measuring spatial dependence among maxima · 2007. 6. 8. · extending the madogram ⇓ ↓ ↑ ⇑...

⇓ ↓ ↑ ⇑ [email protected]

Measuring Spatial Dependenceamong Maxima

• P. Naveau

Laboratoire des Sciences du Climat et de l’Environnement

IPSL, CNRS, France

• Guillou, A.; Cooley, D.; Diebolt, J.

http://amath.colorado.edu/faculty/naveau/



Outline of the Talk

⇓ ↓ ↑ ⇑ [email protected] 2

1. Motivations

2. Maxima distribution

3. Geostatistics

4. Estimation

Spatial Statistics for Extremes

⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 3

10 20 30 40

1020

3040

x

y

−1

01

23

How to describe the

spatial dependence as

a function of the

distance between two

points?



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10 20 30 40

010

2030

40

x

y

How to perform

spatial interpolation

for extreme events?



A few Approaches for modeling spatial extremes

Max-stable processes: Adapting asymptotic results for multivariate ex-

tremes

Schlather & Tawn (2003), Naveau et al. (2007), de Haan & Pereira (2005)

Bayesian or latent models: spatial structure indirectly modeled via the

EVT parameters distribution

Coles & Tawn (1996), Cooley et al. (2005)

Linear filtering: Auto-Regressive spatio-temporal heavy tailed processes,

Davis and Mikosch (2007)

Gaussian anamorphosis: Transforming the field into a Gaussian one

Wackernagel (2003)

Univariate case for Maxima


Convergence of sample maxima

Normal density ⇒

Uniform density ⇒

Cauchy density ⇒

⇐ Gumbel density

⇐ Weibull density

⇐ Frechet density

n = 50 n = 100

Assumptions


Suppose we know the marginal distributions of maxima M(x) with

M(x) = the maximum recorded at the location x from a stationary and

isotropic field.

Without loss of generality, we first assume that the margins follow an

unit Frechet

F (u) = P[M(x) ≤ u] = exp(−1/u)

A central question


For large n

P [Mn(x) < u, Mn(x + h) < v] = ??

Bivariate case for Maxima


Asymptotic theory

If one assumes that we have unit Frechet margins then

limn→∞P

[Mn(x)− an

bn6 u,

Mn(x + h)− an

bn6 v

]= exp [−Vh(u, v)]

where

Vh(u, v) = 2∫ 1

0max

(w

u,1− w

v

)dLh(w)

with Lh(.) a distribution function on [0,1] such that∫ 10 w dLh(w) = 0.5.

Bivariate case (M(x), M(x + h))


Complex non-parametric structure

Vh(u, v) = 2∫ 1

0max

(w

u,1− w

v

)dLh(w)

Special case u = v

Note Vh(u, u) = Vh(1,1)/u

Notations: θ(h) := Vh(1,1)

P [M(x) < u, M(x + h) < u] = exp(−θ(h)/u)

= F (u)θ(h)

because F (u) = exp(−1/u)

θ(h) = Extremal coefficient


P [M(x) < u, M(x + h) < u] = F (u)θ(h)

Interpretation

Independence ⇒ θ(h) = 2M(x) = M(x + h) ⇒ θ(h) = 1Similar to correlation coefficients for Gaussian but ...No characterization of the full bivariate dependence

An important question

Vh(u, v) = 2∫ 10 max

(wu , 1−w

v

)dLh(w) 12

(1) How to estimate θ(h)?

Geostatistics: Variograms


γ(h) = 12E|Z(x + h)− Z(x)|2

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0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

distance

sem

ivar

ianc

e

Finite if light tails

Capture all spatial

structure if {Z(x)}

Gaussian fields

but not well adapted

for extremes

A Different Variogram


|F (M(x + h))− F (M(x))|with F (u) = exp(−1/u)



νh =1

2E |F (M(x + h))− F (M(x))|

with F (u) = exp(−1/u)

Defined for light & heavy tails

Called a Madogram

Nice links with extreme value theory



νh =1

2E |F (M(x + h))− F (M(x))|

Why does it work?

