TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 1
Modelling jointly low, moderate andHEAVYrainfall intensities without a threshold selection
Raphael Huser c©2016(King Abdullah University of Science and Technology, SA)
Agenda
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 2
� KAUST
� Univariate modeling of rainfall intensities (“wet rainfall events”)
Naveau, Huser, Ribereau and Hannart (2016, to appear), Modeling jointly
low, moderate and heavy rainfall intensities without a threshold selection,
Water Resources Research
� Discussion about spatial joint models for extreme and non-extreme data
Krupskii, Huser, Genton (2016, submitted), Factor copula models for spatial
data
KAUST
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 3
� Newly founded (i.e., 6-year old) graduate university in Thuwal, SaudiArabia;
� Located by the Red Sea;
� Highly multi-cultural (more than 100 nationalities on campus);
KAUST
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 4
KAUST
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 5
� Current KAUST President is the former president of CalTech;
� Currently, about 140 faculty, 150 research scientists, 400 postdocs, 850students (academic population: ∼ 1600; campus population: ∼ 6000);
� 3 statistics groups:
– M. G. Genton (Spatio-Temporal Statistics and Data Science)
– Y. Sun (Environmental Statistics)
– R. Huser (Extreme Statistics), http://extstat.kaust.edu.sa
We are looking for talented PhD students and postdocs.
Extremes: Rarity vs Intensity
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 6
LOW – 2011 European Drought HIGH – 2005 European Floods
� An extreme event is rare (by definition), but not necessarily intense.
� We would like to characterize rainfall intensities from low extremes tohigh extremes.
� Extreme-Value Theory is a natural starting point.
Extreme-Value Theory
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 7
Let X ∼ F (x) be a random variable. Under mild assumptions, the tails of Fshould be (asymptotically) Generalized Pareto (GP) distributed, i.e.,
� Pr(X > x | X > u) ≈ 1−Hξ
(
x−uσ
)
, for large x > u, where
1−Hξ(x) =
{
(1 + ξx)−1/ξ+ , ξ 6= 0,
exp(−x), ξ = 0,
and ξ ∈ R is the upper tail shape parameter.
Extreme-Value Theory
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 7
Let X ∼ F (x) be a random variable. Under mild assumptions, the tails of Fshould be (asymptotically) Generalized Pareto (GP) distributed, i.e.,
� Pr(X > x | X > u) ≈ 1−Hξ
(
x−uσ
)
, for large x > u, where
1−Hξ(x) =
{
(1 + ξx)−1/ξ+ , ξ 6= 0,
exp(−x), ξ = 0,
and ξ ∈ R is the upper tail shape parameter.
� Pr(−X > −x | −X > −u) ≈ 1−Hξ
(
−x+uσ
)
, for large −x > −u.
Extreme-Value Theory
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 7
Let X ∼ F (x) be a random variable. Under mild assumptions, the tails of Fshould be (asymptotically) Generalized Pareto (GP) distributed, i.e.,
� Pr(X > x | X > u) ≈ 1−Hξ
(
x−uσ
)
, for large x > u, where
1−Hξ(x) =
{
(1 + ξx)−1/ξ+ , ξ 6= 0,
exp(−x), ξ = 0,
and ξ ∈ R is the upper tail shape parameter.
� Pr(−X > −x | −X > −u) ≈ 1−Hξ
(
−x+uσ
)
, for large −x > −u.With the constraint X > 0, this implies that for small 0 < x < u,
Pr(X < x | X < u) ≈ Cst× xκ
where κ > 0 is the lower tail shape parameter.
