Download A hierarchical model for extremes of ozone, NO and NO2

A hierarchical model for extremes of ozone, NO and
NO2
Emma Eastoe
Department of Mathematics and Statistics
Lancaster University, UK
September 2009
Outline
1
Aims
Data
Questions of interest
2
Background - univariate extremes
Stationary
With covariates
3
Hierarchical model
The model
Simulation
Inference
4
Results
Margins
Joint distribution of NO and NO2
5
Comments
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
2 / 21
The data
Daily maxima of hourly ozone, NO and NO2 concentrations at a site
in central Reading, UK
NO and NO2 are primary pollutants e.g. released during combustion
of fossil fuels
Ozone is a secondary pollutant, i.e. formed in the atmosphere
NO and NO2 are precursors for ozone
Temperature, duration/intensity of sunlight (photochemical reaction)
and wind speed (dispersion) are also important factors in determining
concentrations
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
3 / 21
2001
150
NO2
100
√
√
8
NO2
8 10
1998
1999 2000
Time
2001
O3
10 12 14
2001
O3
10 12 14
1999 2000
Time
0
5
10 15 20 25
√
NO
Emma Eastoe (Lancaster University)
6
4
4
4
6
6
√
1998
8
1999 2000
Time
12
1998
0
0
50
200
50
NO
400
O3
100 150
600
200
800
Data in pictures
0
5
10 15 20 25
√
NO
Hierarchical model for extremes
4
6
8 10
√
NO2
12
September 2009
4 / 21
Notation
Y : d-dimensional random variable, generally observe a sequence Yi
of length n
X : associated n × p matrix of covariates (may be fixed or
time-varying)
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
5 / 21
General questions for MV extremes
The (univariate) marginal probability that one of the components is
large
The probability that all components of the vector are simultaneously
large
Conditional on a subset A of the components being large, the
probability distribution of the d − |A| remaining components
The probability that some function of (a subset of) the components is
large.
Generally, ‘is large’ means exceeds some high level.
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
6 / 21
Stationary univariate extremes
Stationary sequence of random variables {Yt }
Asymptotically motivated peaks over threshold model (e.g. Davison
and Smith, 1990)
For high enough level u model the size and rate of (local maxima of)
threshold exceedances
Size is modelled by a generalised Pareto distribution (GPD), for
y > 0,
ξy −1/ξ
Pr[Yt > u + y |Yt > u] = 1 +
,
ψu +
where ψu > 0 is threshold dependent.
Model the rate φu = Pr[Yt > u] as a Bernoulli random variable.
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
7 / 21
Univariate extremes with covariates
Model (functions of the) parameters as functions of covariates, xt
We consider linear functions only
log ψu (xt ) = ψ ′ xt ,
ξ(xt ) = ξ ′ xt
and
logitφu (xt ) = φ′ xt
where ψ, ξ and φ are vectors of regression coefficients
Inference straightforward through maximum likelihood, which can be
split into two parts - one for (ψ, ξ) and one for φ
Other recent work has looked at modelling covariates using additive
(Chavez-Demoulin and Davison, 2005) or local likelihood (Davison
and Ramesh, 2000 and Hall and Tajvidi, 2000) models.
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
8 / 21
Preprocessing
First model aspects of the underlying process and use these to
preprocess the data, producing an approximately stationary sequence
In the absence of a better model, we suggest the following
λ(xt )
Yt
−1
= µ(xt ) + σ(xt )Zt
λ(xt )
where σ(xt ) > 0 and Zt is a assumed to be stationary
Model the extremes of Zt using a threshold uz and the
GPD(ψu (xt ), ξ(xt )) and rate φu (xt ) model
Functions of location, scale and Box-Cox parameters modelled as
linear functions of covariates (see Eastoe and Tawn, 2009)
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
9 / 21
Model description
Suppose we can order the d-dimensional random variable in some
meaningful way Yt = (Y1t , . . . , Ydt )
Then model Yit conditionally on
Covariates Xt , and
The set Si = {Yjt : j < i}, which is empty for i = 1
In our example, we model NO conditional on covariates, NO2
conditional on NO and covariates and ozone conditional on both NO,
NO2 and covariates
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
10 / 21
Model description 2
At each level we model the sequence {Yit } using the pre-processing
approach, allowing a model on both tails of each component:
1
Estimate the parameters µi (xt , Sit ), σi (xt , Sit ) and λi (xt , Sit )
2
Transform Yit to Zit
3
4
Obtain upper and lower thresholds of Zit , denoted uil (lower) and uiu
(upper)
Estimate the GPD and rate parameters using the data above (below)
the thresholds uiu (uil ) : φli (xt , Sit ), φui (xt , Sit ), etc.
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
11 / 21
Simulation
To estimate extreme probabilities, we use simulation
Denote simulated data by (Y∗ , X∗ )
Since daily data, simulation of N-years of data requires L = 365N
observations
To simulate covariates for day t ∈ {1, . . . , 365} in any given year,
since we have no model, resample across the observed years from
values observed on day t only.
