D2 ), so that E(α∗i ) = α∗ .
This assumption is equivalent to αi ∼ MVLN (α∗ , D2 ), i.e. a bivariate lognormal
distribution with α∗ and D2 being the mean vector and the covariance matrix of
∗
∗
α∗i , respectively. We may then define a population effect as (eγ , eτ )0 , which is
the median of αi under the lognormal distribution. Statistical assumptions for
ij , β i and αi as discussed above, together with Equations (1)-(4), constitute our
longitudinal bent-cable.
3.1
Statistical Inference
We employ a Bayesian approach for statistical inference. The main idea of Bayesian
inference is to combine data and prior knowledge on a parameter to determine its
posterior distribution (the conditional density of the parameter given the data). The
8
prior knowledge is supplied in the form of a prior distribution of the parameter,
which quantifies information about the parameter prior to any data being gathered. For example, recall the argument for the a priori independence of β i and
αi . When little is reliably known about the parameter, it is reasonable to choose a
fairly vague, minimally informative prior. For example, in analyzing the CFC-11
data, we assumed a multivariate normal prior for β with mean 0 and a covariance
matrix with very large variance parameters. This leads to a non-informative prior
for β, meaning the data, assuming a sufficient amount of data is collected, will
primarily dictate β’s resulting posterior distribution.
Any conclusions about the plausible value of the parameter are to be drawn
from the posterior distribution. For example, the posterior mean (median if the
posterior density is noticeably asymmetric) and variance can be considered an estimate of the parameter and the variance of the estimate, respectively. A 100(1−2p)%
Bayesian interval, or credible interval, is [c1 , c2 ], where c1 and c2 are the pth and
(1 − p)th quantiles of the posterior density, respectively. This credible interval has a
probabilistic interpretation. For example, for p = 0.025, the conditional probability
that the parameter falls in the interval [c1 , c2 ] given the data is 0.95.
The Bayesian model formulation of our longitudinal bent-cable model derived
from the above assumptions involves three levels (see Appendix 1 for details of the
first level):
(2) (1)
2
2
∼ MVN µi (β i , αi , φ), σui
Ii ,
Yi |yi , β i , αi , φ, σui



β i |β, D1 ∼ MVN (β, D1 ) 
,


∗
∗

αi |α , D2 ∼ MVLN (α , D2 )



β|h1 , H1 ∼ MVN (h1 , H1 ), α∗ |h2 , H2 ∼ MVN (h2 , H2 ), 




−1
−1
−1
−1
,
D1 ∼ W ν1 , (ν1 A1 ) , D2 ∼ W ν2 , (ν2 A2 ) ,






−2
a ab

φ|h3 , H3 ∼ MVN (h3 , H3 ), σui |a, b ∼ G( 2 , 2 )
(2)
(1)
(5)
(6)
(7)
0
where Yi = (Yi,p+1 , . . . , YP
= (yi1 , . . . , yip )0 , µP
i,ni ) , yi
i (β i , αi , φ) ≡ µi = (µi,p+1 ,
p
0
. . . , µi,ni ) , µij = β0i (1 − P k=1 φk ) + β1i xij + β2i rij + pk=1 φk yi,j−k , xij = tij −
P
p
p
k=1 φk ti,j−k , rij = qij −
k=1 φk qi,j−k (j = p + 1, . . . , ni ), and W and G stand for
Wishart and gamma distributions, respectively. The first two levels are (5) and (6)
which represent, respectively, the within-station and between-station characteristics, and the third level is (7), which is the specification of the prior distributions
with fixed, prespecified hyperparameters.
Bayesian inference is carried out by the Markov Chain Monte Carlo (MCMC)
method (Gilks, Richardson and Spiegelhalter, 1996). More specifically, we generate data by the Hybrid Gibbs (Smith and Roberts, 1993) (Metropolis within Gibbs)
9
algorithm to summarize the posterior distributions. We prefer conjugate priors
at level 3 to facilitate the straightforward implementation of the Gibbs sampler
because of the posterior’s closed form. While occasionally criticized to be too restrictive, conjugate priors are preferred here to reduce the extensiveness of computations involved in the MCMC method. Specification of the priors in Equation (7)
leads to the following full conditionals (see Appendix 2 for derivations):
β i |. ∼ MVN
−2
Mi σui
X0i
zi +
D−1
1
β , Mi ,
−1
β|. ∼ MVN U1 (D−1
β̃
+
H
h
),
U
1
1 ,
1
1
∗
−1
−1
α |. ∼ MVN U2 (D2 α̃ + H2 h2 ), U2 ,
m
hX
i−1 −1
0
D1 |. ∼ W m + ν1 ,
(β i − β) (β i − β) + ν1 A1
,
(8)
(9)
(10)
(11)
i=1
m
hX
i−1 −1
∗
∗
∗
∗ 0
D2 |. ∼ W m + ν2 ,
(αi − α ) (αi − α ) + ν2 A2
,
(12)
i=1
0
ni − p + a (zi − Xi β i (zi − Xi β i + ab
∼G
,
,
2
2
X
m
−2
0
−1
φ|. ∼ MVN V
σui Wi i + H3 h3 , V ,
−2
σui
|.
(13)
(14)
i=1
P
where zi = (zi,p+1 , . . . , zi,ni )0 with zij = yij − pk=1 φk yi,j−k , i = (i,p+1 , . . . , i,ni )0
Pm ∗
Pm
−2
= σui
X0i Xi + D−1
αi , M−1
with ij = yij − fij , β̃ =
1 ,
i
i=1P
i=1 β i , α̃ =
m
−1
−1
−1
−1
−1
−1
−1
−2
0
−1
U1 = mD1 + H1 , U2 = mD2 + H2 , V = i=1 σui Wi Wi + H3 , and




