A framework for interpreting climate model outputs
Nadja A. Leith and Richard E. Chandler
Department of Statistical Science
University College London
Gower Street
London WC1E 6BT, UK
May 13, 2008
Abstract
Projections of future climate are often based on deterministic models of the
earth’s atmosphere and oceans. However, these projections can vary widely between
models, with differences becoming more pronounced at the relatively fine spatial and
temporal scales relevant in many applications. In this paper we suggest that the
resulting uncertainty can be handled in a logically coherent and interpretable way
using a hierarchical statistical model, implemented in a Bayesian framework. Model
fitting using Markov Chain Monte Carlo techniques is feasible but moderately timeconsuming; the computational efficiency can, however, be improved dramatically
by substituting maximum likelihood estimates for the original data. An example,
involving the generation of multivariate atmospheric time series for hydrological
impacts studies, is used to illustrate the methodology.
Some key words: Climate change; Climate model uncertainty; Contemporaneous
ARMA models; Multi-model ensembles; Downscaling; Sufficient statistics
1
Introduction
It is widely acknowledged that human activities have caused changes in the Earth’s climate
(Solomon et al., 2007). Indeed, the Intergovernmental Panel on Climate Change (IPCC)
recently asserted that “the observed pattern of tropospheric warming and stratospheric
cooling is very likely due to the influence of anthropogenic forcing” (Hegerl et al., 2007).
There is also mounting, although less clear-cut, evidence of anthropogenic influences on
Research report No. 294, Department of Statistical Science, University College London.
Date: May 2008.
1
the hydrological cycle (Solomon et al., 2007). To accommodate this possibility therefore,
planners and decision makers are faced with the need to update current infrastructure and
design practice relating to, for example, water resource management and flood defences.
Much of our understanding of climate change is based on deterministic models of the
physical and chemical processes involved. General circulation models (GCMs) produce
output on a coarse grid, with typical resolution around 200 × 200km2 at mid latitudes,
over the entire globe. By contrast, regional climate models (RCMs) operate on a finer
grid resolution (typically around 50 × 50km2 ) over smaller areas such as Europe. RCM
simulations are typically used to add detail to a GCM simulation over a region of interest. To obtain projections of future climate, scenarios of greenhouse gas and other
anthropogenic emissions are used to drive climate model simulations, often to the year
2100. These emissions scenarios are based on “storylines” describing possible patterns of
international economic, industrial and social development (IPCC, 2000).
However, despite continuing improvements in climate models, their use in climate
change impact studies can be problematic. This is particularly true for hydrological
applications, in which hypothetical precipitation sequences are required as input to a hydrological model to determine the response of a system. Many important hydrological
systems respond to relatively localised rainfall events, necessitating the use of precipitation data at a high spatial and temporal resolution (Wheater, 2002): GCMs and even
RCMs often have too coarse a spatial resolution for studies of smaller catchments, which
can be important in, for example, the design of an urban drainage system. Moreover,
impact assessment studies require a large number of potential rainfall sequences which
cannot be provided by climate models due to computational cost. A further complication is that precipitation estimates for climate models are generally acknowledged to
be poor in comparison with other variables such as temperature (Randall et al., 2007;
Christensen et al., 2007). Therefore, in many applications it is common to derive precipitation sequences indirectly from climate model outputs using statistical methods, which
exploit relationships between local-scale precipitation and those large-scale atmospheric
variables that are better reproduced by the climate models. This approach is referred to
as statistical downscaling (Christensen et al., 2007).
Clearly, substantial uncertainty surrounds any projection of future climate: not only
through the inherent unpredictability of the climate system itself, but also the choice of
emissions scenario, errors in the climate model (e.g. incorrect formulation and parameter
values) and, potentially, downscaling method (Haylock et al., 2006; Smith, 2002). In this
paper we focus on the issue of climate model uncertainty, which is now well documented
and has resulted in a shift towards probabilistic climate forecasting based on “ensembles”
of different climate models (McAvaney et al., 2001). A number of articles (e.g., New and
Hulme 2000; Wigley and Raper 2001; Tebaldi et al. 2005; Furrer et al. 2007) have discussed
methods for the probabilistic estimation of the future mean of some climate variable,
often global or regional temperature. However, many applications require consideration
of aspects other than the mean: the response of a hydrological system to a particular
precipitation event will depend on the intensity and duration of the event, which itself is
partly dependent on antecedent weather conditions. In such situations, it is necessary to
represent adequately the detailed structure of the input climate sequences (Wilby et al.,
1998). The problem of characterising uncertainty in this detailed structure has received
2
relatively little attention to date. A simple, and easily interpretable, possibility is to
weight the sequences from different climate models according to some measure of their
