modelling jointly , moderate and heavy rainfall ...toyworkshop,ncar,boulderco april26,2016–slide1...

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


� 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


� 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


KAUST


� 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.

http://extstat.kaust.edu.sa

Extremes: Rarity vs Intensity


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


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

)


1−Hξ(x) =

{

(1 + ξx)−1/ξ+ , ξ 6= 0,

exp(−x), ξ = 0,


� Pr(−X > −x | −X > −u) ≈ 1−Hξ

(

−x+uσ

)

, for large −x > −u.




� Pr(X > x | X > u) ≈ 1−Hξ

(

x−uσ

)


1−Hξ(x) =

{

(1 + ξx)−1/ξ+ , ξ 6= 0,

exp(−x), ξ = 0,


� 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

0.2

0.4

0.6

0.8

Summer data


Den

sity

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

Fall data


Den

sity

0 5 10 15

0.0

0.2

0.4

0.6

0.8

Winter data


Den

sity

0 5 10 15

0.0

0.4

0.8

1.2

Classical approach in EVT


●

●

●●

●●●

●●

●●●●

●●●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●●

●

●

●

●

●●●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●●●●

●

●●

●

●

●

●

●●●●

●

●

●

●

●

●●●●

●●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●●●

●●●

●

●

●

●●●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●

●

●●●●

●

●●

●

●

●

●

●●●

●

●

●●

●

●●●

●

●●●●

●

●●●

●

●

●

●

●●●●

●

●●

●●●●

●●

●●

●

●

●

●●●●●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●●●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●●●●●

●●

●●

●

●●●

●

●●

●

●●

●

●●●●●

●●●

●

●●●●●

●

●●●

●

●

●

●●

●

●

●

●

●

●●●

●●

●●●

●●●●

●

●

●

●

●

●●

●

●●●●●●●●●●●●●●

●

●

●

●

●

●●

●●

●

●●

●

●

●●●

●

●

●

●●

●

●

●●

●●●

●

●

●

●

●

●

●

●●

●

●●●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●●

●

●●●●●

●●

●

●●

●

●

●●

●●

●●

●

●

●

●

●●●●●●

●●●

●●●●●●

●

●●●

●●

●●●●

●●

●●●

●

●●

●●●●●●●●●●●●

●

●●●

●

●●

●●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●●

●

●●

●

●●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●●●●

●●

●

●

●

●●

●

●

●●●

●●●

●●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●●●

●

●●●●

●●

●

●●●

●●

●

●●●

●●●●●●

●

●●

●

●●

●

●

●

●

●

●

●

●

●●●●

●●

●

●

●●●

●●

●

●●●

●●

●●●●

●

●●●●●

●

●

●

●

●

●●

●●●

●

●●●●

●●●

●●●●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●●●●

●

●

●

●

●●

●

●●

●

●

●

●●●

●

●

●●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●●●●●

●

●●●

●

●

●●

●

●●●●

●

●

●●

●

●

●

●

●

●

●●

●●●●●●●

●

●●

●

●

●

●●●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●●●●●

●

●●●

●

●●●

●

●

●●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●●

●

●

●●

●

●●●

●

●●

●●●●●

●

●

●

●

●

●●●

●●●

●

●

0 200 400 600 800

02

46

81

0

Time index

Ra

infa

ll a

mo

un

t [m

m]

Spring data


De

nsity

0 2 4 6 8 100

.00

.40

.81

.2



0 200 400 600 800

02

46

81

0

Time index

Ra

infa

ll a

mo

un

t [m

m]

●

●

●●

●●●

●●

●●●●

●●●

●

●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●●●●●

●

●●●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●●●●

●

●

●

●

●

●

●

●●

●

●●●

●

●●

●

●

●

●

●●

●

●●

●

●●●●

●

●●

●

●

●

●

●●●●

●

●

●

●●●●

●●

●

●

●●●

●

●

●

●●

●

●

●

●●●●

●●●

●

●

●

●●●

●

●

●●

●●

●

●●●

●

●

●

●

●

●●●●

●

●●

●

●

●

●●●

●

●

●●

●●●

●

●●●●●●●

●

●

●

●

●●●●

●●

●●●●

●●

●●

●

●

●

●●●●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●●●

●

●●●

●

●

●

●

●

●

●

●

●●●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●●●●●●●

●

●●●

●

●●

●

●●

●

●●●●●

●●●

●

●●●●●

●

●●●

●

●

●

●●

●

●

●

●

●

●●●

●●

●●●

●●●●

●

●

●

●

●

●●●●●●●●●●●●●●●●

●

●

●

●

●

●●

●●

●

●●

●

●

●●●

●

●

●

●●

●

●●

●●●●

●

●

●

●●

●

●●●●

●

●

●

●

●

●

●

●●●

●

●●●●

●

●●●●●

●●

●

●●

●

●●

●●

●●

●

●

●

●

●●●●●●

●●●

●●●●●●

●

●●●●●●●

●●

●●●

●

●●

●●●●●●●●●●●●

●

●●●

●

●●

●●●

●

●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●●

●

●●

●

●●

●●

●

●

●●

●●

●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●●

●

●●●●

●●

●

●●●

●

●●

●●●

●●●

●

●

●

●

●

●●

●

●●●

●

●●●●●

●

●●●●

●●

●

●●●

●●

●

●●●

●●●●●●

●

●●

●

●●

●

●

●

●●●●●

●●

●●●●

●●

●

●●●

●●

●●●●

●

●●●●●

●

●

●

●

●

●●

●●●

●

●●●●

●●●

●●●●●●

●

●

●

●

●

●

●

●

●

●●●

●●

●●●●

●

●

●●●

●

●●

●

●

●

●●●

●

●

●●

●

●

●●

●●●

●

●

●

●

●

●

●●●●●

●

●●●

●

●●

●

●●●●

●

●●

●

●

●

●

●

●

●●

●●●●●●●

●●

●

●

●

●●●●

●

●●●

●●

●

●

●

●

●●●●●

●

●●●

●

●●●

●

●

●●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●

●●●

●

●●●

●

●

●●

●

●●●

●

●●

●●●●●

●

●

●

●

●●●

●●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●●●●

●

●

●

●

●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

Spring data


De

nsity

0 2 4 6 8 100

.00

.40

.81

.2



0 200 400 600 800

02

46

81

0

Time index

Ra

infa

ll a

mo

un

t [m

m]