Pp
1
−
φ
x
r
i,p+1
i,p+1 
i,p−1
i,1

k=1 k
 i,p





Pp



xi,p+2 ri,p+2 
i,p+1
i,p
i,2 
 1 − k=1 φk


 , Wi = 
Xi = 
.


.
.
.
.
.
.


.
.
.
..
..
.. 

.
. 
 .






P
i,ni −1 i,ni −2 i,ni −p
1 − pk=1 φk
xi,ni
ri,ni
(15)
The full conditional for αi cannot be expressed in a closed form. It is proportional
to the following expression
n
o
n 1
o
1
1
∗
∗
exp − 2 (zi − Xi β i )0 (zi − Xi β i ) exp − (α∗i − α∗ )0 D−1
(α
−
α
)
.
2
i
γi τi
2σui
2
(16)
10
We use a random walk Metropolis (Givens, Hoeting and Sidani, 2005) for αi within
the Gibbs because of its nonstandard full conditional.
As little is known reliably on the station-specific profiles of the CFC-11 data
prior to data collection, values of the hyper-parameters in (7) are chosen to reflect
weak knowledge. We have written our own code in C to generate Monte Carlo
samples, and analyzed these in R using the coda package.
4
Results
240 245 250 255 260 265 270 275
CFC-11 (in ppt)
Figure 4 presents the estimated global CFC-11 concentrations using longitudinal
bent-cable regression assuming the AR(1) structure for within-station dependence.
The global drop (gradual) in CFC-11 took place between January, 1989 and September, 1994 approximately. Estimated incoming and outgoing slopes were 0.65 and
−0.12, respectively. Thus, the average increase in CFC-11 was about 0.65 ppt for a
one-month increase during the incoming phase (January, 1988 - December, 1988),
and the average decrease was about 0.12 ppt during the outgoing phase (October,
1994 - August, 2000). The 95% credible intervals indicated significant slopes for
the incoming/outgoing phases. Specifically, these intervals were (0.50, 0.80) and
(−0.22, −0.01), respectively, for the two linear phases, neither interval including
0. The estimated population critical time point (CTP) was November, 1993, which
implies that,
on average, CFC-11 went from increasing to decreasing around this
time. The corresponding 95% credible interval ranged from December, 1992 to
November, 1994.
CTP = Nov, 1993
Slope = ‐0.12
Slope = 0.65 End of transition = Sep, 1994 Beginning of transition = Jan, 1989 1988
1990
1992
1994
1996
1998
2000
Time
Figure 4: Global fit of the CFC-11 data using longitudinal bent-cable regression
assuming AR(1)
11
The station-specific fits, as well as the population fit are displayed in Figure 5.
It shows that the model fits the data well, with the observed data and individual
fits agreeing quite closely. Table 2 summarizes the fits numerically. We see significant increase/decrease of CFC-11 in the incoming/outgoing phases for all the
stations separately, as well as globally. The rates at which these changes occurred
(Columns 2 and 3) agree closely for the stations. This phenomenon was also evident in the estimate of D, showing small variation of the deviations between the
global and station-specific incoming/outgoing slope parameters (variance ≈ 0.02
for both cases).
1996
2000
1992
1996
2000
280
260
250
230
240
CFC-11 (in ppt)
270
280
270
260
250
CFC-11 (in ppt)
230
1988
1988
1992
1996
2000
1988
1992
1996
Time
Niwot Ridge
(Station 5)
Mace Head
(Station 6)
Cape Grim
(Station 7)
Ragged Point
(Station 8)
1996
Time
2000
1988
1992
1996
2000
270
260
250
CFC-11 (in ppt)
230
1988