performance (e.g., Wilby and Harris 2006). However, in general this will underestimate
the true uncertainty because the results are constrained to lie between the limits set by
the available data: the number of climate models used for any particular study is often
small, raising the possibility that another model will yield more extreme projections. A
further issue is the definition of weights, which can be achieved using heuristic criteria or
by finding a statistical model for which they are in some sense optimal. Indeed, Tebaldi
et al. (2005) showed that the “Reliability Ensemble Averaging” method proposed by
Giorgi and Mearns (2002) can be regarded as a means of computing Bayes estimators
for a particular class of hierarchical model. This work focused on characterising the
uncertainty in regional mean temperatures and has since been extended to deal with
sequences of decadal means (Tebaldi and Sansó, 2008).
In this paper, we also use a hierarchical model to represent climate model uncertainty.
However, our approach differs from that of Tebaldi et al. (2005) in that, rather than
starting from an existing technique and putting it on a formal footing, we attempt to
conceptualise explicitly the way in which climate models work. Our starting point is
to note: first, that in broad terms all climate models represent essentially the same
dynamical processes; and second, that climate model outputs are intended to provide
plausible, rather than exact, scenarios that agree with actual climate statistically rather
than in detail (von Storch and Zwiers 1999, pp. 12, 129; Smith 2002). Thus we may
reasonably expect that the time series outputs from different models will have a similar
structure, which can be described using the same form of statistical model. However,
the parameters of these statistical descriptors will differ between climate models. Our
aim is to establish a distribution for the parameters that describes the “population” of
climate models and hence characterises the uncertainty in their outputs. The idea is
completely generic and is applicable in principle to any type of climate model output
including regional means, spatial fields, time series and space-time sequences.
In this paper we focus on an application involving monthly time series of three variables, which arises in the context of precipitation downscaling for a location in the southern
UK. The methodology is illustrated using data from four GCMs. The data are introduced
in the next section, where it is shown that the different GCMs do indeed yield series with
similar statistical structure. The hierarchical modelling framework is then set out in
Section 3, and inference using Markov Chain Monte Carlo (MCMC) techniques is discussed in Section 4. Although feasible, this MCMC fitting is moderately time-consuming
for the model considered here. In Section 5 therefore, we consider reducing the computational burden by replacing the data with the maximum likelihood estimates (MLEs)
derived from each of the individual GCMs. The rationale for this is that the MLEs are
approximately sufficient statistics in large samples so that the information loss is small.
Results from both fitting methods are discussed in Section 6 and shown to be equivalent
for practical purposes. In Section 7, the resulting hierarchical model is used to generate
future monthly trivariate sequences that take full account of GCM uncertainty. Section 8
provides a summary and discussion of areas for future investigation.
3
Table 1: GCMs used in the study, with acronyms used in the remainder of the paper.
Acronym Institute
Model
CCCma
CSIRO Mk2
MPI
Canadian Centre for Climate Modelling and Analysis
Commonwealth Scientific and Industrial Research
Organisation, Australia
Max Planck Institute, Germany
HC
Hadley Centre, UK
HadCM3
CSIRO
2
CGCM2
ECHAM4
Motivating example
Our work was motivated by the need for future precipitation scenarios in the UK, in applications such as flood risk assessment and water resource management. Such scenarios
are often obtained by downscaling climate model outputs, and it has previously been established on physical and empirical grounds that reasonable downscaling performance can
be achieved by relating local-scale precipitation to coarse-scale atmospheric variables including temperature, pressure and measures of atmospheric moisture content (Wilby and
Wigley, 2000). Leith (2008) developed a statistical downscaling model, relating observed
daily rainfall at a single site to spatially and temporally averaged monthly temperature,
sea-level pressure and relative humidity. These three atmospheric variables were averaged
to a monthly time resolution, and over a spatial region centred on the location for which
precipitation data were required. In the example presented here, this location is a raingauge at Heathrow airport in southern England. Simulated sequences of the atmospheric
variables, for the period 2071-2100, were available from the four GCMs listed in Table
1. All of these future sequences were forced using the IPCC SRES A2 emissions scenario
(IPCC, 2000): uncertainty in future atmospheric composition is not addressed here. The
atmospheric data were standardised, separately for each month of the year, with respect
to the 1961–1990 mean and standard deviation. Such standardisation is routinely applied
in climatology and is an attempt to to adjust for climate model bias (Charles et al., 2007).
Leith (2008) gives further details.
As discussed in Section 1, we might reasonably expect these four GCMs to produce
outputs with a common statistical structure. For i = 1, . . . , 4 and t = 1, . . . , 360, let
Tit , Sit and Rit denote standardised monthly mean temperature, sea-level pressure and
relative humidity at Heathrow respectively, for the ith GCM in month t (t = 1 is January,
2071). Also, let


