1. A spatial peaks-over-threshold model in a
nonstationary climate
Martin Roth (KNMI / EURANDOM)
joint work with A. Buishand (KNMI), G. Jongbloed (TU Delft),
A. Klein Tank (KNMI), and H. v. Zanten (TU Eindhoven)
16 February 2011

2. Goals
estimate site specific quantiles / return levels
assess the temporal trends in these quantiles
reduce the estimation uncertainty by spatial pooling
Inspired by the work of M. Hanel, A. Buishand and C. Ferro (2009)
for the block maxima data.

3. Data
gridded daily precipitation data of the winter (DJF)
season over the Netherlands (E-OBS v. 5.0)
for 1950 – 2010
some disadvantages but convenient, and problems might
be smaller for the Netherlands, because of the high station
density
(a) mean winter maxima (b) event on December 3, 1960

4. Components of the Model
GP
distribution
non-stationary
Index flood
IF POT model
non-stat.
threshold

5. GEV/GP distribution
Generalized Extreme Value (GEV) distribution
P (M ≤ x ) = Hξ ∗ ,σ∗ ,µ∗ (x )
−1/ξ ∗

exp − 1 + ξ ∗ x −∗ µ∗
, ξ ∗ = 0,

σ
=
µ∗
exp − exp(− x −∗ ) , ξ ∗ = 0,

σ
Generalized Pareto (GP) distribution
P (Y ≤ y |Y ≥ 0) = Gξ, σ (y )

−1/ξ
1 − 1 + ξy , ξ = 0,

= σ
1 − exp − y , ξ = 0,
σ

6. GEV/GP Relation
Assume GP distribution for the excesses and that the
exceedance times follow a Poisson process, then the following
relationship between the parameters of the GP and the GEV
distribution can be shown:
u − σ (1 − λ ξ ), ξ = 0,
µ∗ = ξ
u + σ ln(λ), ξ = 0,
(1)
σ∗ = σλξ ,
ξ ∗ = ξ,
where u is the threshold in the POT model, and λ is the mean
number of exceedances in one winter.

7. Index flood for POT I
The index flood method assumes that all site specific
distributions are identical apart from a site specific scaling
factor, the index flood1 , i.e. for exceedances we get
Xs
P ≤ x |Xs ≥ us = ψ (x ) ∀s ∈ S , (2)
ηs
where Xs is a random variable representing the site-specific daily
precipitation, us is the site specific threshold, ηs is the index flood
and ψ does not depend on site s.
1 Hosking and Wallis (1997)

8. Index flood for POT II
Index flood equals threshold
We have
ψ(us /ηs ) = P (Xs ≤ us |Xs ≥ us ) = 0 ∀s ∈ S
and because ψ has a continuous distribution it follows that us /ηs
has to be the lower endpoint of the support of ψ for every s ∈ S .
This can be only true, if the index flood is a multiple of the
threshold. Without loss of generality we can set ηs = us .
Index flood also for the excesses
Ys
P ≤ y |Ys ≥ 0 = ψ (y ) ∀s ∈ S , (3)
ηs
where ψ(y ) := ψ(y + 1) is independent of site s.

9. Index flood for POT III
Site specific threshold
The τ-th quantile (τ >> 0.9) of the daily precipitation
amounts is a natural choice for a site specific threshold.
⇒ λs will be approximately constant over the region.
Restriction on the GP parameters
The distribution of the scaled excesses has the following form:
Ys
P ≤ y |Ys ≥ 0 = Gξ s , us (y ) ≡ ψ(y ).
σ (4)
ηs s
Therefore we have:
σs
≡ γ, ξs ≡ ξ ∀s ∈ S . (5)
us
We refer to γ as the dispersion coefficient.

10. Index flood for BM
Assuming constant λ, γ and ξ gives for the GEV parameters:
∗
ξs ≡ ξ
λξ
∗ σ∗ γ −1 − 1 (1 − λ ξ )
, ξ =0
γs := s =
∗ 1
ξ ≡ γ∗
µs
γ−1 −log(λ)
, ξ = 0.
Therefore the transformed parameters fullfil the IF assumption for
BM data2 . This does not apply for the IF model for POT data
proposed by Madsen and Rosbjerg (1997).
2 Hanel, Buishand and Ferro (2009)

11. Nonstationary Threshold
The threshold is determined as the 0.96 linear
regression quantile3 :
(c) Mean of the threshold for the 1950–2010 pe- (d) Trend in the threshold for the 1950–2010
riod in mm period in %
3 Koenker (2005), Kysel´, Picek and Beranov´ (2010)
y a

12. Nonstationary Version of the IF Model
IF restrictions on the GP parameters
σs (t )
ξ s (t ) ≡ ξ (t ), ≡ γ (t ).
us (t )
Quantile estimates
−1 α
qα (s, t ) = us (t ) + Gξ (t ),σs (t ) 1 −
λ
γ (t )
us ( t ) · 1 − ξ (t )
[ 1 − ( λ ) − ξ (t ) ]
α
, ξ (t ) = 0,
=
us (t ) · 1 + γ(t ) ln(λ/α) , ξ (t ) = 0.
Note the factorization into a time and site dependent index flood
and a quantile function, which depends on time only.