1

2|a− b| = max(a, b)−

1

2(a + b)

a = F (M(x + h)) and b = F (M(x))

Ea = Eb = 1/2

Emax(a, b) = EF (max(M(x + h), M(x)︸︷︷︸max-stable

)) =θ(h)

1 + θ(h)

Madogram νh ⇒ Extremal coeff θ(h)


θ(h) =1 + 2νh

1− 2νh

The madogram νh gives the extremal coefficient θ(h)

Comparisons with other estimators


Gumbel (1960)

P (X ≤ x, Y ≤ y) = exp

−(1

x

)1α+

(1

y

)1α

α

Four estimators

- Pickands’ estimator (1975)

- Deheuvels’ estimator (1991)

- Hall and Tajvidi’s estimator (2000)

- Caperaa et al. (1997) estimator

Comparisons with other estimators


α = 0.3 α = 0.7

Schlather’s models (2003)


10 20 30 40

1020

3040

x

y

−1

01

23

θ(h) = 1 +

√1−

1

2(exp(−h/40) + 1)



Schlather’s fields

Madogram Extremal coeff

0.0

0.2

0.4

0.6

0.8

distance

estim

ated

mad

ogra

m

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1.2

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aHat

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Smith’s models (2003)


10 20 30 40

1020

3040

x

y

01

23

θ(h) = 2Φ(√

hTΣ−1h/2)



Smith’s fields

Madogram Extremal coeff

0.0

0.2

0.4

0.6

0.8

1.0

distance

estim

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mad

ogra

m

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Building valid Extremal coeff


Proposition A

Any extremal coefficient function θ(h) is such that 2 − θ(h) is positive

semi-definite.

Proposition B

Any extremal coefficient function θ(h) satisfies the following inequalities

θ(h + k) ≤ θ(h)θ(k),

θ(h + k)τ ≤ θ(h)τ + θ(k)τ − 1, for all 0 ≤ τ ≤ 1,

θ(h + k)τ ≥ θ(h)τ + θ(k)τ − 1, for all τ ≤ 0.

An important question

Vh(u, v) = 2∫ 10 max

(wu , 1−w

v

)dLh(w) 25

(1) How to estimate θ(h) = Vh(1,1)? Done!!

(2) How to estimate Vh(u, v)?

Note:

Because Vh(u, v) = Vh(u/(u + v), v/(u + v))/(u + v) is sufficient to only

estimate Vh(λ,1− λ) for λ ∈ [0,1].

Extending the madogram


νh(λ) =1

2E∣∣∣∣Fλ(M(x + h))− F1−λ(M(x))

∣∣∣∣

Defined for light & heavy tails

Called a λ-Madogram

Nice links with extreme value theory

νh(0) = νh(1) = 0.25

The λ-madogram

νh(λ) = 12E

∣∣∣Fλ(M(x + h))− F1−λ(M(x))∣∣∣ 27

A fundamental relationship


νh(λ) =Vh(λ,1− λ)

1 + Vh(λ,1− λ)− c(λ), with c(λ) =

3

2(1 + λ)(2− λ)

Conversely,

Vh(λ,1− λ) =c(λ) + νh(λ)

1− c(λ)− νh(λ)

Estimation of Vh(u, v)


Suppose that we have T iid years of daily annual maxima fields with un-

known margins.

How to estimate νh(λ) = 12E

∣∣∣Fλ(M(x + h))− F1−λ(M(x))∣∣∣?

A naive estimator

νh(λ) =1

2T

T∑t=1

∣∣∣Fλn,T (Mn,t(x + h))−G1−λ

n,T (Mn,t(x))∣∣∣

with

Fn,T (u) =1

T

T∑t=1

1l{Mn,t(x+h)≤u} and Gn,T (u) =1

T

T∑t=1

1l{Mn,t(x)≤u}

But, the conditions Eνh(0) = Eνh(1) = 0.25 are not satisfied

Estimation of Vh(u, v)


How to estimate νh(λ) = 12E

∣∣∣Fλ(M(x + h))− F1−λ(M(x))∣∣∣?

A modified estimator

νh(λ) =1

2T

T∑t=1

∣∣∣Fλn,T (Mn,t(x + h))−G1−λ

n,T (Mn,t(x))∣∣∣

−λ

2T

T∑t=1

(1− Fλ

n,T (Mn,t(x + h)))

−1− λ

2T

T∑t=1

(1−G1−λ

n,T (Mn,t(x)))

+1

2

1− λ + λ2

(2− λ)(1 + λ)

Simulations: 300 iid Schalther’s fields


Mean Square Error from simulations

300 iid Schalther’s fields 32

Notations for asymptotic results


Margins of (X, Y ): unknown continuous margins: F, G

Bivariate distribution H and copula:

H(x, y) = C(F (x), G(y)) and C(u, v) = H

(F←(u), G←(v)

)

φ(H)(u, v) := H

(F←(u), G←(v)