Time
240
270
260
250
CFC-11 (in ppt)
230
240
270
260
250
CFC-11 (in ppt)
230
240
270
260
250
240
1992
2000
280
Time
280
Time
280
Time
230
1988
South Pole
(Station 4)
240
270
260
250
CFC-11 (in ppt)
230
240
270
260
250
CFC-11 (in ppt)
240
230
1992
280
1988
CFC-11 (in ppt)
Mauna Loa
(Station 3)
280
Cap Matatula
(Station 2)
280
Barrow
(Station 1)
1992
1996
Time
2000
1988
1992
1996
2000
Time
Figure 5: Station-specific fits (red) and population fit (green) of the CFC-11 data,
with estimated transition marked by vertical lines. The black line indicates observed data.
The above findings support the notion of constant rates of increase and decrease, respectively, before and after the enforcement of the Montréal Protocol, observable despite a geographically spread-out detection network. They also point
to the success of the widespread adoption and implementation of the Montréal
Protocol across the globe. However, the rate by which CFC-11 has been decreasing
(about 0.12 ppt per month, globally) suggests that it will remain in the atmosphere
throughout the 21st century, should current conditions prevail.
Let us turn now to the behavior of the transition of CFC-11 over time. The
transition periods and critical time points varied somewhat across stations (Ta12
Table 2: Estimated station-specific and global concentrtions of CFC-11 assuming
AR(1)
Incoming slope
(95% credible
interval)
Outgoing slope
(95% credible
interval)
Transition period
(Duration)
CTP
(99% credible
interval)
Global
0.65
(0.50, 0.80)
−0.12
(−0.22, −0.01)
Jan, 1989 - Sep, 1994
(69 months)
Nov, 1993
(Aug, 1992 to
May, 1995)
Barrow
(Station 1)
σ12 ≈ 2.97
0.55
(0.39, 0.72)
−0.19
(−0.24, −0.15)
Jan, 1989 - Aug, 1994
(68 months)
Mar, 1993
(Jul, 1992 to
Nov, 1993)
Cap Matatula
(Station 2)
σ22 ≈ 1.01
0.74
(0.56, 0.94)
−0.10
(−0.13, −0.07)
May, 1989 - Jan, 1995
(69 months)
May, 1994
(Oct, 1993 to
Feb, 1995)
Mauna Loa
(Station 3)
σ32 ≈ 1.81
0.67
(0.52, 0.83)
−0.12
(−0.16, −0.09)
Mar, 1989 - Jun, 1994
(64 months)
Aug, 1993
(Dec, 1992 to
May, 1994)
South Pole
(Station 4)
σ42 ≈ 0.30
0.60
(0.42, 0.77)
−0.12
(−0.15, −0.10)
Dec, 1988 - Nov, 1995
(84 months)
Sep, 1994
(Apr, 1994 to
Mar, 1995
Niwot Ridge
(station 5)
σ52 ≈ 0.82
0.56
(0.34, 0.79)
−0.11
(−0.13, −0.08)
Nov, 1988 - Jul, 1994
(69 months)
Aug, 1993
(Dec, 1992 to
May, 1994)
Mace Head
(Station 6)
σ62 ≈ 1.20
0.59
(0.44, 0.74)
−0.11
(−0.13, −0.08)
Sep, 1988 - Jan, 1994
(65 months)
Mar, 1993
(Jul, 1992 to
Dec, 1993)
Cape Grim
(Station 7)
σ72 ≈ 0.29
0.78
(0.68, 0.93)
−0.07
(−0.09, −0.06)
Mar, 1989 - Nov, 1994
(69 months)
Jun, 1994
(Jan, 1994 to
Oct, 1994)
Ragged Point
(Station 8)
σ82 ≈ 2.25
0.70
(0.55, 0.86)
−0.10
(−0.14, −0.07)
Jan, 1989 - Apr, 1994
(64 months)
Aug, 1993
(Nov, 1992 to
Jun, 1994)
ble 2). This may be due to the extended CFC-11 phase-out schedules contained
in the Montréal Protocol – 1996 for developed countries and 2010 for developing
countries. Thus, many countries at various geographical locations continued to
contribute CFCs to the atmosphere during the 152 months in our study period,
while those at other locations had stopped. Overall, the eight transitions began between September, 1988 and May, 1989, a period of only nine months. This reflects
