Spatial Extremes Analyses in Climate Studies P. Naveau Laboratoire - - PowerPoint PPT Presentation

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Spatial Extremes Analyses in Climate Studies P. Naveau Laboratoire - - PowerPoint PPT Presentation

Apr 30, 2023 347 likes •815 views

Spatial Extremes Analyses in Climate Studies P. Naveau Laboratoire des Sciences du Climat et de lEnvironnement IPSL, CNRS, France D. Cooley Applied Math Dept, Colorado University USA P. Poncet Universit e Paris X, France

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naveau@lsce.cnrs-gif.fr

Spatial Extremes Analyses in Climate Studies

  • P. Naveau

Laboratoire des Sciences du Climat et de l’Environnement IPSL, CNRS, France

  • D. Cooley

Applied Math Dept, Colorado University USA

  • P. Poncet

Universit´ e Paris X, France

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Outline of the Talk

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naveau@lsce.cnrs-gif.fr

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  • 1. Motivations
  • 2. Maxima distribution
  • 3. Extremal coefficient
  • 4. Geostatistics
  • 5. Conclusions
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Climate Studies

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Motivations Max Coeff Geostat End

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Climate Studies

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General statistical difficulties (but also true for extremes)

Spatial component: e.g. El-Nino Temporal component: e.g. solar forcing Non-stationary: e.g. release of CO2 Driven by physical processes: e.g. heat equations Multivariate variables: winds, precipitation, temperatures

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Climate Studies

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Motivations Max Coeff Geostat End

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General statistical difficulties (but also true for extremes)

Spatial component: e.g. El-Nino Temporal component: e.g. solar forcing Non-stationary: e.g. release of CO2 Driven by physical processes: e.g. heat equations Multivariate variables: winds, precipitation, temperatures

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Spatial Statistics for Extremes

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Motivations Max Coeff Geostat End

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5500 6000 6500 7000 7500 18500 19000 19500 100 200 300 400 500

How to describe the spatial dependence as a function of the distance between two points?

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Our Data: Daily precipitation

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Motivations Max Coeff Geostat End

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Our Data: Daily precipitation

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Dijon’s mustard region!!

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Our Data: Daily precipitation

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5500 6000 6500 7000 7500 18500 19000 19500 100 200 300 400 500

Maxima over 1970-2004 Data homogenized by O. Mestre Cˆ

  • te d’Or, Bourgogne France

83 locations

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Spatial Statistics for Extremes

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Motivations Max Coeff Geostat End

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A few approaches for modeling spatial extremes

Max-stable processes: Adapting asymptotic results for multivariate ex- tremes Schlather & Tawn (2003), Naveau et al. (2005), 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 (2005) Gaussian anamorphosis: Transforming the field into a Gaussian one Wackernagel (2003)

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Assumptions

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Motiv Maxima distribution Coeff Geostat End

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Suppose we know the marginal distributions of maxima M(x) with M(x) = the maximum recorded at the location x from a stationary field. Without loss of generality, we assume that the margins follow an unit Fr´ echet

F(u) = P[M(x) ≤ u] = exp(−1/u)

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Assumptions

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Motiv Maxima distribution Coeff Geostat End

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Suppose we know the marginal distributions of maxima M(x) with M(x) = the maximum recorded at the location x from a stationary field. Without loss of generality, we assume that the margins follow an unit Fr´ echet

F(u) = P[M(x) ≤ u] = exp(−1/u)

P [M(x) < u1, M(x + h) < u2] = ??

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

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Motiv Maxima distribution Coeff Geostat End

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A well-known non-parametric structure

P [M(x) < u1, M(x + h) < u2] = exp

  • −
  • max
  • g(s, 0)

u1 , g(s, h) u2

  • δ(ds)
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Bivariate case (M(x), M(x + h))

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Motiv Maxima distribution Coeff Geostat End

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A well-known non-parametric structure

P [M(x) < u1, M(x + h) < u2] = exp

  • −
  • max
  • g(s, 0)

u1 , g(s, h) u2

  • δ(ds)
  • Special case u1 = u2 = u

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

= F(u)θ(h), with F(u) = e−1/u

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SLIDE 15

θ(h) = Extremal coefficient

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Motiv Max Extremal coefficient Geostat End

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P [M(x) < u, M(x + h) < u] = F(u)θ(h)

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

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θ(h) = Extremal coefficient

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Motiv Max Extremal coefficient Geostat End

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P [M(x) < u, M(x + h) < u] = F(u)θ(h)