Tit

Yit =  Sit 

Rit


Y i1



Yi =  ... 
,
Yi360
and
so that Y i is the vector of all available data from the ith GCM. Then the statistical
structure for the ith GCM can be summarised by fitting a trivariate time series model to
Y i . This was carried out, separately for each i, using standard modelling techniques for
regression with autoregressive / moving average (ARMA) errors.
4
The results of this exercise indicated that the structure was indeed similar for the four
GCMs. Specifically, the {Yi } were found to be well described by the following model:
Tit = CTt β Ti + eTit
Sit = CSt β Si + eSit
R
R
Rit = CR
t β i + eit
(1)
(2)
(3)
eTit = φTi eTi(t−1) + δitT
(4)
eSit = φSi eSi(t−1) + δitS
(5)
R
R R
eR
it = φi ei(t−1) + δit
(6)
with
and
δitT
 S 
 δit  ∼ MV N (0, Σi) ,
δitR


(7)
where MV N denotes the multivariate normal distribution, {CTt , CSt , CR
t } are vectors of
covariates at time t and Σi is a non-diagonal covariance matrix. Equations (1) to (7) show
that each variable is represented as a sum of a deterministic component and a zero-mean
first-order autoregressive error process; correlations between the variables are induced
through the covariance matrices {Σi } of the innovations. The resulting model can be
regarded as an extension of the class of contemporaneous ARMA models considered by
Camacho et al. (1987): the extension is the inclusion of covariates {CTt , CSt , CR
t }. These
covariates are the same for all GCMs and include a constant term representing mean
change by 2070 (recall that the data are standardised with respect to a 1961–90 baseline),
a linear trend, covariates representing seasonal variation and interactions between trend
and seasonality. A full description of all the covariates is given in Table 2.
As well as having the same overall structure, there are qualitative similarities between
the parameter estimates for the different GCMs. To illustrate this, Table 3 summarises
the maximum likelihood estimates for parameters relating to temperature. According to
these estimates, the following features are common to all GCMs:
1. Temperature will increase in the future (parameter 1 in Table 3), with this increase
continuing over the 2071-2100 period (parameter 2).
2. The annual cycle for temperature will be more pronounced, with warmer summers
and cooler winters than in the 1961–1990 period (this can be deduced by plotting the
contribution of the “seasonal” parameters 3–8). The interaction between trend and
seasonality ensures that this enhancement of the annual cycle continues throughout
the 2071-2100 period.
3. After accounting for linear trend and seasonality in temperature, there remains a
positive autocorrelation between the monthly values (parameter 13).
5
Table 2: Covariates used in the regression models for temperature, sea-level pressure or
relative humidity. Columns 3 to 5 show whether a particular variable was included in CTt ,
CSt and/or CR
t (“Y” indicates that the variable is included, “N” that it is not included).
Parameter
Description
Temperature
Sea-level
pressure
Relative
humidity
Intercept
Mean shift by 2070
Y
Y
Y
t/120
Linear trend, 2071-2100
(units: decades)
Y
N
Y
cos (2tπ/12)
First seasonal harmonic
(cosine component)
First seasonal harmonic
(sine component)
Y
Y
Y
Y
Y
Y
Second seasonal harmonic
(cosine component)
Second seasonal harmonic
(sine component)
Y
Y
Y
Y
Y
Y
Third seasonal harmonic
(cosine component)
Third seasonal harmonic
(sine component)
Y
Y
Y
Y
Y
Y
Fourth seasonal harmonic
(cosine component)
Fourth seasonal harmonic
(sine component)
Y
Y
Y
Y
Y
Y
Biannual cycle
(cosine component)
N
Y
Y
N
Y
Y
Y
N
Y
Y
N
Y
sin (2tπ/12)
cos (2tπ/6)
sin (2tπ/6)
cos (2tπ/4)
sin (2tπ/4)
cos (2tπ/3)
sin (2tπ/3)
cos (2tπ/24)
sin (2tπ/24)
Biannual cycle
(sine component)
t
cos (2tπ/12)
Interaction between
120
first seasonal
harmonic (cosine component)
and linear trend
t
sin
(2tπ/12)
Interaction
between
120
first seasonal
harmonic (sine component)
and linear trend
6
Table 3: Parameter estimates (and standard errors) for equations (1) and (4) for Temperature. Numbers in square brackets correspond to parameter numbers in Figure 2.
Model
CCCma
CSIRO
MPI
HC
Model
CCCma
CSIRO
MPI
HC
Model
CCCma
CSIRO
MPI
HC
Intercept [1]
3.994
5.399
2.756
1.837
(
(
(
(
)
)
)
)
cos(2tπ/6) [5]
0.120
-0.500
-0.418
-0.369
(
(
(
(
0.083
0.070
0.088
0.099
0.053
-0.341
0.144
0.116
(
(
(
(
0.059
0.040
0.060
0.077
0.264
0.423
0.521
0.366
cos(2tπ/12) [3]
sin(2tπ/12) [4]
(
(
(
(
-0.741
-0.453
-0.564
-0.594
-0.884
-0.106
-1.157
-0.497
0.086
0.088
0.107
0.103
)
)
)
)
)
)
)
)
sin(2tπ/6) [6]
) 0.476 ( 0.078
) -0.246 ( 0.062
) 0.107 ( 0.081
) 0.092 ( 0.093
cos(2tπ/3) [9]
)
)
)
)
sin(2tπ/3) [10]
-0.350
-0.325
-0.223
-0.244
(
(
(
(
0.063
0.047
0.056
0.074
)
)
)
)
(
(
(
(
0.221
0.185
0.260
0.217
)
)
)
)
cos(2tπ/4) [7]
0.290
0.068
0.011
-0.255
(
(
(
(
0.067
0.052
0.067
0.080
0.030
-0.026
-0.215
-0.175
(
(
(
(
0.124
0.106
0.141
0.145
0.174
0.173
0.215
0.208
)
)
)
)
sin(2tπ/4) [8]
)
)
)
)
t
cos(2tπ/12) 120
[11]
(
(
(
(
-0.151
-0.151
0.272
-0.057
(
(
(
(
0.071
0.051
0.069
0.088
)
)
)
)
t
sin(2tπ/12) 120
[12]
)
)
)
)
-0.330
-0.138
-0.162
-0.187
(
(
(
(
0.100
0.098
0.122
0.129
)
)
)
)
φT [13]
Model
CCCma
CSIRO
MPI
HC
0.152
0.155
0.195
0.159
t/120 [2]
0.298
0.497
0.415
0.248
(
(
(
(
0.054
0.043
0.048
0.054
)
)
)
)
However, despite these qualitative similarities, Table 3 also shows that the precise
parameter estimates are often significantly different. For example, while the intercept
estimates for the temperature model are all positive, the precise values vary substantially,
showing that GCM uncertainty is by no means negligible. Quantifying these uncertainties
is the subject of the remainder of this paper.
3
Hierarchical model
It has been shown that, in this particular example, the sequences {Yi } from the different
GCMs have the same general structure as described using a statistical model. Thus each
Y i may be represented by some joint distribution with density fy (yi |C, θ i ), say, where
the form of fy is the same for all i but the parameters, {θ i }, differ. For definiteness in
what follows, we define θ i as follows with relation to the parameters in equations (1)–(7):
θ Ti
=
β Ti
φTi
!
,
θ Si
=
β Si
φSi
!
,
θR
i
=
7
βR
i
φR
i
!
,
and



