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[email protected]

Outline of the Talk

1. Motivations 2. Maxima distribution 3. Geostatistics 4. Estimation

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[email protected]

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spatial dependence as

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How to describe the

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Spatial Statistics for Extremes

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Geostat

Estimation

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Spatial Statistics for Extremes

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Spatial Statistics for Extremes

A few Approaches for modeling spatial extremes Max-stable processes: Adapting asymptotic results for multivariate extremes 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)

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Motivations Max

Geostat

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Univariate case for Maxima Convergence of sample maxima Normal density ⇒

⇐ Gumbel density

Uniform density ⇒

⇐ Weibull density

Cauchy density ⇒

⇐ Fr´ echet density

n = 50

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Max Geostat

n = 100

Estimation

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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 Fr´ echet

F (u) = P[M (x) ≤ u] = exp(−1/u)

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Max Geostat

Estimation

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A central question

For large n

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

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Max Geostat

Estimation

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Bivariate case for Maxima Asymptotic theory If one assumes that we have unit Fr´ echet margins then

"

lim P

n→∞

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Mn(x) − an Mn(x + h) − an 6 u, 6 v = exp [−Vh(u, v)] bn bn

where Z 1

w 1−w Vh(u, v) = 2 max , u v 0

dLh(w)

R1 with Lh(.) a distribution function on [0, 1] such that 0 w dLh(w) = 0.5.

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Max Geostat

Estimation

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Bivariate case (M (x), M (x + h)) Complex non-parametric structure

Z 1

w 1−w max Vh(u, v) = 2 , u v 0 Special case u = v

dLh(w)

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) ⇓

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Max Geostat

Estimation

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θ(h) = Extremal coefficient

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

Interpretation

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

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Max Geostat

Estimation

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An important question (1) How to estimate θ(h)?

R1 w 1−w Vh(u, v) = 2 0 max u , v dLh(w)

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Geostatistics: Variograms 2 γ(h) = 1 2 E|Z(x + h) − Z(x)|

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Finite if light tails Capture

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Max

Geostat Estimation

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A Different Variogram

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

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Max

Geostat

Estimation

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A Different Variogram

1 νh = E |F (M (x + h)) − F (M (x))| 2 with F (u) = exp(−1/u)

Defined for light & heavy tails Called a Madogram Nice links with extreme value theory

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Max

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A Different Variogram

1 νh = E |F (M (x + h)) − F (M (x))| 2 Why does it work?

1 1 |a − b| = max(a, b) − (a + b) 2 2

a = F (M (x + h)) and b = F (M (x)) Ea = Eb = 1/2 θ(h) E max(a, b) = EF (max(M (x {z + h), M (x)})) = | 1 + θ(h) max-stable

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Madogram νh ⇒ Extremal coeff θ(h)

1 + 2νh θ(h) = 1 − 2νh The madogram νh gives the extremal coefficient θ(h)

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Comparisons with other estimators Gumbel (1960)

1 1 α P (X ≤ x, Y ≤ y ) = exp − + x

! 1 α 1 α y

Four estimators - Pickands’ estimator (1975) - Deheuvels’ estimator (1991) - Hall and Tajvidi’s estimator (2000) - Cap´ era` a et al. (1997) estimator

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Comparisons with other estimators

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α = 0.7

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1 1 − (exp(−h/40) + 1) 2 Max

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Madogram νh ⇒ Extremal coeff θ(h) Schlather’s fields Extremal coeff

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Max

p

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Madogram νh ⇒ Extremal coeff θ(h) Smith’s fields

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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.

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An important question (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].

R1 w 1−w Vh(u, v) = 2 0 max u , v dLh(w)

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Extending the madogram

1 λ 1−λ νh(λ) = E F (M (x + h)) − F (M (x)) 2

Defined for light & heavy tails Called a λ-Madogram Nice links with extreme value theory νh(0) = νh(1) = 0.25

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Geostat Estimation

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The λ-madogram

λ 1 1−λ νh(λ) = 2 E F (M (x + h)) − F (M (x))

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A fundamental relationship Vh(λ, 1 − λ) 3 νh(λ) = − c(λ), with c(λ) = 1 + Vh(λ, 1 − λ) 2(1 + λ)(2 − λ) Conversely, Vh(λ, 1 − λ) =

