On generalized max-linear models
8th Conference on Extreme Value Analysis (EVA 2013),
Shanghai, China
Michael Falk, Martin Hofmann, Maximilian Zott
University of Wuerzburg, Germany
July 9, 2013
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Our favorite max-stable random vector
We call a rv X = (X1 , . . . , Xd ) standard-max-stable, if it is
max-stable and
P(Xi ≤ x) = exp(x),
x ≤ 0, i = 1, . . . , d.
In this case
X =D n max X (i) ,
1≤i≤n
n ∈ N,
where X (1) , X (2) , . . . are independent copies of X . Obviously,
X ≤ 0 with probability one. (All operations such as max, ≤, . . .
are meant componentwise.)
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Characterization of standard max-stable rv
Theorem (de Haan and Resnick (1977), Pickands (1981))
A rv X in Rd is standard-max-stable iff there exists c ≥ 1 and a
rv Z = (Z1 , . . . , Zd ) with E(Zi ) = 1 and Zi ∈ [0, c] a. s.,
i = 1, . . . , d, such that
P(X ≤ x) = exp (− kxkD ) := exp −E max (|xi | Zi )
i=1,...,d
for x ≤ 0 ∈ Rd .
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Characterization of standard max-stable rv
Theorem (de Haan and Resnick (1977), Pickands (1981))
A rv X in Rd is standard-max-stable iff there exists c ≥ 1 and a
rv Z = (Z1 , . . . , Zd ) with E(Zi ) = 1 and Zi ∈ [0, c] a. s.,
i = 1, . . . , d, such that
P(X ≤ x) = exp (− kxkD ) := exp −E max (|xi | Zi )
i=1,...,d
for x ≤ 0 ∈ Rd .
k·kD is called D-norm on Rd with generator Z .
3 / 16
Characterization of standard max-stable rv
Theorem (de Haan and Resnick (1977), Pickands (1981))
A rv X in Rd is standard-max-stable iff there exists c ≥ 1 and a
rv Z = (Z1 , . . . , Zd ) with E(Zi ) = 1 and Zi ∈ [0, c] a. s.,
i = 1, . . . , d, such that
P(X ≤ x) = exp (− kxkD ) := exp −E max (|xi | Zi )
i=1,...,d
for x ≤ 0 ∈ Rd .
k·kD is called D-norm on Rd with generator Z .
k·kD = k·k1 =⇒ independence of the margins
k·kD = k·k∞ =⇒ complete dependence of the margins
k·k1 ≤ k·kD ≤ k·k∞ .
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Our favorite max-stable process
We call a stochastic process η = (ηt )t∈[0,1] with paths in C[0, 1]
a standard max-stable process (SMSP), if it is max-stable and
P(ηt ≤ x) = exp(x),
x ≤ 0, t ∈ [0, 1].
In this case
η =D n max η (i) ,
1≤i≤n
n ∈ N,
where η (1) , η (2) , . . . are independent copies of η. One can
show η < 0 with probability one, cf. Aulbach, Falk and Hofmann
(2012).
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Characterization of SMSP
Some function spaces:
E[0, 1] := set of those bounded functions f : [0, 1] → R that
have only finitely many discontinuities.
Ē − [0, 1] := {f ∈ E[0, 1] : f ≤ 0}.
C̄ + [0, 1] := {f ∈ C[0, 1] : f ≥ 0}.
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Characterization of SMSP
Some function spaces:
E[0, 1] := set of those bounded functions f : [0, 1] → R that
have only finitely many discontinuities.
Ē − [0, 1] := {f ∈ E[0, 1] : f ≤ 0}.
C̄ + [0, 1] := {f ∈ C[0, 1] : f ≥ 0}.
Theorem (Giné, Hahn and Vatan (1990))
A stochastic process η in C[0, 1] is an SMSP iff there exists
c ≥ 1 and a process Z = (Zt )t∈[0,1] in C̄ + [0, 1] with E(Zt ) = 1
and Zt ≤ c a. s., t ∈ [0, 1], such that
!!
P(η ≤ f ) = exp(− kf kD ) := exp −E
sup (|f (t)| Zt )
t∈[0,1]
for f ∈ Ē − [0, 1].