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
12 / 21
Simulation 2
Simulate Yt in the order Y1t , . . . , Ydt . For each i , conditional on
∗ , . . . , Y∗
∗
(Y1t
(i −1)t , Xt ),
1
Simulate a Ut ∼ Uniform(0, 1)
2
If Ut > φui (x∗t , Sit∗ ), then simulate Zit∗ from the upper tail model
3
If Ut < φli (x∗t , Sit∗ ), then simulate Zit∗ from the lower tail model
4
5
If φli (x∗t , Sit∗ ) ≤ Ut ≤ φui (x∗t , Sit∗ ), then resample Zit∗ from
{Zit : uil ≤ Zit ≤ uiu }
Back transform to the original scale
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
13 / 21
Inference
Adopt a Bayesian framework : posterior distributions, better able to
quantify uncertainty
Requires MCMC : Gibbs updates for location and rate parameters,
Metropolis-Hastings random walk for the rest
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
14 / 21
4
2
0
0
0
5
5
Wind
10
15
20
Temp
10 15 20 25 30
Sun
6 8 10 12 14
Covariates Xt
1998
1999 2000
Time
2001
1998
1999 2000
Time
2001
1998
1999 2000
Time
2001
Also include first order interactions and indicators for
Weekend
Year
Season
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
15 / 21
Preprocessing results
50
Z3t
-4 -3 -2 -1 0 1
Y3t
100 150 200
2 3
Recalling that Y1t is NO, Y2t is NO2 and Y3t is ozone, and using 90%
thresholds:
2 3
1998 1999 2000 2001
Time
Z3t
-4 -3 -2 -1 0 1
Z3t
-4 -3 -2 -1 0 1
2 3
1998 1999 2000 2001
Time
-3 -2 -1 0 1 2 3 4
Z1t
Emma Eastoe (Lancaster University)
-4
Hierarchical model for extremes
-2
0
Z2t
2
4
September 2009
16 / 21
95%
99% CR
0
200
400
Empirical
600
800
0
95%
99% CR
0
0
0
50
500
Model
1000
95%
99% CR
Model
50 100 150 200 250 300
Model
100 150 200 250
1500
Goodness of fit
50
100
Empirical
150
50
100 150
Empirical
200
QQ plots with 95% (99%) credibility regions. Simulated data sets of the
same length as the observed data for each of the samples from the
posterior. These were ordered and the median (quantiles) found.
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
17 / 21
Prediction
Estimated 5-, 10- and 100-year return levels, with 95% credibility regions.
NO
NO2
O3
5-year
706
(596,910)
156
(146,175)
218
(197,247)
10-year
819
(670,1143)
167
(152,194)
234
(208,271)
100-year
1346
(914,3381)
210
(174,364)
289
(239,387)
5-year levels exceeded only 2 or 3 times for each series
Only 10-year level for NO2 exceeded (once)
None of 100-year levels exceeded
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
18 / 21
Conditional joint distribution
250
Conditional on ozone achieving it’s 10-year marginal return level. This
condition deceases the probability that both NO and NO2 exceed their
marginal threshold level, but increases the probability that at least one
does.
0
50
100
NO2
150
200
E1
0
Emma Eastoe (Lancaster University)
100
200
300
NO
400
Hierarchical model for extremes
500
September 2009
19 / 21
Comments
Presented an approach for the prediction of the probabilities of joint
and marginal extreme events for a multivariate, non-stationary data
set
Proposed method is a hierarchical extension to the pre-processing
method previously described in the univariate case
Drawback - relies on the assumption that we can extrapolate the
response-covariate regression model
Method also relies on finding a sensible ‘hierarchy’
Future work - how well does the model capture levels of asymptotic
(in)dependence?
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
20 / 21
References
Chavez-Demoulin, V. and Davison, A.C. Generalized additive modelling of sample
extremes. Appl. Statist. 54, 207-222.
Davison, A.C. and Smith, R.L. (1990) Models for exceedances over high
Thresholds, with discussion. J. Roy. Statist. Soc. B., 52, 393-442.
Davison, A.C. and Ramesh, N.I. (2000) Local likelihood smoothing of extremes.
J. R. Statist. Soc. B, 62, 191-208.
Eastoe, E.F. (2008) A hierarchical model for non-stationary multivariate
extremes: a case study of surface-level ozone and NOX data in the UK.
Environmetrics, 20:428-444.
Eastoe, E.F. and Tawn, J.A. (2009) Modelling non-stationary extremes with
application to surface-level ozone. Appl. Statist. 58, 25-45.
Hall, P. and Tajvidi, N. (2000) Nonparametric analysis of temporal trend when
fitting parametric models to extreme-value data. Statist. Sci., 2, 153-167.
Emma Eastoe (Lancaster University)
Hierarchical model for extremes
September 2009
21 / 21