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Max

c(λ) + νh(λ) 1 − c(λ) − νh(λ)

Geostat Estimation

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Estimation of Vh(u, v) Suppose that we have T iid years of daily annual maxima fields with unknown margins. λ 1 1−λ (M (x)) ? How to estimate νh(λ) = 2 E F (M (x + h)) − F

A naive estimator T 1 X λ 1−λ ν ˆh(λ) = Fn,T (Mn,t(x + h)) − Gn,T (Mn,t(x)) 2T t=1

with T T 1 X 1 X Fn,T (u) = 1l and Gn,T (u) = 1l T t=1 {Mn,t(x+h)≤u} T t=1 {Mn,t(x)≤u}

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

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Geostat Estimation

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Estimation of Vh(u, v) λ 1 1−λ How to estimate νh(λ) = 2 E F (M (x + h)) − F (M (x)) ?

A modified estimator T 1 X λ 1−λ ν ˆh(λ) = Fn,T (Mn,t(x + h)) − Gn,T (Mn,t(x)) 2T t=1 T λ X λ 1 − Fn,T (Mn,t(x + h)) − 2T t=1 T 1−λ X 1−λ − 1 − Gn,T (Mn,t(x)) 2T t=1

1 1 − λ + λ2 + 2 (2 − λ)(1 + λ)

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Simulations: 300 iid Schalther’s fields

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Mean Square Error from simulations

300 iid Schalther’s fields

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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 T 1 X HT (u, v) = 1l{Xt≤u,Yt≤v} T t=1

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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 associated copula C whose partial derivatives are continuous. Then, the process {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 ∂C ∂C NC (u, v) = BC (u, v) − BC (u, 1) (u, v) − BC (1, v) (u, v), ∂u ∂v where BC is a Brownian bridge on [0, 1]2 with covariance function

E BC (u, v) · BC (u0, v 0) = C(u ∧ u0, v ∧ v 0) − C(u, v) · C(u0, v 0)

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Max

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Estimation

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Convergence of the λ−madogram (X1, Y1), ..., (XT , YT ) T bivariate rv with unknown margins F and G λ 1−λ T 1 X F (Xt) − G (Yt) νbT (λ) := T T 2T t=1

Proposition 2. Under the assumptions of Proposition 1, let J be a function of bounded variation, continuous. Then, we have T X

1 √ J FT (Xt), GT (Yt) − EJ F (X), G(Y ) T t=1

d

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Z [0,1]2

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

λ − y 1−λ|, provides the weak convergence of The special case, J(x, y) := 1 |x 2 the λ−madogram estimator

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Max

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Estimation

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

Mn(x), Mn(x + h) =

max Zi(x), max Zi(x + h)

i=1,...,n

i=1,...,n

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 T T 1 X 1 X Fn,T (u) = 1l{Mn,t(x+h)≤u} and Gn,T (u) = 1l{Mn,t(x)≤u} T t=1 T t=1 T 1 X λ 1−λ νbn,T (h, λ) = Fn,T Mn,t(x + h) − Gn,T (Mn,t(x)) 2T t=1

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Max

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Estimation

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Madogram & EVT (cont’d) Proposition 3. Let (Mn,t(x), Mn,t(x + h)) be a sample of T bivariate vectors with that satisfies the assumptions of Proposition 1 and such that Mn,t(x)−an Mn,t(x+h)−an , bn bn

converges in distribution to a bivariate EV distri-

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

√ 1 T νbn,T (h, λ) − E|F λ(M (x + h)) − F 1−λ(M (x))| 2 d

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Z [0,1]2

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

where n tends to ∞ as T goes to ∞

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Estimation

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An application: in Bourgogne (Dijon)

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

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

X 1 1 1 − λ + λ2 λ 1−λ (M (xi)) + F (M (xj )) − F 2|Nh| (x ,x )∈N 2 (2 − λ)(1 + λ) i

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X X λ 1−λ λ 1−λ − 1 − F (M (xi)) − 1−F (M (xi)) 2|Nh| (x ,x )∈N 2|Nh| (x ,x )∈N i

j

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Max

Geostat

Estimation

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λ−madogram

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

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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 ν ˆh(λ)

Future research Develop spatial interpolation methods for maxima Derive statistical schemes for downscaling for maxima

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Max

Geostat

Estimation

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