Rainfall data in Lyon, France
(1996–2011)Spring data
Rainfall amounts [mm]
Den
sity
0 2 4 6 8 10
0.0
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0.8
Summer data
Rainfall amounts [mm]
Den
sity
0 2 4 6 8 10
0.0
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Fall data
Rainfall amounts [mm]
Den
sity
0 5 10 15
0.0
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0.6
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Winter data
Rainfall amounts [mm]
Den
sity
0 5 10 15
0.0
0.4
0.8
1.2
Classical approach in EVT
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 9
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0 200 400 600 800
02
46
81
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Time index
Ra
infa
ll a
mo
un
t [m
m]
Spring data
Rainfall amounts [mm]
De
nsity
0 2 4 6 8 100
.00
.40
.81
.2
Classical approach in EVT
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 10
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Spring data
Rainfall amounts [mm]
De
nsity
0 2 4 6 8 100
.00
.40
.81
.2
Classical approach in EVT
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 11
0 200 400 600 800
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81
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Spring data
Rainfall amounts [mm]
De
nsity
0 2 4 6 8 100
.00
.40
.81
.2
Classical approach in EVT
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 12
0 200 400 600 800
02
46
81
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Spring data
Rainfall amounts [mm]
De
nsity
0 2 4 6 8 100
.00
.40
.81
.2
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Our goal: To construct simple parametric models for the full range of rainfallintensities
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Our goal: To construct simple parametric models for the full range of rainfallintensities, continuous (i.e., bypassing threshold selection to separate low,moderate and large rainfall)
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Our goal: To construct simple parametric models for the full range of rainfallintensities, continuous (i.e., bypassing threshold selection to separate low,moderate and large rainfall), easy to simulate from
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Our goal: To construct simple parametric models for the full range of rainfallintensities, continuous (i.e., bypassing threshold selection to separate low,moderate and large rainfall), easy to simulate from, Gamma-like
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Our goal: To construct simple parametric models for the full range of rainfallintensities, continuous (i.e., bypassing threshold selection to separate low,moderate and large rainfall), easy to simulate from, Gamma-like, butheavy-tailed
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Our goal: To construct simple parametric models for the full range of rainfallintensities, continuous (i.e., bypassing threshold selection to separate low,moderate and large rainfall), easy to simulate from, Gamma-like, butheavy-tailed, in compliance with Extreme-Value Theory on the left and righttails.
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Our goal: To construct simple parametric models for the full range of rainfallintensities, continuous (i.e., bypassing threshold selection to separate low,moderate and large rainfall), easy to simulate from, Gamma-like, butheavy-tailed, in compliance with Extreme-Value Theory on the left and righttails.Furthermore, inference should be simple.
Modeling of rainfall intensities
LOW – 2011 European Drought HIGH – 2005 European Floods
Our goal: To construct simple parametric models for the full range of rainfallintensities, continuous (i.e., bypassing threshold selection to separate low,moderate and large rainfall), easy to simulate from, Gamma-like, butheavy-tailed, in compliance with Extreme-Value Theory on the left and righttails.Furthermore, inference should be simple. And return levels easy to compute.
Literature review
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 14
� Wilks (2006, Elsevier): Use of Gamma distribution for rainfall data;
� Katz et al. (2002, Ad. Water Res.): Use of GPD for rainfall extremes;
� Carreau and Bengio (2006, Extremes): Mixtures of hybridGaussian–Pareto densities for moderate and extreme data;
� Frigessi et al. (2002, Extremes): Mixture of light-tailed and GP densities,with mixture proportion depending smoothly on the event “extremeness”;
� Carvalho et al. (2013, Tech Rep): Non-stationary extreme-value mixturemodel via P-splines;
� Papastathopoulos and Tawn (2013, JSPI): GP distribution extensions toimprove estimation of the right tail.
Rainfall intensities modelling
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 15
� Let U ∼ Unif(0, 1). Then, X = σH−1ξ (U) is distributed according the
the GP distribution with scale σ and shape ξ.
Rainfall intensities modelling
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 15
� Let U ∼ Unif(0, 1). Then, X = σH−1ξ (U) is distributed according the
the GP distribution with scale σ and shape ξ.
� To enrich this family of distributions, let X = σH−1ξ (V ), where
V = G−1(U) and G is a continuous distribution on [0, 1].
Rainfall intensities modelling
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 15
� Let U ∼ Unif(0, 1). Then, X = σH−1ξ (U) is distributed according the
the GP distribution with scale σ and shape ξ.
