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 naveau@lsce.cnrs-gif.fr ⇓ ↓ ↑ ⇑
Outline of the Talk 1. Motivations 2. Maxima distribution 3. Extremal coefficient 4. Geostatistics 5. Conclusions naveau@lsce.cnrs-gif.fr 2 ⇓ ↓ ↑ ⇑
Climate Studies Motivations Max Coeff Geostat End 3 ⇓ ↓ ↑ ⇑
Climate Studies 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 CO 2 Driven by physical processes : e.g. heat equations Multivariate variables : winds, precipitation, temperatures Motivations Max Coeff Geostat End 3 ⇓ ↓ ↑ ⇑
Climate Studies 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 CO 2 Driven by physical processes : e.g. heat equations Multivariate variables : winds, precipitation, temperatures Motivations Max Coeff Geostat End 3 ⇓ ↓ ↑ ⇑
Spatial Statistics for Extremes 500 19500 How to describe the 400 spatial dependence as 19000 300 a function of the 200 distance between two 18500 100 points? 5500 6000 6500 7000 7500 Motivations Max Coeff Geostat End 4 ⇓ ↓ ↑ ⇑
Our Data: Daily precipitation Motivations Max Coeff Geostat End 5 ⇓ ↓ ↑ ⇑
Our Data: Daily precipitation Dijon’s mustard region!! Motivations Max Coeff Geostat End 6 ⇓ ↓ ↑ ⇑
Our Data: Daily precipitation 500 19500 400 Maxima over 1970-2004 Data homogenized 19000 300 by O. Mestre Cˆ ote d’Or, Bourgogne France 200 83 locations 18500 100 5500 6000 6500 7000 7500 Motivations Max Coeff Geostat End 7 ⇓ ↓ ↑ ⇑
Spatial Statistics for Extremes 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) Motivations Max Coeff Geostat End 8 ⇓ ↓ ↑ ⇑
Assumptions 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 ) Motiv Maxima distribution Coeff Geostat End 9 ⇓ ↓ ↑ ⇑
Assumptions 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 ) < u 1 , M ( x + h ) < u 2 ] = ?? Motiv Maxima distribution Coeff Geostat End 9 ⇓ ↓ ↑ ⇑
Bivariate case ( M ( x ) , M ( x + h )) A well-known non-parametric structure � � � � � g ( s, 0) , g ( s, h ) P [ M ( x ) < u 1 , M ( x + h ) < u 2 ] = exp max δ ( ds ) − u 1 u 2 Motiv Maxima distribution Coeff Geostat End 10 ⇓ ↓ ↑ ⇑
Bivariate case ( M ( x ) , M ( x + h )) A well-known non-parametric structure � � � � � g ( s, 0) , g ( s, h ) P [ M ( x ) < u 1 , M ( x + h ) < u 2 ] = exp max δ ( ds ) − u 1 u 2 Special case u 1 = u 2 = u P [ M ( x ) < u, M ( x + h ) < u ] = exp( − θ ( h ) /u ) = F ( u ) θ ( h ) , with F ( u ) = e − 1 /u Motiv Maxima distribution Coeff Geostat End 10 ⇓ ↓ ↑ ⇑
θ ( h ) = Extremal coefficient P [ M ( x ) < u, M ( x + h ) < u ] = F ( u ) θ ( h ) with F ( u ) = P [ M ( x ) < u ] = P [ M ( x + h ) < u ] Motiv Max Extremal coefficient Geostat End 11 ⇓ ↓ ↑ ⇑
θ ( h ) = Extremal coefficient 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 Motiv Max Extremal coefficient Geostat End 11 ⇓ ↓ ↑ ⇑
Geostatistics: Variograms 2 E | Z ( x + h ) − Z ( x ) | 2 if { Z ( x ) } stationary field s.t. E | Z ( x ) | 2 < ∞ γ ( h ) = 1 ● ● 1.0 Finite if light tails ● ● ● ● 0.8 Capture all spatial ● semivariance 0.6 ● ● ● structure if { Z ( x ) } ● 0.4 Gaussian fields ● 0.2 ● but not well adapted 0.0 for extremes 0.0 0.2 0.4 0.6 0.8 distance Motiv Max Coeff Geostatistics End 12 ⇓ ↓ ↑ ⇑
A Different Variogram 1 E | M ( x + h ) − M ( x ) | 2 =??? 2 where { M ( x ) } stationary max-stable field with unit-Fr´ echet margins Motiv Max Coeff Geostatistics End 13 ⇓ ↓ ↑ ⇑