θi = 


θ Ti
θ Si
θR
i
Σi



.


One could imagine that, given data from more GCMs, the structure of the sequences
would remain essentially the same but that no two GCMs would yield identical θs. Thus,
from the notional “population” of GCMs one would obtain a complete distribution of θ
values, from which the individual {θ i } are drawn. Denote the density of this distribution
by fθ (θ|Θ), where Θ is a vector of hyper-parameters. Then fθ (θ|Θ) provides a concise and
interpretable summary of uncertainty in the GCM outputs. The idea is clearly generic,
and could be applied to other forms of GCM output as discussed in Section 1.
Although this framework is conceptually simple, inference is a formidable task without
recourse to Bayesian methods that make use of MCMC techniques. We therefore take
a Bayesian approach here, which necessitates the specification of a prior distribution for
Θ. The aim then is to derive the posterior distribution of the model parameters {θ, Θ}
given the data y. This is achieved via the usual Bayes factorisation (Gelman et al., 2003,
p. 124)
fθ,Θ (θ, Θ|y) ∝ fy (y|θ, Θ)fθ,Θ (θ, Θ) = fy (y|θ)fθ|Θ (θ|Θ) fΘ (Θ) .
Table 4 presents the different components of the model developed here for the Heathrow
GCM data. Its hierarchical nature is clear. Following the terminology of Gelman et al.
(2003), we refer to the three levels of the hierarchy as the “data level” (i.e. the likelihood fy (y|θ)), the “population level” fθ|Θ (θ|Θ) and the “hyper-prior level”, fΘ (Θ).
Formulation of the latter two levels is discussed below. The data level is defined by the
multivariate time series model presented in Section 2, noting that at t = 1 it is necessary
to use the marginal, rather than conditional, covariance matrix of the innovation vector
T
S
R 0
δ 1t = (δ1t
, δ1t
, δ1t
) due to the lack of any observations at the preceding time point. This
marginal covariance matrix, given in Table 4, is just the unconditional covariance matrix
for this particular stationary vector autoregression.
3.1
3.1.1
Population level
Regression parameters
0
The regression parameters β i = (β Ti , βSi , β R
i ) are assumed to follow a joint multivariate
normal distribution, as proposed by Gelman et al. (2003, Chapter 15), with mean µβ and
non-diagonal covariance matrix Σβ . This allows for correlations between the parameters,
an important aspect of the model. For instance, since temperature sequences exhibited
a significant interaction between annual cycle and linear trend (see Section 2), there is
likely to be dependence between the parameters for the annual cycle and the overall
shift in temperature. One might also expect to find dependence between the parameters
corresponding to different atmospheric variables: if a particular GCM suggests a large
shift in temperature (the first component of β Ti ), it is likely also to have a large shift
in relative humidity (the first component of β R
i ). However, for simplicity, the regression
parameters β i are assumed to be independent of the autoregressive parameters Φi and
the covariance matrix Σi.
8
9
Table 4: The hierarchical multivariate time series model.
Regression components
Hyper-prior
level
fΘ (Θ)
Autoregressive components
Covariance components
−1
Σ−1
β ∼ W ish(Rβ , νβ )
σφT ∼ Unif (0, LφT )
σφS ∼ Unif (0, LφS ) σφR ∼ Unif (0, LφR )
µβ ∼ MV N(0, Rβ )
µφT ∼ N(mφT , σφ2T )
µφS ∼ N(mφS , σφ2S )
µφR ∼ N(mφR , σφ2R )

β Ti
 β Si  = β i ∼ MV N(µβ , Σβ )
βR
i
ziT ∼ N(µφT , σφ2T )
ziS ∼ N(µφS , σφ2S )
ziR ∼ N(µφR , σφ2R )
−1
Σ−1
Σ ∼ W ish(RΣ , νΣ )
Population
level
fθ|Θ (θ|Θ)