� To enrich this family of distributions, let X = σH−1ξ (V ), where
V = G−1(U) and G is a continuous distribution on [0, 1].
– The distribution F of X is F (x) = G{Hξ(x/σ)}.
Rainfall intensities modelling
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 15
� Let U ∼ Unif(0, 1). Then, X = σH−1ξ (U) is distributed according the
the GP distribution with scale σ and shape ξ.
� To enrich this family of distributions, let X = σH−1ξ (V ), where
V = G−1(U) and G is a continuous distribution on [0, 1].
– The distribution F of X is F (x) = G{Hξ(x/σ)}.
– One can easily simulate from F , whenever G−1 is available.
Rainfall intensities modelling
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 15
� Let U ∼ Unif(0, 1). Then, X = σH−1ξ (U) is distributed according the
the GP distribution with scale σ and shape ξ.
� To enrich this family of distributions, let X = σH−1ξ (V ), where
V = G−1(U) and G is a continuous distribution on [0, 1].
– The distribution F of X is F (x) = G{Hξ(x/σ)}.
– One can easily simulate from F , whenever G−1 is available.
– Similarly, quantiles (i.e., return levels) may be readily obtained.
Rainfall intensities modelling
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 15
� Let U ∼ Unif(0, 1). Then, X = σH−1ξ (U) is distributed according the
the GP distribution with scale σ and shape ξ.
� To enrich this family of distributions, let X = σH−1ξ (V ), where
V = G−1(U) and G is a continuous distribution on [0, 1].
– The distribution F of X is F (x) = G{Hξ(x/σ)}.
– One can easily simulate from F , whenever G−1 is available.
– Similarly, quantiles (i.e., return levels) may be readily obtained.
– G must be such that
⊲ The right-tail of F behaves like Hξ (x → ∞);
⊲ The left-tail of F behaves like xκ (x → 0).
Sufficient conditions on G
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 16
(A) For some finite a > 0, one has
limv↓0
1−G(1−v)v = a;
(B) For any positive function w(v) such that w(v) = 1 + o(v) as v → 0, onehas for some finite b > 0
limv↓0
G{v w(v)}G(v) = b;
(C) For some finite c > 0, one has
limv↓0
vκ
G(v) = c.
Parametric models for G
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 17
� G(v) = vκ, κ > 0: obeys conditions (A), (B), and (C).
� F (x) has only three parameters (κ, σ and ξ) and corresponds to theextended GP model Type III in Papastathopoulos and Tawn (2013, JSPI).
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
Model (i): Case G(v) = vκ
x
De
nsity f
(x)
κ = 1 (GP)
κ = 2κ = 5Gamma
Parametric models for G
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 18
� G(v) = pvκ1 + (1− p)vκ2 , p ∈ [0, 1], κ1, κ2 > 0:obeys conditions (A), (B), and (C) (with κ = min(κ1, κ2)).
� F (x) has five parameters (p, κ1, κ2, σ and ξ).
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
Model (ii): Case G(v) = (vκ1 + v
κ2) / 2
x
De
nsity f
(x)
κ1 = 2 , κ2 = 2κ1 = 2 , κ2 = 5κ1 = 2 , κ2 = 10
Gamma
Parametric models for G
� G(v) = 1−Qδ{(1− vδ)}, where δ > 0 and Qδ is the Beta distributionwith parameters 1/δ, 2: obeys conditions (A), (B), and (C) (with κ = 2).
� F (x) has only three parameters (δ, σ and ξ).
� One may write f(x) = Cst× σ−1hξ(x/σ)[1− {1−Hξ(x/σ)}δ], which
makes a link with skewed distributions.
� One can generalize the model to G(v) = [1−Qδ{(1− vδ)}]κ/2 withκ > 0.
0.0
0.2
0.4
0.6
0.8
1.0
Model (iii): Case G(v) = 1 − Qδ{(1 − v)δ}
De
nsity f
(x)
δ = ∞ (GP)
δ = 5δ = 1Gamma
Inference
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 20
� Maximum Likelihood Estimator (MLE)
– based on the density f(x) = σ−1g{Hξ(x/σ)}hξ(x/σ), x > 0.
