Measuring Spatial Dependence among Maxima · 2007. 6. 8. · Extending the madogram ⇓ ↓ ↑ ⇑...
Transcript of 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?
Spatial Statistics for Extremes
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 4
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10 20 30 40
010
2030
40
x
y
How to perform
spatial interpolation
for extreme events?
Spatial Statistics for Extremes
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 5
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 6
Convergence of sample maxima
Normal density ⇒
Uniform density ⇒
Cauchy density ⇒
⇐ Gumbel density
⇐ Weibull density
⇐ Frechet density
n = 50 n = 100
Assumptions
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 7
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 8
For large n
P [Mn(x) < u, Mn(x + h) < v] = ??
Bivariate case for Maxima
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 9
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))
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 10
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 11
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 13
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 14
|F (M(x + h))− F (M(x))|with F (u) = exp(−1/u)
A Different Variogram
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 15
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
A Different Variogram
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 16
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)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 17
θ(h) =1 + 2νh
1− 2νh
The madogram νh gives the extremal coefficient θ(h)
Comparisons with other estimators
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 18
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 19
α = 0.3 α = 0.7
Schlather’s models (2003)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 20
10 20 30 40
1020
3040
x
y
−1
01
23
θ(h) = 1 +
√1−
1
2(exp(−h/40) + 1)
Madogram νh ⇒ Extremal coeff θ(h)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 21
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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Smith’s models (2003)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 22
10 20 30 40
1020
3040
x
y
01
23
θ(h) = 2Φ(√
hTΣ−1h/2)
Madogram νh ⇒ Extremal coeff θ(h)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 23
Smith’s fields
Madogram Extremal coeff
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0.8
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estim
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mad
ogra
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Building valid Extremal coeff
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 24
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 26
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 28
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)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 29
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)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 30
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 31
Mean Square Error from simulations
300 iid Schalther’s fields 32
Notations for asymptotic results
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 33
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 34
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 35
(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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 36
• (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)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 37
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)
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 38
Locations (in Lambert coordinates). Pre-processed 30-year maxima of daily precipitation
λ−madogram
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 39
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
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 40
Estimated λ−madogram for the field of maxima of daily precipitation over 1970-1999
Take-home messages
⇓ ↓ ↑ ⇑ Motivations Max Geostat Estimation 41
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