φTi =
exp 2ziT −1
exp 2ziT +1
φSi =
exp 2ziS −1
exp 2ziS +1
φR
i =
Σ−1
i ∼ W ish(ΣΣ , J)
exp 2ziR −1
exp 2ziR +1
Data level
fy|θ (y|θ)
Tit = CTt β Ti + eTit
Sit = CSt β Si + eSit
R
R
Rit = CR
t β i + eit

eTit = φTi eTi(t−1) + δitT 
S
S S
S
eit = φi ei(t−1) + δit
R R
R 
eR
it = φi ei(t−1) + δit

eTit = δitT 
eSit = δitS
R 
eR
it = δit
0
Φi = (φTi , φSi , φR
i )
for t > 1
for t = 1
 T 
 δit
 δitS  ∼ MV N (0, Σi)

δitR
 T 
 δit
 δitS  ∼ MV N (0, Σi1 )

δitR
where
Σ [k, l] = Σi [k,l]
i1
1−Φi [k]Φi [l]
3.1.2
Autoregressive parameters
The autoregressive parameters φTi , φSi and φR
i are assumed to be independent of each
other, as well as of Σi. Since the error processes {eT }, {eS } and {eR } represent residuals
from regression models, it is reasonable to assume that they should be stochastically
stationary. This imposes restrictions on the autoregressive parameters (Priestley, 1981),
which appear plausible according to the initial modelling exercise, (see, for example,
Table 3): specifically, if φi denotes any particular autoregressive parameter for climate
model i, it is required that |φi | < 1. When fitting the hierarchical model using MCMC
techniques, it is convenient to reparameterise to ensure that these restrictions are satisfied
automatically. We have achieved this by working with Fisher’s z-transform:
!
1
1 + φi
zi = ln
,
2
1 − φi
and by taking the {zi } to be normally distributed (Table 4). An alternative approach
would have been to use a truncated normal distribution for the {φi }, as done by Tonellato
(2001).
3.1.3
Covariance parameters
For t > 1, the covariance matrix of the white noise innovations {δ it = (δitT , δitS , δitR )0 }
is taken to follow an Inverse-Wishart distribution with scale matrix ΣΣ and J degrees
of freedom where, for ease of application, J is considered known (see Section 4). The
parameterisation used here is that of Gelman et al. (2003). It has been argued that the
Inverse-Wishart family lacks flexibility as J is the only “tuning parameter” available to
express uncertainty in the elements of Σi (Liechty, 2004). However, this is not considered
a particular limitation here, as the available data are unlikely to support more complicated
models for the covariance structure.
3.2
Hyper-prior level
The model specification is completed by using our knowledge of the climate system, obtained from historical climate information, to determine plausible constraints on different
aspects of future climate. These constraints are then used to inform our choices of hyperprior distributions. Such a procedure is justified, since the climate system operates within
physical limits: it is also necessary since, due to the sparsity of the data and the relative
complexity of the model, the use of non-informative hyper-priors could be problematic.
The historical information used consists of atmospheric data for the 1961–1990 period
from the NCEP/NCAR reanalysis data set (Kalnay et al., 1996). Further information
could be obtained from palaeoclimatic reconstructions (Hegerl et al., 2007), but this has
not been attempted here.
10
3.2.1
Regression parameters
The mean of the regression parameters, µβ , was assigned a multivariate normal hyperprior with mean 0 (giving no prior information about the sign of any parameter and
hence the direction of any future change in climate) and covariance Rβ . The covariance
matrix of the regression parameters, Σβ , was assigned an Inverse-Wishart hyper-prior
with scale matrix R−1
β and νβ = p + 2 degrees of freedom, where p = 37 is the number
of elements in β i (see Table 2). This is the smallest value of νβ for which a proper
distribution is obtained, and hence represents the least informative Inverse-Wishart prior
possible (Gelman et al., 2003). This choice of prior gives
E(Σβ ) = Rβ .
The matrix Rβ was itself chosen so that β i would remain in a plausible range, without
being too restrictive. More specifically, the diagonal elements of Rβ were chosen so that
when Σβ is set at its expected value, the central 95% portion of β i ∼ MV N(0, 2Rβ )
approximately matches the range of values that might reasonably be entertained in the
absence of any GCM data. To illustrate, consider the first component of β i , corresponding
to the mean shift in standardized temperature by 2070. On the basis of historical information alone, we consider it unlikely that annual mean temperature in 2070 will lie outside
the range of monthly temperatures observed between 1961 and 1990. Although the idea
is conceptually simple, its implementation is complicated slightly by the fact that the
temperatures have been standardized separately for each month of the year (see Section
2). Our approach was to start with the unstandardized monthly temperature time series
for 1961–90 and to centre this by subtracting the overall mean: each value was then scaled
by the appropriate monthly standard deviation so as to convert the original temperatures
into the same units of measurement as the GCM temperatures, while preserving the seasonal cycle. Let M T denote the maximum, in absolute value, of the resulting
series.
q
q The
first diagonal element of Rβ , denoted Rβ1 , was defined so that (−1.96 2Rβ1 , 1.96 2Rβ1 )
was approximately equal to (−M T , M T ). To define the second diagonal element of Rβ ,
corresponding to the parameter for decadal linear trend in temperature, it was noted that
if mean shift in temperature by 2070 is unlikely to lie outside the range (−M T , M T ),
decadal linear trend in temperature is unlikely to lie outside (−M T /10, M T /10) as there
are approximately 10 decades between the mid-point of the 1961–1990 period and 2070.
A similar approach was taken for the parameters corresponding to the first seasonal harmonic of standardized temperature. Here, the corresponding diagonal elements of Rβ
were calculated on the basis that the amplitudes of both components of the first seasonal