the success and acceptability of the protocol all over the globe. Durations of the
13
transition periods are very similar among stations except for South Pole. Thus, it
took almost the same amount of time in different parts of the world for CFC-11
to start dropping linearly with an average rate of about 0.12 ppt per month. The
last column of the table indicates that the global estimate of the CTP is contained
in all the station-specific 99% credible intervals except for those of South Pole and
Cape Grim, with the lower bound for South Pole coming three months later than
that for Cape Grim, making South Pole a greater outlier. Specifically, the transition
for South Pole was estimated to take place over 84 months, an extended period
compared to the other stations. This could be due to the highly unusual weather
conditions specific to the location. CFCs are not disassociated during the long
winter nights in the South Pole. Only when sunlight returns in October does ultraviolet light break the bond holding chlorine atoms to the CFC molecule (Ozone
Facts, NASA, “http://ozonewatch.gsfc.nasa.gov/facts/hole.html”). For this reason, it may be expected for CFCs to remain in the atmosphere over the South Pole
for a longer period of time, and hence, an extended transition period. Indeed,
our findings for South Pole are very similar to those reported by Ghude, Jain and
Arya (2009). To evaluate the trend, the authors used the NASA EdGCM model –
a deterministic global climate model wrapped in a graphical user interface. They
found the average growth rate to be 9 ppt per year for 1983-1992, and about −1.4
ppt per year for 1993-2004, turning negative in the mid 1990’s. With our statistical
modeling approach, we estimated a linear growth rate of 0.6 ppt per month (7.2
ppt per year) prior to December, 1988, a transition between December, 1988 and
November, 1995, and a negative linear phase (−0.12 ppt per month, or −1.44 ppt
per year) after November, 1995.
The eight estimates of within-station variability (σi2 ) are given in the first column of Table 2. We noticed earlier from the profile plot (Figure 2) that Barrow
measurements are more variable, whereas Cape Grim and South Pole show little
variation over time. This is reflected in their within-station variance estimates of
2.97, 0.29, and 0.30, respectively. This also explains, at least partially, as to why
the credible intervals of the South Pole and Cape Grim CTPs did not contain the
estimate of the global CTP. As expected, we found a high estimated correlation between consecutive within-station error terms (AR(1) parameter ρ ≈ 0.81 with 95%
credible interval (0.77, 0.85)).
In summary, our methodology provides a satisfactory fit of the longitudinal
bent-cable model to the CFC-11 data under the assumption of an AR(1) withinstation correlation structure. We are currently investigating the goodness of this
fit in comparison to a higher order AR model. The generality of our methodology
with respect to the order of the AR process makes it a flexible tool in analyzing
different types of longitudinal changepoint data.
5
Conclusion
CFC-11 is a major source for the depletion of stratospheric ozone around the globe.
