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Outline Representations of Max–Stable Processes Structure and Classification Examples
Outline Representations of Max–Stable Processes Structure and Classification Examples
On the structure of maxstable processes Yizao Wang and Stilian Stoev - - PowerPoint PPT Presentation
On the structure of maxstable processes Yizao Wang and Stilian Stoev - - PowerPoint PPT Presentation
Outline Representations of MaxStable Processes Structure and Classification Examples On the structure of maxstable processes Yizao Wang and Stilian Stoev University of Michigan, Ann Arbor Graybill VIII: Extreme Value Analysis Fort
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Preliminaries
Motivation: Consider i.i.d. processes {Y (i)
t }t∈T. If for some
an ∼ n1/αℓ(n) (α > 0), we have 1 an
- 1≤i≤n
Y (i)
t
- t∈T
f .d.d.
− → {Xt}t∈T, (n → ∞), then X = {Xt}t∈T is max–stable (α−Fr´ echet). Def The process X = {Xt}t∈T is max–stable (α−Fr´ echet) if: {X (1)
t
∨ · · · ∨ X (n)
t
}t∈T
d
= n1/α{Xt}t∈T, for all n ∈ N, where X (i) = {X (i)
t }t∈T are independent copies of X and
where α > 0.
- The margins of X are then α−Fr´
echet (α > 0), namely: P{Xt ≤ x} = exp{−σα
t x−α},
x > 0, with σα
t > 0.
Fine print: For simplicity, we focus on max–stable processes with α−Fr´
echet marginals. By transforming the margins, our theory applies to the most general definition of max–stable processes where the margins can be extreme value distributions of different types.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Spectral Representations
For an α−Fr´ echet process (α > 0) X = {Xt}t∈T, we have: (a) de Haan’s spectral representation: Xt =
∞
- i=1
ft(Ui)/Γ1/α
i
, (t ∈ T) for a Poisson point process (Γi, Ui) on (0, ∞) × U with intensity dx × µ(du). (b) Extremal integrals: Xt =
- e
U
ft(u)Mα(du), (t ∈ T) for an α−Fr´ echet random sup–measure Mα(du) on (U, µ).
- The deterministic functions ft(u) ≥ 0 are called spectral functions of X
and satisfy:
- U
ft(u)αµ(du) < ∞, (t ∈ T). Fine print: The measure space (U, µ) can be chosen ([0, 1], dx) if the process {Xt}t∈T is separable in
probability, in particular, continuous in probability when T is a separable metric space.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Extremal integrals
Let Mα be a random α−Fr´ echet sup–measure on (U, µ).
- For simple functions f (u) = n
i=1 ai1Ai(u), f (u) ≥ 0:
- e
U
f (u)Mα(du) =
- 1≤i≤n
aiMα(Ai).
- The def of
- e
U fdMα extends to all f ∈ Lα +(µ) and
P{
- e
U
fdMα ≤ x} = exp{−
- U
f αdµx−α}, x > 0.
- For f , g ∈ Lα
+(µ):
- e
U
(af ∨ bg)dMα = a
- e
U
fdMα ∨ b
- e
U
gdMα (max–linearity)
- e
U fdMα and
- e
U gdMα are independent if and only if fg = 0, (mod µ).
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Benefits: For any ft ∈ Lα
+(µ), t ∈ T, we get a max–stable process:
Xt :=
- e
U
ftdMα
- For the finite–dimensional distributions, we have:
P{Xti ≤ xi, 1 ≤ i ≤ d} = P{
- e
U
(∨1≤i≤dx−1
i
fti)dMα ≤ 1} = exp{−
- U
(∨f α
ti /xα i )dµ}.
Examples:
- (moving maxima) Xt :=
- e
R f (t − x)Mα(dx), with (U, µ) ≡ (R, dx) and
f ∈ Lα
+(dx).
- (mixed moving maxima) With (U, µ) = (R × V , dx × dν):
Xt :=
- e
R×V
f (t − x, v)Mα(dx, dv), f (x, v) ∈ Lα
+(dx, dν).
- (Brown–Resnick) With (U, µ) a probability space and {wt(u)}t∈R a
standard Brownian motion on (U, µ): Xt :=
- e
U
ewt(u)−|t|/2αMα(du), t ∈ R.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Max-linear isometries
Consider a max–stable process: Xt =
- e
U
ftdMα, (t ∈ T). Some Natural Questions:
- How does the structure of the ft(u)’s determine the structure of
X = {Xt}t∈T and vice versa?