harmonic would lie between zero and twice the amplitude of the average seasonal cycle
(before standardization) for the 1961–1990 period. The resulting value was also used
for the diagonal elements of Rβ corresponding to all other harmonics, since the annual
harmonic of standardized temperature is expected to dominate. The same approach was
used for both sea-level pressure and relative humidity. Off-diagonal elements of Rβ were
set to zero for convenience. Note, however, that this does not constrain the posterior
distribution of µβ to be diagonal.
11
3.2.2
Autoregressive parameters
Hyper-priors are also required for the parameters of the normal distributions for Fishertransformed autoregressive coefficients. For each atmospheric variable (we omit superscripts here for convenience), the mean µφ was taken to be normal with mean mφ and
standard deviation σφ , itself assigned a uniform distribution on the range (0, Lφ ). The
parameters mφ and Lφ were chosen so that, when σφ is set to its expected value of Lφ /2,
the central 95% portion of the distribution for φi corresponds to (0, 2φobs ), where φobs > 0
denotes the corresponding autogressive parameter estimate obtained from 1961–90 data.
Given that, unconditionally, zi ∼ N(mφ , L2φ /2) when σφ is set to Lφ /2, and
1 + 2φobs
1
P (0 ≤ φi ≤ 2φobs ) = 0.95 ⇐⇒ P 0 ≤ zi ≤ ln
2
1 − 2φobs
!!
= 0.95,
it can be seen that,
1
1 + 2φobs
mφ ≈ ln
4
1 − 2φobs
!
and
!
1
1 + 2φobs
Lφ ≈ √ ln
.
1 − 2φobs
32
This choice of Lφ ensures that, within a reasonable range, the prior does not dominate
the data and also safeguards against an unrealistically long right tail in the posterior for
σφ , which can be a problem when noninformative uniform priors are used in such settings
(Gelman, 2006).
3.2.3
Covariance parameters
The final hyper-prior to be determined is for the scale matrix of the Inverse-Wishart
distribution for Σi , ΣΣ . This is itself assigned an Inverse-Wishart distribution, with scale
matrix R−1
Σ and degrees of freedom νΣ . In the absence of any GCM data it seems sensible
to set the unconditional expectation of Σi equal to Σobs , the covariance matrix estimated
from the 1961–90 data. This gives
R−1
Σ =
(J − 4)Σobs
.
νΣ
Furthermore, extremely large values of Σi were considered unlikely as this would result
in standardised future atmospheric sequences dominated by their stochastic rather than
deterministic components. Thus it was assumed that Σi would not vary greatly about
Σobs . A simulation study showed this could be achieved by setting degrees of freedom
J = 10 and νΣ = 30, providing highly informative priors which ensure limited variability
of Σi .
4
Inference using complete time series data
The hierarchical model presented in Table 4 was fitted using OpenBUGS (Thomas et al.,
2006). OpenBUGS is a program for Bayesian analysis of complex statistical models using
12
MCMC techniques, which outputs a sample from the joint posterior of all model parameters. Unfortunately, OpenBUGS does not allow prior distributions to be specified
directly for the parameters of Wishart distributions. Thus, in order to place prior distributions on ΣΣ (see the “Covariance components” column in Table 4), dummy variables
{Xij , j = 1, ..., J} were introduced for GCM i, with each Xij drawn from a multivariate
PJ
0
normal distribution with mean 0 and covariance matrix ΣΣ . Setting Σ−1
i =
j=1 Xij Xij
ensured that, by definition, Σ−1
i ∼ W ish(ΣΣ , J) (Krzanowski, 1990, p. 209).
Joint posterior distributions of all the model parameters were obtained from three
independent MCMC chains, each consisting of 15,000 iterations after an initial burn-in
period of 10,000 iterations. The three chains were initialised from a range of starting
values and inspection of the trace plots and the Gelman-Rubin statistics (Gelman and
Rubin, 1992) suggested that convergence to the posterior had been reached before the
end of the burn in period. Using an Intel Core 2 Duo 1.66 GHz CPU (1GB RAM), the
total time taken to obtain the three chains of 25,000 iterations each was approximately
53 hours. In the next section we consider whether this computational burden can be
reduced.
5
Inference using maximum likelihood estimates
This section proposes an alternative approach to estimating the joint posterior fθ,Θ|y (θ, Θ|y),
which aims to reduce the computing time from that reported in the previous section. This
is achieved by reducing the dimensionality of the data used to fit the model: the {Yi }
are replaced with maximum likelihood estimates (MLEs) of the GCM-specific parameters
{θ i }. The simplification is possible because, in general, MLEs are asymptotically sufficient
statistics and so, in large samples, capture almost all of the available information about
the {θ i } (Casella and Berger, 1990, p. 246). With 360 observations for each atmospheric
variable and each climate model, it is reasonable to assume that the asymptotic results
will hold for practical purposes. Denoting the MLEs by θ̂ therefore, we can use fθ̂ (θ̂|θ)
rather than fy (y|θ) at the data level of the hierarchical model. This drastically simplifies
the calculation of the posterior distributions since, in many large-sample settings, the
density fθ̂ (θ̂|θ) can be taken as multivariate normal. For previous applications of this
idea in the climate change literature, see Berliner et al. (2000) and Lee et al. (2005).
For univariate regression models with first-order autoregressive errors, Hildreth (1969)
showed that the asymptotic distribution of the MLEs is indeed multivariate normal and
gave expressions for their large-sample covariance matrix. Camacho et al. (1987) extended
these results to the multivariate setting, but in the absence of regression components.
They also considered what happens if, instead of fitting the full model to all series simultaneously, the appropriate univariate models are fitted separately to each individual series