Since the Montréal Protocol came into effect, a global decrease in the CFC-11 has
14
been observed, a finding confirmed by our analysis. Our analysis, using the longitudinal bent-cable model assuming an AR(1) within-station dependence structure,
revealed a gradual rather than abrupt change, the latter of which is assumed by
most standard changepoint models. This makes scientific sense due to the fact
that CFC molecules can stay in the upper atmosphere for about a century, and
their breakdown does not take place instantaneously. The substantial decrease in
global CFC-11 levels after the gradual change shown by our analysis suggests that
the Montréal Protocol can be regarded as a successful international agreement to
reduce the negative impact of CFCs on the ozone layer.
One possible extension of the proposed bent-cable model for longitudinal data
is to incorporate individual-specific covariate(s) (e. g. effects of instrumentations
specific to different stations in measuring the CFC-11 data) to see if they could
partially explain the variations within and between the individual profiles. One
may incorporate this by modeling the individual-specific coefficients θ i ’s to vary
systematically with the covariate(s) plus a random component. This specification
makes the estimation method more complicated, and is currently in progress.
Acknowledgement
This work is partially supported by NSERC through Discovery Grants to G. Chiu
(RGPIN261806-05) and J.A. Dubin (RGPIN327093-06), and by the Government of
Ontario through an Ontario Graduate Scholarship to S.A. Khan (000113006). We
extend our appreciation to the NOAA/ESRL and ALE/GAGE/AGAGE global
network program, who made their CFC-11 data available to the public. We thank
Geoffrey S. Dutton, NOAA/ESRL, and Derek Cunnold, ALE/GAGE/AGAGE global
network program, for clarifications of their data used in this article. We also thank
Peter X. Song, Professor, Department of Statistics and Actuarial Science, University
of Waterloo, Waterloo, Ontario, for his temporary support through NSERC.
15
Appendix 1: Level 1 of the Hierarchical Model
We can write the first level of our hierarchical model as in Equation (5). This is
formulated from the regression model represented by Equations (1)-(3), and under
the assumptions for ij ’s, i.e.
yij = fij + ij , ij =
p
X
2
φk i,j−k + uij , uij ∼ N (0, σui
).
k=1
Here,
yij = fij + ij
= β0i + β1i tij + β2i qij + ij
= β0i + β1i tij + β2i qij + φ1 i,j−1 + φ2 i,j−2 + . . . + φ1 i,j−p + uij
= β0i + β1i tij + β2i qij + φ1 (yi,j−1 − β0i − β1i ti,j−1 − β2i qi,j−1 )+
φ2 (yi,j−2 − β0i − β1i ti,j−2 − β2i qi,j−2 ) + . . . +
φp (yi,j−p − β0i − β1i ti,j−p − β2i qi,j−p ) + uij
p
p
p
X
X
X
= β0i 1 −
φk + β1i tij −
φk ti,j−k + β2i qij −
φk qi,j−k +
k=1
k=1
k=1
p
X
φk yi,j−k + uij
k=1
= β0i 1 −
p
X
φk + β1i xij + β2i rij +
k=1
p
X
φk yi,j−k + uij
k=1
= µij + uij .
Therefore, we specify the first level as
(2)
(1)
2
2
Yi | yi , β i , αi , σui
, φ ∼ MVN (µi , σui
Ii ).
Appendix 2: Full Conditionals
−1
−2
Full conditionals for β i , β, α∗ , D−1
1 , D2 , σui , φ and αi are given in Equations
(8)-(14) and (16), respectively. Here we will present the derivations for β i , β, D−1
1 ,
−2
−1
∗
σui , φ and αi . Full conditionals for α and D2 can be derived in the same way