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

Interpretation Independence ⇒ θ(h) = 2 M(x) = M(x + h) ⇒ θ(h) = 1 Do not completely characterize the full bivariate dependence structure

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Geostatistics: Variograms

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Motiv Max Coeff Geostatistics End

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γ(h) = 1

2E|Z(x + h) − Z(x)|2 if {Z(x)} stationary field s.t. E|Z(x)|2 < ∞

  • 0.0

0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 distance semivariance

Finite if light tails Capture all spatial structure if {Z(x)} Gaussian fields but not well adapted for extremes

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A Different Variogram

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Motiv Max Coeff Geostatistics End

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1 2

E |M(x + h) − M(x)|2

=???

where {M(x)} stationary max-stable field with unit-Fr´ echet margins

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A Different Variogram

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1 2

E |M(x + h) − M(x)|2

=???

where {M(x)} stationary max-stable field with unit-Fr´ echet margins

M(x) unit-Fr´ echet ⇒ EM(x) = ∞

E |M(x + h) − M(x)|2 not finite

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A Different Variogram

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Motiv Max Coeff Geostatistics End

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|F(M(x + h)) − F(M(x))|

with F(u) = exp(−1/u)

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A Different Variogram

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ν(h) = 1 2

E |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

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A Different Variogram

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

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A Different Variogram

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ν(h) = 1 2E |F(M(x + h)) − F(M(x))| Why does it work? 1 2|a − b| = max(a, b) − 1 2(a + b)

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A Different Variogram

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ν(h) = 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 E max(a, b) = EF(max(M(x + h), M(x)

  • max-stable

)) = θ(h) 1 + θ(h)

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Madogram ν(h) ⇒ Extremal coeff θ(h)

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Motiv Max Coeff Geostatistics End

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θ(h) = 1 + 2ν(h) 1 − 2ν(h)

The madogram ν(h) gives the extremal coefficient θ(h)

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Madogram ν(h) ⇒ Extremal coeff θ(h)

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Motiv Max Coeff Geostatistics End

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θ(h) = 1 + 2ν(h) 1 − 2ν(h)

The madogram ν(h) gives the extremal coefficient θ(h) The madogram ν(h) = 1

2E |F(M(x + h)) − F(M(x))| is easy to estimate:

ˆ ν(h) = 1 Nh

  • ||xi−xj||=h
  • F(M(xi) − F(M(xj))
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Schlather’s models (2003)

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Motiv Max Coeff Geostatistics End

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10 20 30 40 10 20 30 40 x y −1 1 2 3

θ(h) = 1 +

  • 1 − 1

2 (exp(−h/40) + 1)

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Madogram ν(h) ⇒ Extremal coeff θ(h)

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Schlather’s fields Madogram Extremal coeff

0.0 0.2 0.4 0.6 0.8 distance estimated madogram

  • 1

4 6 8 10 12 14 16 18 20

  • 1.0

1.2 1.4 1.6 1.8 2.0 distance thetaHat

  • 1

4 6 8 10 12 14 16 18 20

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Precipitation

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Motiv Max Coeff Geostatistics End

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Histogram Madogram

normalized data Density 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5

  • ● ●
  • ● ●
  • ●
  • ● ● ● ● ● ●
  • ●
  • 200

400 600 800 1000 1200 1400 0.00 0.05 0.10 0.15 distance madogram

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Building valid θ(h)

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Proposition A Any extremal coefficient function θ(h) is such that 2 − θ(h) is positive semi-definite.

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Building valid θ(h)

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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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Complete bivariate structure

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Special case u1 = u2 = u

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

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Complete bivariate structure

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Special case u1 = u2 = u

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

General case

P [M(x) < u1, M(x + h) < u2] = exp(−θλ(h)/u1), with λ = u1

u2

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Complete bivariate structure

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Special case u1 = u2 = u

θ(h) = 1 + 2ν(h) 1 − 2ν(h)

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Complete bivariate structure

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Special case u1 = u2 = u

θ(h) = 1 + 2ν(h) 1 − 2ν(h)

General case

θλ(h) = cλ + νλ(h) 1 − νλ(h) − cλ , with cλ = 1 + 3λ 4(1 + λ) and νλ(h) = 1 2E |F(M(x + h)) − F(λM(x))|

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Complete bivariate structure

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Motiv Max Coeff Geostatistics End