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Characterization of SMSP
Theorem (Giné, Hahn and Vatan (1990))
A stochastic process η in C[0, 1] is an SMSP iff there exists
c ≥ 1 and a process Z = (Zt )t∈[0,1] in C̄ + [0, 1] with E(Zt ) = 1
and Zt ≤ c a. s., t ∈ [0, 1], such that
!!
P(η ≤ f ) = exp(− kf kD ) := exp −E
sup (|f (t)| Zt )
t∈[0,1]
for f ∈ E − [0, 1].
Important: P supt∈[0,1] Zt ≤ c = 1 can be weakened to
E supt∈[0,1] Zt < ∞, see de Haan and Ferreira 2006,
Corollary 9.4.5.
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Characterization of SMSP
Theorem (Giné, Hahn and Vatan (1990))
A stochastic process η in C[0, 1] is an SMSP iff there exists
c ≥ 1 and a process Z = (Zt )t∈[0,1] in C̄ + [0, 1] with E(Zt ) = 1
and Zt ≤ c a. s., t ∈ [0, 1], such that
!!
P(η ≤ f ) = exp(− kf kD ) := exp −E
sup (|f (t)| Zt )
t∈[0,1]
for f ∈ E − [0, 1].
Again: k·kD is called D-norm on E[0, 1] with generator
process Z .
For example k·kD = k·k∞ (generated by Z ≡ 1).
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Aim of this talk
Problems:
1
Given a standard max-stable rv on Rd , how can one
construct a whole SMSP of this rv?
8 / 16
Aim of this talk
Problems:
1
2
Given a standard max-stable rv on Rd , how can one
construct a whole SMSP of this rv?
A whole SMSP η in C[0, 1] can not be observed in
practice. Given some multivariate observations, is it
possible to approximate/reconstruct η by another SMSP η̂
in an appropriate way?
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Generalized max-linear models
Let X = (X1 , . . . , Xd ) a standard max-stable rv with D-norm
k·kD1,...,d generated by Z = (Z1 , . . . , Zd ).
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Generalized max-linear models
Let X = (X1 , . . . , Xd ) a standard max-stable rv with D-norm
k·kD1,...,d generated by Z = (Z1 , . . . , Zd ).
Choose deterministic functions g1 , . . . , gd ∈ C̄ + [0, 1] with
k(g1 (t), . . . , gd (t))kD1,...,d = 1,
t ∈ [0, 1].
(1)
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Generalized max-linear models
Let X = (X1 , . . . , Xd ) a standard max-stable rv with D-norm
k·kD1,...,d generated by Z = (Z1 , . . . , Zd ).
Choose deterministic functions g1 , . . . , gd ∈ C̄ + [0, 1] with
k(g1 (t), . . . , gd (t))kD1,...,d = 1,
t ∈ [0, 1].
(1)
Then the process
ηt := max
i=1,...,d
Xi
,
gi (t)
t ∈ [0, 1]
(2)
defines an SMSP on C[0, 1] with generator
t ∈ [0, 1].
Zt0 = max gi (t)Zi ,
i=1,...,d
We call model (2) generalized max-linear model refering to the
max-linear model established by Wang and Stoev (2011).
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Reconstruction of SMSP
Suppose one knows an SMSP η in C[0, 1] is underlying,
e. g. as a model for the maximum sea-level along a coastal
area. Now observe this process at some points
0 = s1 < · · · < sd = 1.
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Reconstruction of SMSP
Suppose one knows an SMSP η in C[0, 1] is underlying,
e. g. as a model for the maximum sea-level along a coastal
area. Now observe this process at some points
0 = s1 < · · · < sd = 1.
=⇒ (ηs1 , . . . , ηsd ) is a standard max-stable rv.
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Reconstruction of SMSP
Suppose one knows an SMSP η in C[0, 1] is underlying,
e. g. as a model for the maximum sea-level along a coastal
area. Now observe this process at some points
0 = s1 < · · · < sd = 1.
=⇒ (ηs1 , . . . , ηsd ) is a standard max-stable rv.