– straightforward to apply whenever the density g(v) is available.
– very efficient (especially for large sample sizes), but may suffer frommodel misspecification.
� Probability Weighted Moments (PWMs)
– equate empirical and theoretical probability weighted moments of theform E[X{1− F (X)}s], s = 0, 1, . . ..
– need explicit expression of theoretical moments.
– usually fast to compute.
– typically more robust than MLEs.
– less efficient (in large samples) but have good small sample properties.
Simulation study
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 21
Setting: X1, . . . , Xniid∼ F (x) = G{Hξ(x/σ)}, x > 0.
� Sample size: n = 300 (similar to the rainfall data)
� GP scale parameter σ = 1, GP shape parameter ξ = 0.1, 0.2, 0.3.
� Four different models for G:
(i) G(v) = vκ, with lower tail parameter κ ∈ {1, 2, 5, 10};
(ii) G(v) = 1−Qδ
{
(1− v)δ}
, with skewness parameter δ ∈ {0.5, 1, 2, 5}
(iii) G(v) = pvκ1 + (1− p)vκ2 , with p = 0.4, κ1 ∈ {1, 2, 5, 10} andκ2 ∈ {2, 5, 10, 20}, with κ1 < κ2;
(iv) G(v) = [1−Qδ
{
(1− v)δ}
]κ/2, with δ ∈ {0.5, 1, 2, 5}, andκ ∈ {1, 2, 5, 10}.
� Replicates: R = 105
Selected simulation results
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 22
Model (i) G(v) = vκ
PWM ML
12
34
56
κ = 2
PWM ML
0.5
1.0
1.5
σ = 1
PWM ML
0.0
0.2
0.4
0.6
ξ = 0.2
PWM ML
51
01
52
0
0.99−quantile
Application to the rainfall data in
Lyon, France (1996–2011)Spring data
Rainfall amounts [mm]
Den
sity
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
Summer data
Rainfall amounts [mm]
Den
sity
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
Fall data
Rainfall amounts [mm]
Den
sity
0 5 10 15
0.0
0.2
0.4
0.6
0.8
Winter data
Rainfall amounts [mm]
Den
sity
0 5 10 15
0.0
0.4
0.8
1.2
Application to the rainfall data in
Lyon, France (1996–2011)
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 24
� Hourly rainfall data (1996–2011)
Application to the rainfall data in
Lyon, France (1996–2011)
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 24
� Hourly rainfall data (1996–2011)
� Temporal dependence
⇒ Declustering: keep every third observation
Application to the rainfall data in
Lyon, France (1996–2011)
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 24
� Hourly rainfall data (1996–2011)
� Temporal dependence
⇒ Declustering: keep every third observation
� A high proportion of zeros!
⇒ Focus on wet rainfall events
Application to the rainfall data in
Lyon, France (1996–2011)
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 24
� Hourly rainfall data (1996–2011)
� Temporal dependence
⇒ Declustering: keep every third observation
� A high proportion of zeros!
⇒ Focus on wet rainfall events
� Discretization (rainfall measured to the closest 0.1mm)
⇒ Use censored ML and PWMs (below 0.5mm)
⇒ About 400 fully informative observations (above this threshold)
Application to the rainfall data in
Lyon, France (1996–2011)
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 24
� Hourly rainfall data (1996–2011)
� Temporal dependence
⇒ Declustering: keep every third observation
� A high proportion of zeros!
⇒ Focus on wet rainfall events
� Discretization (rainfall measured to the closest 0.1mm)
⇒ Use censored ML and PWMs (below 0.5mm)
⇒ About 400 fully informative observations (above this threshold)
� We fit separate models for each season.
Application to the rainfall data in
Lyon, France (1996–2011)
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 24
� Hourly rainfall data (1996–2011)
� Temporal dependence
⇒ Declustering: keep every third observation
� A high proportion of zeros!
⇒ Focus on wet rainfall events
� Discretization (rainfall measured to the closest 0.1mm)
⇒ Use censored ML and PWMs (below 0.5mm)
⇒ About 400 fully informative observations (above this threshold)
� We fit separate models for each season.