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Estimators of νλ(h) = 1

2E |F(M(x + h)) − F(λM(x))|

The “naive” estimator is ˆ νλ(h) = 1 Nh

  • ||xi−xj||=h
  • F(M(xi) − F(λM(xj))
  • but it does not satisfy ˆ

θ0(h) = ˆ θ∞(h) = 1

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Complete bivariate structure

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Estimators of νλ(h) = 1

2E |F(M(x + h)) − F(λM(x))|

The “naive” estimator is ˆ νλ(h) = 1 Nh

  • ||xi−xj||=h
  • F(M(xi) − F(λM(xj))
  • but it does not satisfy ˆ

θ0(h) = ˆ θ∞(h) = 1 An unbiased estimator satisfying the above condition is ˆ ˆ νλ(h) = ˆ νλ(h) − (1 − ω(λ))1 n

  • F(λM(xj)) − ω(λ)1

n

  • F(λM(xj)) + 1

4 with ω(λ) = λ/(1 + λ)

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λ-madogram versus Caperaa et al. 1997

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Motiv Max Coeff Geostatistics End

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G(u1, u2) = exp

  • −
  • 1

u1 + 1 u2

  • A
  • u1

u1 + u2

  • with A(w) = (1 − t1)w + (1 − t2)(1 − w) + [(t1w)(1/α) + (t2(1 − w))(1/α)]α

α = .7, t1 = t2 = 1 α = .3, t1 = t2 = 1 α = .2, t1 = 0.8, t2 = 0.5

w, (lambda) A(w) 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 1.0 (Inf) (4) (3/2) (2/3) (1/4) (0)

  • w, (lambda)

A(w) 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 1.0 (Inf) (4) (3/2) (2/3) (1/4) (0)

  • w, (lambda)

A(w) 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 1.0 (Inf) (4) (3/2) (2/3) (1/4) (0)

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λ−Madogram for spatial fields

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Schlather’s fields

λ = 1 , 2/3 ,..., 1/4 ,..., 1/99 λ = 1 , 3/2 ,..., 4 ,..., 99

5 10 15 20 0.00 0.05 0.10 0.15 0.20 0.25 distance madogram

  • ● ●
  • 5

10 15 20 0.00 0.05 0.10 0.15 0.20 0.25 distance madogram

  • ● ●
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Take-home messages

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Madogram ν(h) ⇒ Extremal coeff θ(h) Extremal coeff θ(h) ⇒ Info about the spatial dependence λ-madogram captures bivariate structure

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SLIDE 41

Take-home messages

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Madogram ν(h) ⇒ Extremal coeff θ(h) Extremal coeff θ(h) ⇒ Info about the spatial dependence λ-madogram captures bivariate structure Future research Derive limiting results for unknown marginals Find “madograms” for exceedances Develop spatial interpolation methods for maxima Derive statistical schemes for downscaling of extremes Acknowledgement NSF-GMC (ATM-0327936) E2C2: Extreme Events, Cause and Consequences: Post-docs

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To learn more about this topic ...

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

  • Naveau, P., Poncet, P., and Cooley, D. (2005). First-order variograms for extreme

bivariate random vectors (submitted).

  • Schlather, M. and Tawn, J. (2003).

A dependence measure for multivariate and spatial extreme values: Properties and inference.

  • Davis, R. and Resnick, S. (1993).

Prediction of stationary max-stable processes,

  • Ann. of Applied Prob 3, 497–525
  • Capeera, P., Foug´

eres A.L., and Genest, C (1997). A non-parametric estimation procedure for bivariate extreme value copulas.

  • Smith, R. (1990). Max-stable processes and spatial extremes.

“Better to have the approximate solution to the correct problem than the exact solution to the wrong problem” -J. Tukey

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Spatial Statistics for Extremes

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5500 6000 6500 7000 7500 18500 19000 19500 0.0 0.2 0.4 0.6 0.8

Our “renormalized” random field

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SLIDE 44

Smith’s models (2003)

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10 20 30 40 10 20 30 40 x y 1 2 3

θ(h) = 2Φ

  • hTΣ−1h/2
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SLIDE 45

Madogram ν(h) ⇒ Extremal coeff θ(h)

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Motiv Max Coeff Geostatistics End

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Smith’s fields Madogram Extremal coeff

0.0 0.2 0.4 0.6 0.8 1.0 distance estimated madogram

  • 1

4 6 8 10 12 14 16 18 20

  • 1.0

1.2 1.4 1.6 1.8 2.0 distance thetaHat

  • 1

4 6 8 10 12 14 16 18 20