Applying the generalized max-linear model (2) to this rv
with some specific deterministic functions g1 , . . . , gd with
(1) leads to
ηsi−1
ηsi
ηbt := k(si − t, t − si−1 )kDi−1,i max
,
(3)
si − t t − si−1
for t ∈ [si−1 , si ], i = 2, . . . , d, where k·kDi−1,i is the D-norm
pertaining to (ηsi−1 , ηsi ).
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Discretized version of SMSP
We call the SMSP η̂ = (η̂t )t∈[0,1] defined in (3) discretized
version of η with grid {s1 , . . . , sd }.
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Discretized version of SMSP
We call the SMSP η̂ = (η̂t )t∈[0,1] defined in (3) discretized
version of η with grid {s1 , . . . , sd }.
It interpolates the rv (ηs1 , . . . , ηsd ):
0 = s1
s2
s3
s4
s5
0.2
0.4
0.6
0.8
s6 = 1
1
ηs4
ηs1
ηs2
|
k·kD
{z
1,2
}|
=k·k1
ηs3
k·kD
{z
2,3
=k·kD
3,4
ηs5
ηbt
=k·k2
ηs6
}|
k·kD
4,5
{z
=k·kD
5,6
=k·k10
}
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Uniform approximation by the discretized version
Why is the discretized version of an SMSP considered a
reasonable approximation?
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Uniform approximation by the discretized version
Why is the discretized version of an SMSP considered a
reasonable approximation?
Take an SMSP η with generator Z .
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Uniform approximation by the discretized version
Why is the discretized version of an SMSP considered a
reasonable approximation?
Take an SMSP η with generator Z .
Now choose a sequence of discretized versions η̂ (d) with
(d)
generators Ẑ
and grids Gd , d ∈ N, where the fineness of
Gd converges to zero for d → ∞.
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Uniform approximation by the discretized version
Why is the discretized version of an SMSP considered a
reasonable approximation?
Take an SMSP η with generator Z .
Now choose a sequence of discretized versions η̂ (d) with
(d)
generators Ẑ
and grids Gd , d ∈ N, where the fineness of
Gd converges to zero for d → ∞.
Theorem
b (d) , d ∈ N, converge uniformly to η
The processes ηb(d) and Z
(d)
b
and Z on [0,
1] , respectively, i. e. η − η ∞ →d→∞ 0 and
(d)
b − Z →d→∞ 0.
Z
∞
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The mean squared error of the prediction
(d)
Considering η̂t a predictor for the true value ηt at a fixed index
t ∈ [0, 1], what is the pointwise mean squared error of this
prediction?
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The mean squared error of the prediction
(d)
Considering η̂t a predictor for the true value ηt at a fixed index
t ∈ [0, 1], what is the pointwise mean squared error of this
prediction?
Theorem
Let η and η̂ (d) , d ∈ N, be as above, and denote by k·kD (d) the
t
(d)
D-norm pertaining to the standard max-stable rv (ηt , ηbt ). The
(d)
mean squared error of η̂t is given by
MSE
(d)
η̂t
:= E
ηt −
(d) 2
η̂t
Z
=2 2−
0
∞
1
du
k(1, u)kD (d)
!
t
→d→∞ 0.
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By the way: The entire concept of generalized max-linear
models and the discretized versions carries over to the case of
generalized Pareto distributed rv and generalized Pareto
processes!
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Thank you very much for your attention!
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Some sources
AULBACH , S., FALK , M., H OFMANN , M. (2012). On
max-stable processes and the functional D-norm.
Extremes. To appear.
G INÉ , E., H AHN , M., AND VATAN , P. (1990). Max-infinitely
divisible and max-stable sample continous processes.
Probab. Theory Related Fields.
H AAN , L., AND F ERREIRA , A. (2006). Extreme Value
Theory: An Introduction. Springer, New York.
DE
DE H AAN , L. AND R ESNICK , S. (1977). Limit theory for
multivariate sample extremes. Probab. theory Related
Fields.
WANG , Y., AND S TOEV, S. A. (2010). Conditional sampling
for spectrally discrete max-stable random fields. Adv. in
Appl. Probab.
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