A Different Variogram 1 E | M ( x + h ) − M ( x ) | 2 =??? 2 where { M ( x ) } stationary max-stable field with unit-Fr´ echet margins M ( x ) unit-Fr´ echet ⇒ E M ( x ) = ∞ E | M ( x + h ) − M ( x ) | 2 not finite Motiv Max Coeff Geostatistics End 13 ⇓ ↓ ↑ ⇑
A Different Variogram | F ( M ( x + h )) − F ( M ( x )) | with F ( u ) = exp( − 1 /u ) Motiv Max Coeff Geostatistics End 14 ⇓ ↓ ↑ ⇑
A Different Variogram ν ( h ) = 1 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 Motiv Max Coeff Geostatistics End 15 ⇓ ↓ ↑ ⇑
A Different Variogram ν ( h ) = 1 2 E | F ( M ( x + h )) − F ( M ( x )) | Why does it work? Motiv Max Coeff Geostatistics End 16 ⇓ ↓ ↑ ⇑
A Different Variogram ν ( h ) = 1 2 E | F ( M ( x + h )) − F ( M ( x )) | Why does it work? 1 2 | a − b | = max( a, b ) − 1 2( a + b ) Motiv Max Coeff Geostatistics End 16 ⇓ ↓ ↑ ⇑
A Different Variogram ν ( h ) = 1 2 E | F ( M ( x + h )) − F ( M ( x )) | Why does it work? 2 | a − b | = max( a, b ) − 1 1 2( a + b ) a = F ( M ( x + h )) and b = F ( M ( x )) E a = E b = 1 / 2 θ ( h ) E max( a, b ) = E F (max( M ( x + h ) , M ( x ) )) = 1 + θ ( h ) � �� � max-stable Motiv Max Coeff Geostatistics End 16 ⇓ ↓ ↑ ⇑
Madogram ν ( h ) ⇒ Extremal coeff θ ( h ) θ ( h ) = 1 + 2 ν ( h ) 1 − 2 ν ( h ) The madogram ν ( h ) gives the extremal coefficient θ ( h ) Motiv Max Coeff Geostatistics End 17 ⇓ ↓ ↑ ⇑
Madogram ν ( h ) ⇒ Extremal coeff θ ( h ) θ ( h ) = 1 + 2 ν ( h ) 1 − 2 ν ( h ) The madogram ν ( h ) gives the extremal coefficient θ ( h ) The madogram ν ( h ) = 1 2 E | F ( M ( x + h )) − F ( M ( x )) | is easy to estimate: ν ( h ) = 1 � � � � � ˆ � F ( M ( x i ) − F ( M ( x j )) � N h || x i − x j || = h Motiv Max Coeff Geostatistics End 17 ⇓ ↓ ↑ ⇑
Schlather’s models (2003) 40 3 30 2 1 y 20 0 10 −1 10 20 30 40 x � 1 − 1 θ ( h ) = 1 + 2 (exp( − h/ 40) + 1) Motiv Max Coeff Geostatistics End 18 ⇓ ↓ ↑ ⇑
Madogram ν ( h ) ⇒ Extremal coeff θ ( h ) Schlather’s fields Madogram Extremal coeff 0.8 2.0 ● ● ● ● ● ● ● ● ● 1.8 ● ● 0.6 ● estimated madogram ● 1.6 ● ● thetaHat 0.4 ● ● ● ● ● 1.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● 1.2 ● ● ● ● ● ● ● 0.0 1.0 1 4 6 8 10 12 14 16 18 20 1 4 6 8 10 12 14 16 18 20 distance distance Motiv Max Coeff Geostatistics End 19 ⇓ ↓ ↑ ⇑
Precipitation Histogram Madogram 1.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.15 ● ● ● ● ● ● 1.0 ● madogram 0.10 Density ● 0.5 0.05 0.00 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0 200 400 600 800 1000 1200 1400 normalized data distance Motiv Max Coeff Geostatistics End 20 ⇓ ↓ ↑ ⇑
Building valid θ ( h ) Proposition A Any extremal coefficient function θ ( h ) is such that 2 − θ ( h ) is positive semi-definite. Motiv Max Coeff Geostatistics End 21 ⇓ ↓ ↑ ⇑
Building valid θ ( h ) 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 ) 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 . θ ( h + k ) τ ≥ Motiv Max Coeff Geostatistics End 21 ⇓ ↓ ↑ ⇑
Complete bivariate structure Special case u 1 = u 2 = u P [ M ( x ) < u, M ( x + h ) < u ] = exp( − θ ( h ) /u ) Motiv Max Coeff Geostatistics End 22 ⇓ ↓ ↑ ⇑
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