●

●

●●●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●●

●●

●

●

●

●

●●

●●

●●

●

●●●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●●●

●●●

●

●

●

●

●

●

●

●

●

●

●●

●●●●

●

●

●

●

●

●

●

●

●●

●

●●

●●

●●

●

●

●

●●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●●●

●

●

●●●●

●●●

●●●●●●

●●●●●●

●●●●●●●●●●●●●●●

●●●●●●●●●●●●●

●●●●●

●

●●●●●

●●●●●●

●

●●●●●●●

●

●●●●●●●●●

●●●●●

●●●●●●●●●

●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●●●●

●

●●●

●

●●●●●●●●

●

●●●●●●●

●●●●●●●●●●

●

●

●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●

●●●

●

●●●●●●●●●●●●●●●●●●

●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●●●●●●

●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●

●

●●●●●●

●●●

●●●●●●●

●●●●●

●

●●●●●

●

●●●●●●●●●●●●●●

●●●●

●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●

●●●●●●●●●

●

●

●

●

●●●●●●●●●●●●●●●●●●●●●●●●

●●●

●

●●●●●●●●●●

●

●●●●●●●●●●

●●●●●●●●●●

●●●●●●●

●

●●●●●●●

●

●

●

●●●●●●●●●

●●●●●●●●

●●●

●

●●

●●●●

●●●●●●●●●●●●

●●●●●

Spring data


De

nsity

0 2 4 6 8 100

.00

.40

.81

.2



0 200 400 600 800

02

46

81

0

Time index

Ra

infa

ll a

mo

un

t [m

m]

●

●

●●●●●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●●

●

●●●

●

●

●

●

●●●

●

●

●

●

●●●●

●

●

●

●

●●●

●●●

●

●

●●●

●

●●●

●●

●●

●

●●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●●

●

●●

●●

●

●

●

●●

●

●●●●

●

●●

●●

●

●

●

●

●●●●

●

●●●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●●●●

●

●

●

●●

●

●●

●●

●●

●●●

●

●

●

●●●

●

●●

●●●●

●

●

●

●

●●

●●

●●

●●

●

●

●

●●●

●●

●

●

●

●●

●

●

●●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●●●

●

●

●●●●

●●●

●●●●●●

●●●●●●

●●●●●●●●●●●●●●●

●●●●●●●●●●●●●

●●●●●

●

●●●●●

●●●●●●

●

●●●●●●●

●

●●●●●●●●●

●●●●●

●●●●●●●●●

●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●●●●

●

●●●

●

●●●●●●●●

●

●●●●●●●

●●●●●●●●●●

●

●

●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●

●●●

●

●●●●●●●●●●●●●●●●●●

●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●●●●●●

●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●

●

●●●●●●

●●●

●●●●●●●

●●●●●

●

●●●●●

●

●●●●●●●●●●●●●●

●●●●

●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●

●●●●●●●●●

●

●

●

●

●●●●●●●●●●●●●●●●●●●●●●●●

●●●

●

●●●●●●●●●●

●

●●●●●●●●●●

●●●●●●●●●●

●●●●●●●

●

●●●●●●●

●

●

●

●●●●●●●●●

●●●●●●●●

●●●

●

●●

●●●●

●●●●●●●●●●●●

●●●●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●●●●

●

●

●

●

●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

Spring data


De

nsity

0 2 4 6 8 100

.00

.40

.81

.2

Modeling of rainfall intensities




Our goal: To construct simple parametric models for the full range of rainfallintensities



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)



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



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



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



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.



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.



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


� 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


� 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.









– 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


(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


� 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



� 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


� 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


� 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


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


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


Den

sity

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

Summer data


Den

sity

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

Fall data


Den

sity

0 5 10 15

0.0

0.2

0.4

0.6

0.8

Winter data


Den

sity

0 5 10 15

0.0

0.4

0.8

1.2


Lyon, France (1996–2011)


� 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.












� We fit separate models for each season.

� Uncertainty assessment: using non-parametric bootstrap.

Parameter estimates for model (i)


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


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)


Fall data


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


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


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]

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●

●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●

●●

●

●

●

●

●

●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●

●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●

●●

●

●

●

●

●

●


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


Fitte

d q

uantile

s [m

m]

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●● ●●

●●●●●●

●●●

●●

●●●

●

●

●

●

●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●● ●●

●●●●●●

●●

●●●

●●●

●

●

●

●

●

Diagnostics for model (i)

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

Spring


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


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).




� 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










– 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 .


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.


X(s) = Z(s) + V, s ∈ Rd,

where Z(s) ∼ GP(0, 1, ρ(h)) ⊥⊥ V ∼ fV .



� If suitably chosen, V yields tail dependence and tail asymmetry!


X(s) = Z(s) + V, s ∈ Rd,

where Z(s) ∼ GP(0, 1, ρ(h)) ⊥⊥ V ∼ fV .



� 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


� 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.









� 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!


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

modelling jointly , moderate and heavy rainfall ...toyworkshop,ncar,boulderco april26,2016–slide1...

Documents