� Uncertainty assessment: using non-parametric bootstrap.
Parameter estimates for model (i)
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 25
Spring data
Method Estimated parametersκ σ ξ
ML 1(0.6,2) × 103 0.00(0.00,0.00) 0.81(0.78,0.85)ML-c 0.59(0.51,0.73) 1.44(1.09,1.65) 0.03(0.00,0.14)PWMs 0.61(0.56,0.67) 1.47(1.27,1.61) 0.03(0.00,0.11)PWMs-c 0.50(0.44,0.57) 1.55(1.31,1.73) 0.00(0.00,0.08)
Summer data
Method Estimated parametersκ σ ξ
ML 0.7(0.4,2) × 103 0.00(0.00,0.00) 0.91(0.87,0.96)ML-c 0.51(0.41,0.64) 1.94(1.36,2.65) 0.18(0.03,0.33)PWMs 0.56(0.51,0.63) 1.82(1.46,2.16) 0.20(0.11,0.30)PWMs-c 0.43(0.37,0.53) 2.16(1.61,2.66) 0.12(0.01,0.25)
Parameter estimates for model (i)
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 26
Fall data
Method Estimated parametersκ σ ξ
ML 1(0.5,2) × 103 0.00(0.00,0.00) 0.85(0.81,0.88)ML-c 0.80(0.64,1.02) 1.00(0.73,1.34) 0.29(0.16,0.40)PWMs 0.66(0.60,0.72) 1.32(1.11,1.53) 0.20(0.11,0.28)PWMs-c 0.62(0.52,0.75) 1.22(0.93,1.48) 0.22(0.12,0.31)
Winter data
Method Estimated parametersκ σ ξ
ML 3(2,8) × 103 0.00(0.00,0.00) 0.67(0.63,0.70)ML-c 0.84(0.64,1.14) 0.63(0.45,0.84) 0.23(0.09,0.34)PWMs 0.59(0.54,0.65) 1.07(0.92,1.17) 0.04(0.00,0.18)PWMs-c 0.63(0.46,0.82) 0.73(0.52,1.04) 0.20(0.00,0.33)
Diagnostics for model (i)
Spring data
Winter data
Rainfall amounts [mm]
Density
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
0 2 4 6 8 10
02
46
810
Empirical quantiles [mm]
Fitte
d q
uantile
s [m
m]
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
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Rainfall amounts [mm]
Density
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6 8 10 12 14
−2
02
46
810
12
Empirical quantiles [mm]
Fitte
d q
uantile
s [m
m]
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Diagnostics for model (i)
0.0 0.5 1.0 1.5
0.0
0.5
1.0
1.5
Spring
Empirical quantiles [mm]
Fitte
d q
ua
ntile
s [m
m]
MLE
PWMs
GPD
0.0 0.5 1.0 1.5
0.0
0.5
1.0
1.5
Summer
Empirical quantiles [mm]
Fitte
d q
ua
ntile
s [m
m]
MLE
PWMs
GPD
Towards spatial modeling of
extreme and non-extreme data
� Skar’s Theorem: F (x1, . . . , xD) = C{F1(x1), . . . , FD(xD)}.C is the copula (uniquely defined if F is continuous).
Towards spatial modeling of
extreme and non-extreme data
� Skar’s Theorem: F (x1, . . . , xD) = C{F1(x1), . . . , FD(xD)}.C is the copula (uniquely defined if F is continuous).
� Copula families:
– Gaussian: tractable but tail independent
Towards spatial modeling of
extreme and non-extreme data
� Skar’s Theorem: F (x1, . . . , xD) = C{F1(x1), . . . , FD(xD)}.C is the copula (uniquely defined if F is continuous).
� Copula families:
– Gaussian: tractable but tail independent
– Trans-Gaussian: same as Gaussian copula
Towards spatial modeling of
extreme and non-extreme data
� Skar’s Theorem: F (x1, . . . , xD) = C{F1(x1), . . . , FD(xD)}.C is the copula (uniquely defined if F is continuous).