and the covariance matrix Σ is estimated from the resulting innovations. In this setting,
the ‘univariate’ MLEs of the mean vector and innovation covariance matrix have the same
asymptotic distribution as the corresponding full MLEs so that the univariate fitting can
be regarded as fully efficient for these parameters. Although it is beyond the scope of the
present paper to extend this result to the case when there are regression components, we
conjecture that it remains valid so that, if β̄ i , Φ̄i and Σ̄i denote the ‘univariate’ MLEs in
13
our model, then β̄ i and Σ̄i are asymptotically efficient. This is useful because software
is not readily available to carry out full likelihood-based inference for the multivariate
model. Therefore, although we anticipate some loss of efficiency in estimating Φi , for
computational convenience here we have used the univariate MLEs. The results in the
next section show that, in the present application, any loss of efficiency is negligible.
The results of Hildreth (1969) and Camacho et al. (1987) suggest that in large samples
and
β̄ i ∼ MV N(β i , Vβ̄i )
Φ̄i ∼ MV N(Φi , VΦ̄i )
vec(Σ̄i ) ∼ MV N(vec(Σi ), VΣ̄i )
approximately, with independence between β̄ i , Φ̄i and vec(Σ̄i ). Here vec(Σi ) is a vector
of the six unique elements of Σi . The covariance matrices of the estimators, here denoted
Vβ̄i , VΦ̄i and VΣ̄i , depend in theory on the parameters β i , Φi and Σi . However, for
convenience in our MCMC implementation they have been taken as fixed and replaced
with consistent estimators. This can be justified on the basis of their connection with the
Fisher information matrix which, if standard asymptotic theory is to apply, must vary
slowly within the parameter region of interest (Davison, 2003, p. 118). VΣ̄i was estimated
using a result from Camacho et al. (1987), who show that the element corresponding to
kl
rs
the covariance between Σ̄i and Σ̄i is given by
sl
ks rl
Σkr
i Σ i + Σi Σi
N
where, for example, Σkl
i denotes the element in the kth row and lth column of Σi and
N is the number of observations. The results of Camacho et al. were not followed,
however, to estimate Vβ̄i and VΦ̄i . Instead, here we exploited the fact that the univariate
estimation can be regarded as maximum likelihood estimation under a misspecified model,
because the univariate estimators can be regarded as maximising an “independence” loglikelihood formed by summing the individual univariate contributions (which have no
parameters in common) and ignoring the correlation between the atmospheric variables.
Standard results for misspecified models show that both Vβ̄i and VΦ̄i can be written in
“information sandwich” form (Davison, 2003, p. 147), involving the information matrix
derived from the independence log-likelihood and the covariance matrix of the associated
score vector. The information matrix is block diagonal, with one block corresponding
to each atmospheric variable. The score vector is a sum of uncorrelated contributions
from each time point (Leith 2008, Appendix C gives precise details) so that an empirical
estimator of its covariance matrix, of the form given by Davison (2003, p. 148), can
be constructed. The reason for estimating Vβ̄i and VΦ̄i in this way is that the blocks
of the information matrix, along with the innovations necessary to calculate the score
contributions, can be extracted directly from the output of univariate fitting routines,
such as arima in R (R Development Core Team, 2006), that provide estimated covariance
matrices for the parameters. Implementation is therefore straightforward using existing
software.
Having calculated the MLEs and estimated their covariance matrices, implementation
in OpenBUGS is straightforward. The computational savings for our example were dra14
6
4
2
0
−2
−4
−6
Prior
Posterior:
Full time series
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9
Posterior:
MLEs
10
11
11
12
Component of µβ
Figure 1: Prior distributions (left) and posterior distributions, fitted using the full data
(middle) and the marginal MLEs (right), for the first 12 elements of µβ which correspond
to temperature components.
matic: three chains of 25,000 MCMC iterations took approximately 50 minutes, less than
one sixtieth of the time taken in the previous section using the full time series data.
6
Results
Figure 1 presents boxplots of 999 samples from the prior and posterior (output directly by
OpenBUGS) distributions for the hyper-parameters associated with regression coefficients
in the temperature models. The differences between prior and posterior distributions
show that there is considerable information in the output from the four GCMs. The
figure also shows that the posterior distributions are extremely similar whether calculated
using the full data or the marginal MLEs. These results are typical: in particular, for all
three variables the posterior distributions for the autoregressive parameters were virtually
indistinguishable from each other. This justifies our earlier claim that negligible loss of
efficiency arises from the use of the {Φ̄i } in place of the {Φ̂i }.
Given the joint posterior distribution for the hyper-parameters, fΘ|y (Θ|y), it is possible
to simulate from the posterior predictive distribution for θ, corresponding to a “potential”,
but as yet unobserved, GCM (Gelman et al., 2003). In other words, we can draw values
of θ from the population of GCMs. This was done by repeatedly sampling from the joint
posterior for the hyper-parameters, fΘ|y (Θ|y), and then using these values to simulate
from fθ (θ|Θ). Figure 2 shows the resulting distributions for θ T (again based on a sample of
size 999), with parameter numbers corresponding to those given in Table 3. The parameter
estimates from the four GCMs are also shown. The predictive posterior distributions