as for β and D−1
1 , respectively.
In general, full conditionals are derived from the joint distribution of the vari
0
(2)0
(2)0
ables of interest (Gilks, 1996). Letting Y(2) = Y1 , . . . , Ym , the joint distribu-
16
tion of our hierarchical model (5)-(7) is given by
−2
−2
−1
π(Y(2) , β 1 , . . . , β m , α1 , . . . , αm , σu1
, . . . , σum
, β, α∗ , D−1
1 , D2 , φ)
−1
−2
−2
, β, α∗ , D−1
, . . . , σum
= π Y(2) | β 1 , . . . , β m , α1 , . . . , αm , σu1
1 , D2 , φ) ×
−2
−2
−1
π(β 1 , . . . , β m | α1 , . . . , αm , σu1
, . . . , σum
, β, α∗ , D−1
1 , D2 , φ) ×
−1
−2
−2
, β, α∗ , D−1
, . . . , σum
π(α1 , . . . , αm | σu1
1 , D2 , φ) ×
2
2
−1
π(σu1
, . . . , σum
| β, α∗ , D−1
1 , D2 , φ) ×
−1
∗
−1
−1
π(β| α∗ , D−1
1 , D2 , φ) × π(α | D1 , D2 , φ) ×
−1
−1
π(D−1
1 | D2 , φ) × π(D2 | φ) × π(φ)
=
m
hY
m
i hY
i
(2)
2
Ii ) ×
MVN (Yi | µi , σui
MVN (β i | β, D1 ) ×
i=1
i=1
m
hY
m
i
hY
i
−2 a ab
MVLN (αi | α∗ , D2 ) ×
G σui
| ,
×
2
2
i=1
i=1
MVN β| h1 , H1 × MVN α∗ | h2 , H2 ×
−1
−1
W D−1
× W D−1
× MVN φ| h3 , H3 ) .
1 | ν1 , (ν1 A1 )
2 | ν2 , (ν2 A2 )
Now, to construct the full conditional of a variable, say β i , we need only to pick
out the terms in the joint density which involve β i . Note that any term which does
not depend on β i can be taken as a proportionality constant in the full conditional.
Full Conditional for β i
Picking out the terms in the joint density which involve β i , we get
π(β i | .) ∝ exp
n
−
o
0 (2)
1
1
(2)
0
−1
y
−
µ
y
−
µ
−
(β
−
β)
D
(β
−
β)
.
i
i
i
1
i
i
2
2σui
2 i
Here,
yij − µij = yij − β0i 1 −
p
X
φk − β1i xij − β2i rij −
k=1
p
p
X
φk yi,j−k
k=1
X = zij − β0i 1 −
φk − β1i xij − β2i rij for j = p + 1, . . . , ni .
k=1
17
(2)
In vector-matrix notation, yi − µi = zi − Xi β i . Therefore,
0
o
1 −2
0
−1
zi − Xi β i zi − Xi β i + (β i − β) D1 (β i − β)
σ
π(β i | .) ∝ exp −
2 ui
n 1
−2 0
−2 0
−2 0
−2 0
= exp −
zi Xi β i − σui
β i X0i zi + σui
β i X0i Xi β i +
σui
zi zi − σui
2
o
0
0
0
0
−1
−1
−1
−1
β i D1 β i − β i D1 β − β D1 β i + β D1 β
n
n 1
−2 0
−2 0
−2 0
∝ exp −
− σui
zi Xi β i − σui
β i X0i zi + σui
β i X0i Xi β i +
2
o
0
0
0
−1
−1
−1
β i D1 β i − β i D1 β − β D1 β i
−2 0
zi zi and β 0 D−1
proportionality follows because σui
1 β do not depend on β i
n 1
o
0
0
−2 0
−2 0
0
0
−1
−1
= exp −
− 2 σui β i Xi zi + σui β i Xi Xi β i + β i D1 β i − 2 β i D1 β
2
−2 0
σui zi Xi β i and β 0 D−1
1 β i are scalars
n 1
o
0
0
−2
−2
0
−1
0
−1
− 2β i σui Xi zi + D1 β + β i σui Xi Xi + D1 β i
= exp −
2
n 1
o
0
0
−2
−1
0
−1
− 2β i σui Xi zi + D1 β + β i Mi β i
= exp −
2
−2
0
−1
where M−1
i = σui Xi Xi + D1
n 1
0
−2
−2
0
−1
∝ exp −
− β 0i σui
X0i zi + D−1
1 β − σui Xi zi + D1 β β i +
2
0
o
−2
−2
0
−1
0
−1
β 0i M−1
β
+
σ
X
z
+
D
β
M
σ
X
z
+
D
β
i
i
i i
1
i i
1
i
ui
ui
h
−2
−2
β 0i σui
X0i zi + D−1
is a scalar, and so is β 0i σui
X0i zi + D−1
1 β
1 β =
0
0
−2
−2
0
−1
σui
X0i zi + D−1
1 β β i . Proportionality follows because σui Xi zi + D1 β Mi
−2
σui
X0i zi + D−1
1 β
does not depend on β i
i
n 1
0 −1 o
−2
−2
0
−1
= exp −
β i − Mi σui
X0i zi + D−1
β
M
β
−
M
σ
X
z
+
D
β
,
i
i
1
i i
1
i
ui
2
which is proportional to the
pdf of a trivariate normal distribution with mean vec−2
−1
0 ∗
tor Mi σui Xi yi + D1 β and covariance matrix Mi . Therefore,
−2
0
−1
β i | . ∼ MVN Mi σui Xi zi + D1 β , Mi .