� Copula families:
– Gaussian: tractable but tail independent
– Trans-Gaussian: same as Gaussian copula
– V-transformed Gaussian copula (χ2 processes): intractable and tailindependent
Towards spatial modeling of
extreme and non-extreme data
� Skar’s Theorem: F (x1, . . . , xD) = C{F1(x1), . . . , FD(xD)}.C is the copula (uniquely defined if F is continuous).
� Copula families:
– Gaussian: tractable but tail independent
– Trans-Gaussian: same as Gaussian copula
– V-transformed Gaussian copula (χ2 processes): intractable and tailindependent
– Student t: tail dependent but tail-symmetric
Towards spatial modeling of
extreme and non-extreme data
� Skar’s Theorem: F (x1, . . . , xD) = C{F1(x1), . . . , FD(xD)}.C is the copula (uniquely defined if F is continuous).
� Copula families:
– Gaussian: tractable but tail independent
– Trans-Gaussian: same as Gaussian copula
– V-transformed Gaussian copula (χ2 processes): intractable and tailindependent
– Student t: tail dependent but tail-symmetric
– Extreme-value copulas: justified for extremes only, computationallydemanding to fit
Towards spatial modeling of
extreme and non-extreme data
� Skar’s Theorem: F (x1, . . . , xD) = C{F1(x1), . . . , FD(xD)}.C is the copula (uniquely defined if F is continuous).
� Copula families:
– Gaussian: tractable but tail independent
– Trans-Gaussian: same as Gaussian copula
– V-transformed Gaussian copula (χ2 processes): intractable and tailindependent
– Student t: tail dependent but tail-symmetric
– Extreme-value copulas: justified for extremes only, computationallydemanding to fit
– Vine copulas: (too?) flexible but lack interpretability
Towards spatial modeling of
extreme and non-extreme data
� Skar’s Theorem: F (x1, . . . , xD) = C{F1(x1), . . . , FD(xD)}.C is the copula (uniquely defined if F is continuous).
� Copula families:
– Gaussian: tractable but tail independent
– Trans-Gaussian: same as Gaussian copula
– V-transformed Gaussian copula (χ2 processes): intractable and tailindependent
– Student t: tail dependent but tail-symmetric
– Extreme-value copulas: justified for extremes only, computationallydemanding to fit
– Vine copulas: (too?) flexible but lack interpretability
� Factor copula models: combine tractability, interpretability, flexibility, andyield tail dependence if the factor is suitably chosen.
Factor copula models� Model:
X(s) = Z(s) + V, s ∈ Rd,
where Z(s) ∼ GP(0, 1, ρ(h)) ⊥⊥ V ∼ fV .
Factor copula models� Model:
X(s) = Z(s) + V, s ∈ Rd,
where Z(s) ∼ GP(0, 1, ρ(h)) ⊥⊥ V ∼ fV .
� The random variable V is the latent random factor.
⇒ Interpretation: An unobserved random phenomenon affects thejoint dependence of variables.
Factor copula models� Model:
X(s) = Z(s) + V, s ∈ Rd,
where Z(s) ∼ GP(0, 1, ρ(h)) ⊥⊥ V ∼ fV .
� The random variable V is the latent random factor.
⇒ Interpretation: An unobserved random phenomenon affects thejoint dependence of variables.
� If suitably chosen, V yields tail dependence and tail asymmetry!
Factor copula models� Model:
X(s) = Z(s) + V, s ∈ Rd,
where Z(s) ∼ GP(0, 1, ρ(h)) ⊥⊥ V ∼ fV .
� The random variable V is the latent random factor.
⇒ Interpretation: An unobserved random phenomenon affects thejoint dependence of variables.
� If suitably chosen, V yields tail dependence and tail asymmetry!
� Copula density for D locations s1, . . . , sD ∈ Rd:
c(u1, . . . , uD) =fXD {F−1
1 (u1), . . . , F−1D (uD)}
fX1 {F−1
1 (u1)} × · · · × fXD {F−1
D (uD)},
where
fXD =
∫ ∞
−∞φD(x1 − v, . . . , xD − v; Σ)fV (v)dv,
and φD is the multivariate standard Gaussian density with covariancematrix Σ = {ρ(si − sj)}
Di,j=1.