14. Composite Likelihood
simplified (not true) likelihoods4 (e.g. independence
likelihood5 or pairwise likelihood)
allows to assess for spatial dependence
specify a certain structure for the parameters, e.g.
¯
γ(t ) = γ1 + γ2 · (t − t ), ξ (t ) = ξ 1 .
maximize:
S T
I (θ ) = ∑ ∑ log fγ(t )us (t ),ξ (t ) (ys (t )) ,
s =1 t =1
ys (t )≥0
where fσ,ξ (y ) is the density of the GP distribution.
4 Varin, Reid and Firth (2011)
5 Chandler and Bate (2007)

15. Asymptotic Normality
ˆ
θI is asymptotically normal with mean θ and
covariance matrix G −1 (θ )
Godambe (sandwich) information
G ( θ ) = H ( θ )J −1 ( θ )H ( θ )
H (θ ) is the expected negative Hessian of I ( θ, Y )
Fisher information or sensitivity matrix
J (θ ) is the covariance matrix of the score θ I ( θ, Y )
referred to as variability matrix
In the independent case we have
J (θ ) = H (θ ) ⇒ G (θ ) = H (θ )

16. Composite Information Criteria
Composite likelihood adaptations of the
Akaike information criterion (AIC) and the
Bayesian information criterion (BIC)
AIC ˆ
= −2 I (θI , Y ) + 2 dim(θ ),
BIC ˆ
= −2 I (θI , Y ) + log(n) dim(θ ),
where dim(θ ) is an effective number of parameters, given by
dim(θ ) = tr H (θ )G (θ )−1 ,
which is the true number of parameters in the case of
independence.

17. Composite Likelihood Ratio (CLR) Test
Cl adaptation of the likelihood ratio test
W =2 I
ˆ
θM1 ; y − I
ˆ
θM0 ; y .
The asymptotic distribution of W is given by a linear combination
of independent χ2 variables, and can be determined using the
Godambe information.
Bootstrap
transform to standard exponentials using the full model M1
sample monthly blocks of the whole region
transform the sampled data back using the nested model M0

18. Composite Likelihood Ratio (CLR) Test
Cl adaptation of the likelihood ratio test
W =2 I
ˆ
θM1 ; y − I
ˆ
θM0 ; y .
The asymptotic distribution of W is given by a linear combination
of independent χ2 variables, and can be determined using the
Godambe information.
Bootstrap
transform to standard exponentials using the full model M1
sample monthly blocks of the whole region
transform the sampled data back using the nested model M0

19. Application I – Models and Information Criteria
Table: IF models used
Model dispersion γ shape ξ
no trend γ1 ξ1
trend in dispersion ¯
γ1 + γ2 ∗ (t − t ) ξ1
trend in shape γ1 ¯
ξ 1 + ξ 2 ∗ (t − t )
Table: Information criteria for the IF models
Model AIC BIC
no trend 78387.28 78715.59
trend in dispersion 78435.60 78880.41
trend in shape 78333.28 78748.95

20. Application I – Models and Information Criteria
Table: IF models used
Model dispersion γ shape ξ
no trend γ1 ξ1
trend in dispersion ¯
γ1 + γ2 ∗ (t − t ) ξ1
trend in shape γ1 ¯
ξ 1 + ξ 2 ∗ (t − t )
Table: Information criteria for the IF models
Model AIC BIC
no trend 78387.28 78715.59
trend in dispersion 78435.60 78880.41
trend in shape 78333.28 78748.95

21. Application II – Shape parameter
Figure: shape parameter for different models (dotted – constant, dashed
– linear trend, solid red – 20 year window estimates)

22. Application III – Significance Tests
Trend in the GP parameters
Table: p-values of the CLR-test against the IF model without trend
Model asymptotic bootstrap
trend in dispersion 82.9% 81.3%
trend in shape 26.7% 12.2%
Index flood assumption
We compare the model without trend in the parameters with a
model with site specific dispersion coefficient and common shape
parameter with the bootstrap. We obtain a p-value of 0.103, i.e.
the IF assumption must not be rejected.

23. Application IV – Uncertainty
Figure: Estimated return levels and 95% pointwise confidence bands of
the excesses distribution at the gridbox around De Bilt in 1980 (black –
at site, red – IF)

24. Conclusions
positive trends in the threshold are observed
this leads to an increase in the scale parameter
no change in the dispersion coefficient
negative trend in the shape parameter not significant
the uncertainty in the excesses distribution is cut in half

25. Further Research
apply the method to climate model data
validity of the bootstrap might be questionable
use of pairwise likelihood could further decrease the
uncertainty
assess the uncertainty in the threshold and excess distribution
together

26. Literature
R.E. Chandler and S. Bate (2007),
Inference for clustered data using the independence loglikelihood.
Biometrika, 94(1):167–183.
M. Hanel, T. A. Buishand, and C. A. T. Ferro (2009),
A nonstationary index flood model for precipitation extremes in transient regional climate model
simulations.
J. Geophys. Res., 114(D15):D15107.
J. R. M. Hosking and J. R. Wallis (1997),
Regional frequency analysis.
Cambridge University Press.
Koenker, R. (2005)
Quantile Regression.
Cambridge University Press
Kysel´, J., J. Picek, and R. Beranov´ (2010)
y a
Estimating extremes in climate change simulations using the peaks-over-threshold with a nonstationary
threshold.
Global and Planetary Change 72, 55-68
H. Madsen and D. Rosbjerg (1997),
The partial duration series method in regional index-flood modeling.
Water Resour. Res., 33(4):737–746.
C. Varin, N. Reid, and D. Firth (2011),
An overview of composite likelihood methods.
Statistica Sinica, 21:5–42.