18
Full Conditional for β
Picking out the terms in the joint density which involve β, we get
m
Y
o
n 1
o
1
0
−1
(β i − β)0 D−1
(β
−
β)
exp
−
(β
−
h
)
H
(β
−
h
)
1
1
i
1
1
2
2
i=1
m
io
n 1 hX
0
−1
(β i − β)0 D−1
(β
−
β)
+
(β
−
h
)
H
(β
−
h
)
= exp −
1
1
i
1
1
2 i=1
m
n 1 hX
0
0
0
−1
−1
−1
= exp −
(β 0i D−1
1 β i − β i D1 β − β D1 β i + β D1 β)+
2 i=1
io
0
−1
0
−1
0
−1
β 0 H−1
β
−
β
H
h
−
h
H
β
+
h
H
h
1
1
1
1
1
1
1
1
π(β| .) ∝
exp
n
−
m
n 1 hX
0
0
−1
−1
∝ exp −
(−β 0i D−1
1 β − β D1 β i + β D1 β)+
2 i=1
io
0
0
−1
−1
0
−1
β H1 β − β H1 h1 − h1 H1 β
0
−1
proportionality follows because β 0i D−1
1 β i and h1 H1 h1 do not depend on β
m
io
n 1h
X
0
0
0
−1
−1
−1
− 2β 0 D−1
β
+
m
β
D
β
+
β
H
β
−
2β
H
h
= exp −
1
i
1
1
1
1
2
i=1
0 −1
0
0
−1
−1
0
−1
β i D1 β and h01 H−1
1 β are scalars, and so are β i D1 β = β D1 β i , h1 H1 β =
β 0 H1−1 h1
n 1
o
0
−1
−1
= exp −
− 2 β 0 (D−1
β̃
+
H
h
)
+
β
U
β
1
1
1
1
2
m
X
−1
−1
where β̃ =
β i , and U−1
1 = m D1 + H1
i=1
n 1
0
−1
−1
−1
0
−1
− β 0 (D−1
∝ exp −
1 β̃ + H1 h1 ) − (D1 β̃ + H1 h1 ) β + β U1 β +
2
o
−1
0
−1
−1
(D−1
β̃
+
H
h
)
U
(D
β̃
+
H
h
)
1
1
1
1
1
1
1
h
0
−1
−1
−1
β 0 (D−1
1 β̃ + H1 h1 ) is a scalar, and so is β (D1 β̃ + H1 h1 ) =
0
−1
−1
0
−1
(D−1
1 β̃ + H1 h1 ) β. Proportionality follows because (D1 β̃ + H1 h1 ) U1
i
−1
(D−1
β̃
+
H
h
)
does
not
depend
on
β
1
1
1
n 1
o
−1
−1
−1
−1
0
−1
= exp −
β − U1 (D1 β̃ + H1 h1 )] U1 [β − U1 (D1 β̃ + H1 h1 ) ,
2
19
which is proportional to the pdf of a trivariate normal distribution with mean vec−1
tor U1 D−1
and covariance matrix U1 . Therefore,
1 β̃ + H1 h1
−1
β| . ∼ MVN U1 D−1
β̃
+
H
h
,
U
1
1 .
1
1
Full Conditional for D−1
1
Picking out the terms in the joint density which involve D1 , we get
π(D−1
1 | .) ∝
m
Y
−1
MVN (β i | β, D1 ) W(D−1
1 | ν1 , (ν1 A1 ) )
i=1
m Y
n 1
o
1
0
−1
∝
exp − (β i − β) D1 (β i − β)
1/2
|D
|
2
1
i=1
n ν
o
ν1 −3−1
1
−1
2
|D−1
|
exp
−
tr(A
D
)
1
1
1
2
m
1
1 X
0
−1
=
exp −
(β − β) D1 (β i − β)
|D1 |m/2
2 i=1 i
o
n ν
ν −3−1
1
−1
−1 1 2
tr(A1 D1 )
exp −
|D1 |
2
m
i
m+ν1 −3−1
1 hX
−1
0
−1
−1
2
= |D1 |
exp −
(β − β) D1 (β i − β) + tr(ν1 A1 D1 )
2 i=1 i
m
i
(m+ν1 )−3−1
1 hX
0
−1
−1
−1
2
exp −
tr (β i − β) (β i − β) D1 + tr(ν1 A1 D1 )
= |D1 |
2 i=1
if we let d1i = (β i − β)0 D−1
and d2i = (β i − β), then d1i d2i = tr(di1 di2 ) =
1
tr(d2i d1i ) by the property of the trace of a matrix
m
i
(m+ν1 )−3−1
1 h X
0
−1
−1
−1
2
exp −
tr
(β i − β) (β i − β) D1 + tr(ν1 A1 D1 )
= |D1 |
2
i=1
m
h
i
X
(m+ν1 )−3−1
1
−1
0
−1
2
= |D1 |
exp −
tr
(β i − β) (β i − β) + ν1 A1 D1
,
2
i=1
which is proportional to a Wishart pdf with df m+ν1 and scale matrix
−1
(β i − β)0 + ν1 A1 . Therefore,
D−1
1 |
. ∼ W m + ν1 ,
m
hX
i−1 (β i − β) (β i − β) + ν1 A1
.
0
i=1
20
Pm
i=1
(β i − β)
−2
Full Conditional for σui
2
Picking out the terms in the joint density which involve σui
, we get
−2
π(σui
|
.) ∝
∝
=
∝
=
a ab Ii ) G
,
2 2
n
0 (2)
o −2 a ab 1
1
(2)
G σui ,
exp − 2 yi − µi yi − µi
2
|σui
Ii |1/2
2σui
2 2
o
n
0
1
1
−2 a ab
z
−
X
β
z
−
X
β
G
σ
,
exp
−
i
i i
i
i i
ui
2
2
|σui
Ii |1/2
2σui
2 2
n
0
o
ni2−p
1
−2
σui
exp − 2 zi − Xi β i zi − Xi β i
2σui
a
ab −2 −2 2 −1
σui
σ
exp −
2 ui
0
ni −p+a
zi − Xi β i zi − Xi β i + ab −2
−1
−2
2
σui ,
σui
exp −
2
(2)
MVN (Yi |
2
µi , σui
−2 σui