Factor copula models
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 31
� If V is chosen to be Gaussian, X(s) will be tail independent.
Factor copula models
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 31
� If V is chosen to be Gaussian, X(s) will be tail independent.
� Proposition: Let Pr(V > v) ∼ Kvβ exp(−θvα), 0 ≤ α < 2, θ > 0,K > 0, as v → ∞.
– 0 < α < 1 or α = 0, β < 0: compete (upper) tail dependence
– α = 1: (upper) tail dependence
– α > 1: (upper) tail independence
Factor copula models
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 31
� If V is chosen to be Gaussian, X(s) will be tail independent.
� Proposition: Let Pr(V > v) ∼ Kvβ exp(−θvα), 0 ≤ α < 2, θ > 0,K > 0, as v → ∞.
– 0 < α < 1 or α = 0, β < 0: compete (upper) tail dependence
– α = 1: (upper) tail dependence
– α > 1: (upper) tail independence
� Proposition: if V ∼ Exp(λ), the process X(s) is in the max-domain ofattraction of the Brown-Resnick process.(⇒ X(s) lives in a penultimate world).
Factor copula models
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 31
� If V is chosen to be Gaussian, X(s) will be tail independent.
� Proposition: Let Pr(V > v) ∼ Kvβ exp(−θvα), 0 ≤ α < 2, θ > 0,K > 0, as v → ∞.
– 0 < α < 1 or α = 0, β < 0: compete (upper) tail dependence
– α = 1: (upper) tail dependence
– α > 1: (upper) tail independence
� Proposition: if V ∼ Exp(λ), the process X(s) is in the max-domain ofattraction of the Brown-Resnick process.(⇒ X(s) lives in a penultimate world).
� By setting V = V1 − V2 with V1, V2 ≥ 0, one can model tail asymmetry.
Factor copula models
TOY workshop, NCAR, Boulder CO April 26, 2016 – slide 31
� If V is chosen to be Gaussian, X(s) will be tail independent.
� Proposition: Let Pr(V > v) ∼ Kvβ exp(−θvα), 0 ≤ α < 2, θ > 0,K > 0, as v → ∞.
– 0 < α < 1 or α = 0, β < 0: compete (upper) tail dependence
– α = 1: (upper) tail dependence
– α > 1: (upper) tail independence
� Proposition: if V ∼ Exp(λ), the process X(s) is in the max-domain ofattraction of the Brown-Resnick process.(⇒ X(s) lives in a penultimate world).
� By setting V = V1 − V2 with V1, V2 ≥ 0, one can model tail asymmetry.
� If V1 ∼ Exp(λ1) ⊥⊥ V2 ∼ Exp(λ2), the likelihood can be obtained inclosed-form!
Factor copula models
0.001 0.100
0.0
0.5
1.0
Exponential factor
λ Lq
0.001 0.100
0.0
0.5
1.0
Pareto factor
0.001 0.100
0.0
0.5
1.0
Weibull factor
0.001 0.100
0.0
0.5
1.0
q
λ Uq
0.001 0.100
0.0
0.5
1.0
q
0.001 0.100
0.0
0.5
1.0
q
More information in Krupskii, Huser and Genton (2016, submitted).
Discussion
� We have proposed univariate models for the full range of rainfallintensities, which bypass the need to select suitable thresholdsdistinguishing between low, moderate and heavy rainfall.
� These models are in compliance with extreme-value theory on both tails.
� Simulation and inference are simple.
� ML performs slightly better when the model is well-specified, but PWMsare much more robust to model misspecification.
� Spatial (or multivariate) modeling of extreme and non-extreme data maybe achieved by combining our univariate models with factor copulamodels.
� Open questions:
– Correlation between lower and upper shape parameters?
– More flexible models for the bulk (Doug’s approach?)
– Modeling of rainfall occurrences (wet/dry events)
– Non-stationarity