cover a range well beyond that of the observed estimates for {θ i , i = 1, ..., 4}. This is not
15
10
CCCma
MPI
HC
Posterior distribution
−5
0
5
CSIRO
1
2
3
4
5
6
7
8
9
10
11
12
13
Parameter number
Figure 2: Predictive posterior distributions for θ T . Symbols show show the marginal
MLEs for the four available GCMs. Parameter numbers correspond to Table 3. The
dashed line at 0 is provided only as a reference value.
surprising given that data from only four climate models were used to update the relatively
vague prior distributions. Nevertheless, the effect of the climate model data is clear. For
instance, the mean shift in temperature (i.e. parameter 1 in Figure 2) has an interquartile
range well above zero, because all GCMs yielded positive estimates for this parameter.
Similarly, the distributions for the parameters corresponding to annual seasonality in
temperature (i.e. parameters 3 and 4 in Figure 2) both exhibit an interquartile range
below zero.
7
Simulation of atmospheric sequences
Having fitted a hierarchical multivariate time series model to describe the uncertainty in
GCM-simulated sequences of atmospheric variables, we are in a position to simulate a
full range of atmospheric sequences consistent with the chosen emissions scenario, and
hence to address the main aim of this work. To illustrate this, 200 parameter sets (i.e.
values of θ) were sampled from the predictive posterior distributions discussed in Section
6. For each parameter set, one multivariate sequence was then simulated for the 20712100 period. For clarity of display, all sequences were then aggregated to an annual time
scale: Figure 3 shows the resulting distributions of annual mean standardised temperature
for each year of the simulation period. The hierarchical model clearly, and reasonably,
indicates a far greater range of uncertainty than that suggested by the GCMs alone. Such
simulations can be used to drive statistical downscaling models, and hence a provide a
means of accounting for climate model uncertainty in hydrological, and other, impacts
16
MPI
CCCma
CSIRO
6
4
2
−2
0
Standardized temperature
8
10
HC
2070
2075
2080
2085
2090
2095
2100
Year
Figure 3: Posterior predictive distributions of annual mean standardised temperature,
2071–2100. Bands correspond to the 5th, 10th, 25th, 50th, 75th, 90th, and 95th percentiles
of these distributions. Lines show the sequences obtained directly from each of the climate
models.
studies. An example of this is presented in Leith (2008).
8
Discussion
We have proposed a coherent framework within which to think about climate model
uncertainty which, conditional on a particular emissions scenario, provides a logically
consistent way for analysts to combine information from climate models with their own
assessments of likely future changes. Our approach attempts to conceptualise the way that
climate models work, based on the premise that the outputs from different models should
have similar structure because they represent the same underlying processes. Although
we have focused on a specific application involving multivariate monthly time series, the
fundamental idea is generic and could be applied to other types of model output. The
approach could also find applications in many other areas where deterministic models are
used.
The results above show that the proposed methodology can be employed effectively to
simulate future climatic sequences that take account of GCM uncertainty. If the posterior
distributions are based on the MLEs rather than on the full data then the computational
cost is relatively modest, so that the methodology is suitable for routine implementation.
There remain, however, a number of issues that require further attention. Firstly, the
results presented here are based on the output from only four climate models: it would
be informative to refit the model using a comprehensive set of GCM experiments, such
17
as the data from phase 3 of the World Climate Research Programme’s Coupled Model
Intercomparison Project (Meehl et al., 2007a). Furthermore, because so few climate
models were used, the results are likely to be sensitive to the choice of prior distribution:
although we have illustrated some general principles for choosing priors in this kind of
situation, there is of course no single “correct” prior and an analysis investigating the
effect of different choices would be extremely informative. Another possibility would to
use reconstructions of palaeoclimate to inform the choice of prior distributions (Hegerl
et al., 2007).
A final issue, which is arguably the most important, is that the climate models are
regarded as exchangeable in the methodology described here, so that no account is taken
of known differences between them in terms of past performance. Ongoing work aims
to improve on this, by incorporating information both from atmospheric observations
and from climate models forced using historical emissions. This information effectively
provides a handle on the relationship between climate models and the true climate system,
and enables us to determine which climate models are best able to reproduce particular
features of past climate. By extending the framework set out in this paper to incorporate
such information in a structured way, we hope to reduce the uncertainty in future scenarios
by downweighting under-performing climate models, and to allow for the communal bias
in all climate models that arises from knowledge being shared between modelling groups
(Meehl et al., 2007b; Tebaldi et al., 2005).
Acknowledgements
This work was funded by the UK Department for the Environment, Food and Rural
Affairs (R&D project FD2113). We are grateful to Rob Wilby, David Stephenson and
Nick Reynard for providing data and advice to the project, and to our colleagues Valerie
Isham, Christian Onof and Howard Wheater for helpful discussions.
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