which is proportional to a gamma pdf with shape parameter (ni − p + a)/2 and rate
0
parameter zi − Xi β i zi − Xi β i + ab /2. Therefore,
−2
σui
|
0
ni − p + a (zi − Xi β i (zi − Xi β i + ab
.∼G
,
.
2
2
Full Conditional for φ
Picking out the terms in the joint density which involve φ, we get
π(φ| .) ∝
m
Y
(2)
2
MVN (yi | µi , σui
Ii ) MVN (φ| h3 , H3 )
i=1
m
Y
n
n 1
o
0 (2)
o
1
(2)
0
−1
∝
exp − 2 yi − µi yi − µi exp − (φ − h3 ) H3 (φ − h3 ) .
2σui
2
i=1
Here, for j = p + 1, p + 2, . . . , ni ,
yij − µij = yij − β0i 1 −
p
X
φk − β1i xij − β2i rij −
k=1
= yij −
p
X
φ yi,j−k − β0i 1 −
β2i qij −
p
X
k=1
p
X
φk qi,j−k
φ yi,j−k
k=1
k=1
p
X
k=1
21
φk
p
X
− β1i tij −
φk ti,j−k −
k=1
= (yij − β0i − β1i tij − β2i qij ) − φ1 (yi,j−1 − β0i − β1i ti,j−1 − β2i qi,j−1 )−
φ2 (yi,j−2 − β0i − β1i ti,j−2 − β2i qi,j−2 ) − · · · −
φp (yi,j−p − β0i − β1i ti,j−p − β2i qi,j−p )
= ij − φ1 i,j−1 − φ2 i,j−2 − . . . − φp i,j−p
where ij = yij − β0i − β1i tij − β2i qij for j = p + 1, p + 2, . . . , ni . In vector-matrix
(2)
notation, yi − µi = i − Wi φ, where i = (i,p+1 , i,p+2 , . . . , i,ni )0 , and Wi is a
(ni − p) × 3 matrix as given in Equation (15). Thus,
m
Y
n
n 1
o
0 (2)
o
1
(2)
(φ
−
h
)
exp − 2 yi − µi yi − µi exp − (φ − h3 )0 H−1
3
3
2σui
2
i=1
m
n 1
o
n
Y
0
o
1
(φ
−
h
)
=
exp − 2 i − Wi φ i − Wi φ exp − (φ − h3 )0 H−1
3
3
2σui
2
i=1
m
i
0
1 h X −2
0
−1
= exp −
σ
i − Wi φ i − Wi φ + (φ − h3 ) H3 (φ − h3 )
2 i=1 ui
m
1 h X −2 0
σui i i − 0i Wi φ − φ0 W0i i + φ0 W0i Wi φ + φ0 H−1
= exp −
3 φ−
2 i=1
i
0
−1
0
−1
0
−1
φ H3 h3 − h3 H3 φ + h3 H3 h3
π(φ| .) ∝
m
i
1 h X −2
0
0
0
0
0
0
−1
−1
σ
− 2 φ Wi i + φ Wi Wi φ + φ H3 φ − 2φ H3 h3
∝ exp −
2 i=1 ui
0
0
0
0
0
−1
i Wi φ and h03 H−1
3 φ are sclars, and so are i Wi φ = φ Wi i , h3 H3 φ =
−2 0
0
−1
φ0 H−1
3 h3 . Proportionality follows because σui i i and h3 H3 h3 do not depend on φ
m
m
X
X
1h
−2
−2
− 2 φ0
σui
W0i i + φ0
σui
W0i Wi φ + φ0 H−1
= exp −
3 φ−
2
i=1
i=1
i
2φ0 H−1
3 h3
m
m
X
X
i
1h
0
0
−2
−2
0
−1
0
−1
= exp −
−2φ
σui Wi i + H3 h3 + φ
σui Wi Wi + H3 φ
2
i=1
i=1
m
X
i
1h
0
0
−2
0
−1
−1
−2φ
σui Wi i + H3 h3 + φ V φ
= exp −
2
i=1
22
−1
where V
=
m
X
−2
σui
W0i Wi + H−1
3
i=1
m
m
X
X
0
1h
0
−2
−2
0
−1
σui Wi i + H3 h3 −
−φ
σui
W0i i + H−1
h
∝ exp −
φ+
3
3
2
i=1
i=1
m
m
X
0 X
i
0
−2
−2
−1
0
−1
0
−1
φ V φ+
σui Wi i + H3 h3 V
σui Wi i + H3 h3
i=1
i=1
m
m
h X
X
0
−2
−2
−1
0
φ0
σui
W0i i + H−1
h
=
h
σ
+
H
W
3
3 φ because
3
3
i i
ui
i=1
φ0
m
X
i=1
−2
σui
W0i i + H−1
3 h3 is a scalar. Proportionality follows because
i=1
m
X
−2
σui
W0i
i +
H−1
3
h3
0
i=1
V
m
X
−2
σui
W0i i + H−1
3 h3
i
does not depend on φ
i=1
m
X
i0 −1
1h
−2
φ−V
σui
W0i i + H−1
h
V
= exp −
3
3
2
i=1
m
h
X
i
−2
0
−1
φ−V
σui Wi i + H3 h3
,
i=1
which isPproportional to the pdfof a p-variate normal distribution with mean vecm
−2
−1
0
tor V
and covariance matrix V. Therefore,
i=1 σui Wi i + H3 h3
X
m
−2
0
−1
φ| . ∼ MVN V
σui Wi i + H3 h3 , V .
i=1
Full Conditional for αi
Picking out the terms in the joint density which involve αi , we can express the full
conditional for αi only up to a proportionality constant, that is,
n
o
1
1
0
exp − 2 (zi − Xi β i ) (zi − Xi β i ) ×
π(αi | .) ∝
γi τi
2σui
n 1
o
∗
∗ 0
−1
∗
∗
exp − (αi − α ) D2 (αi − α ) .
2
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