)Bivariate empirical process ZT (u, v):

ZT (u, v) :=√

T

φ(HT )(u, v)− φ(H)(u, v)

with

HT (u, v) =1

T

T∑t=1

1l{Xt≤u,Yt≤v}

Asymptotic properties


ZT (u, v) =√

T

(φ(HT )(u, v)− φ(H)(u, v)

)Proposition 1. Let (Xt, Yt)t=1,...,T be a sample of T bivariate random

vectors with df H, continuous margins F and G, and with its associ-

ated copula C whose partial derivatives are continuous. Then, the pro-

cess {ZT (u, v),0 ≤ u, v ≤ 1} converges weakly to the Gaussian process

{NC(u, v),0 ≤ u, v ≤ 1} in `∞([0,1]2) that is defined as

NC(u, v) = BC(u, v)− BC(u,1)∂C

∂u(u, v)− BC(1, v)

∂C

∂v(u, v),

where BC is a Brownian bridge on [0,1]2 with covariance function

E[BC(u, v) · BC(u′, v′)

]= C(u ∧ u′, v ∧ v′)− C(u, v) · C(u′, v′)

Convergence of the λ−madogram


(X1, Y1), ..., (XT , YT ) T bivariate rv with unknown margins F and G

νT (λ) :=1

2T

T∑t=1

∣∣∣∣(FT (Xt))λ−(

GT (Yt))1−λ∣∣∣∣

Proposition 2. Under the assumptions of Proposition 1, let J be a function

of bounded variation, continuous. Then, we have

1√T

T∑t=1

J

(FT (Xt), GT (Yt)

)− EJ

(F (X), G(Y )

)d−→

∫[0,1]2

NC(u, v)dJ(u, v)

The special case, J(x, y) := 12|x

λ−y1−λ|, provides the weak convergence of

the λ−madogram estimator

Madogram & EVT


• (Z(x), Z(x + h))= precip. measurements at two nearby locations(Mn(x), Mn(x + h)

)=(

maxi=1,...,n

Zi(x), maxi=1,...,n

Zi(x + h))

n = recording unit, either hourly, daily or monthly

• Suppose that such bivariate vectors can be computed for a series

of years and that these vectors are assumed to be iid in time

Fn,T (u) =1

T

T∑t=1

1l{Mn,t(x+h)≤u} and Gn,T (u) =1

T

T∑t=1

1l{Mn,t(x)≤u}

νn,T (h, λ) =1

2T

T∑t=1

∣∣∣Fλn,T

(Mn,t(x + h)

)−G1−λ

n,T (Mn,t(x))∣∣∣

Madogram & EVT (cont’d)


Proposition 3. Let (Mn,t(x), Mn,t(x + h)) be a sample of T bivariate vec-

tors with that satisfies the assumptions of Proposition 1 and such that(Mn,t(x)−an

bn,Mn,t(x+h)−an

bn

)converges in distribution to a bivariate EV distri-

bution with an extremal function defined by Vh(., .). Then, we have

√T

(νn,T (h, λ)−

1

2E|Fλ(M(x + h))− F1−λ(M(x))|

)d−→

∫[0,1]2

NC(u, v)dJ(u, v)

where n tends to ∞ as T goes to ∞

An application: in Bourgogne (Dijon)


Locations (in Lambert coordinates). Pre-processed 30-year maxima of daily precipitation

λ−madogram


Our estimator of our λ−madogram ν(h, λ)

1

2|Nh|∑

(xi,xj)∈Nh

∣∣∣Fλ(M(xj))− F1−λ(M(xi))∣∣∣+ 1

2

1− λ + λ2

(2− λ)(1 + λ)

−λ

2|Nh|∑

(xi,xj)∈Nh

(1− Fλ(M(xi))

)−

1− λ

2|Nh|∑

(xi,xj)∈Nh

(1− F1−λ(M(xi))

)

where Nh is the set of sample pairs lagged by the distance h.

λ−madogram


Estimated λ−madogram for the field of maxima of daily precipitation over 1970-1999

Take-home messages


Fields of maxima 6= Gaussian ones

Spatial structure defined by the function Vh(u, v)

λ−Madogram νh ⇒ dependence function Vh(u, v)

We have proposed and study an estimator νh(λ)

Future research

Develop spatial interpolation methods for maxima

Derive statistical schemes for downscaling for maxima

measuring spatial dependence among maxima · 2007. 6. 8. · extending the madogram ⇓ ↓ ↑ ⇑...

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