- Given another representation {gt} ⊂ Lα
+(V , ν)
{Xt}t∈T
d
=
e V
gtd Mα
- ,
what is the relationship between {ft}t∈T and {gt}t∈T? Answer: There exists a max–linear isometry I : Lα
+(U, µ) → Lα +(V , ν),
such that I(ft) = gt, for all t ∈ T.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
That is: I(a1f1∨a2f2) = a1I(f1)∨a2I(f2), for all fi ∈ Lα
+(U, µ), ai ≥ 0,
(max–linear) and
- V
I(f )αdν =
- U
f αdµ (isometry).
- Conversely, any max–linear isometry I : Lα
+(U, µ) → Lα +(V , ν) yields an
equivalent spectral represenation gt := I(ft) of the process X over (V , ν).
- Understanding the structure of the max–linear isometries is key!
- In Wang & S., 2009, we extend results of Hardin, 1981/82 on linear
isometries to max–linear isometries.
- The role of these results in the theory of max–stable processes is best
understood via the notion of a minimal representation.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Minimal spectral representations Def The spectral representation {ft}t∈T ⊂ Lα
+(U, µ) of X is minimal if:
(i) (full support) supp{ft, t ∈ T} = U (mod µ) (ii) (non–redundance) For any measurable A ⊂ U, there exists B ∈ ρ{ft, t ∈ T} ≡ σ{ft/fs, t, s ∈ T}, such that µ(A∆B) = 0.
- This def is identical to the one of Rosi´
nski (1995) in the sum–stable case and similar to Hardin (1982) and to the proper rep in de Haan and Pickands (1986).
- Why are minimal representations minimal?
- The full–support condition is natural.
- Consider the process
Xt :=
- e
[0,1]
t2 sin2(u)Mα(du) = t2Z, where Z =
- e
[0,1] sin2(u)Mα(du).
- This representation is clearly redundant. Note that
ρ(ft, t ∈ T) = {∅, [0, 1]} ≃ B[0,1].
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Outline Representations of Max–Stable Processes Structure and Classification Examples
- We have a natural, simpler representation:
Xt
d
=
- e
U
ft(u) Mα(du), with ft(u) = t2, and trivial U = {0}, and µ(du) = cδ0(du), c :=
- [0,1] sin2α(x)dx.
- The ratio σ−algebra ρ(ft, t ∈ T) captures best the ’minimal
information’ needed to represent the process.
- What are the benefits of minimal representations?
Thm 1. (Wang & S.(2009)) Let {ft}t∈T ⊂ Lα
+(µ) be a minimal
measurable rep of X. If {gt}t∈T ⊂ Lα
+(V , ν) is another measurable rep of
X, then: gt(v) = h(v)ft(φ(v)), ν − a.e. for some measurable h ≥ 0 and φ : V → U. The map φ is unique (mod ν). If {gt} is also minimal, then φ is bimeasurable, ν ∼ µ ◦ φ and dµ ◦ φ dν (v) = hα(v) > 0.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Minimal representations with standardized support Consider the sets SI,N where I = 0, 1 and 0 ≤ N ≤ ∞:
- If I = 1, set S1,N = (0, 1) ∪ {1, · · · , N},
(0 ≤ N ≤ ∞)
- If I = 0, set S0,N = {1, · · · , N},
(0 ≤ N ≤ ∞)
- By convention: S1,0 = (0, 1), S0,∞ = N and S0,0 = ∅.
- Equip SI,N with the measure
λI,N(x) = dx +
N
- i=1
δ{i}(dx). Fine print: Every standard Lebesgue space is isomorphic to some (SI,N, λI,N). Def A minimal representation {ft}t∈T ⊂ Lα
+(U, µ) is said to have
standardized support if, for some I, N: (U, µ) ≡ (SI,N, λI,N). Thm 2. (Wang & S., 2009) Every separable in probability α−Fr´ echet process X has a minimal representation with standardized support: {Xt}t∈T
d
=
e SI,N
ftdMα
- t∈T.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Continuous–Discrete Decomposition Consider an α−Fr´ echet process X with the minimal rep of standardized support: {Xt}t∈T
d
=
e SI,N
ftdMα
- t∈T.
By setting X I
t :=
- e
SI,N∩(0,1)
ftdMα and X N
t :=
- e
SI,N∩N
ftdMα, we obtain the continuous–discrete decomposition: {Xt}t∈T
d
= {X I
t ∨ X N t }t∈T.
The components X I
t and X N t
are independent. Intuition: Suppose I = 1 and N > 0. Then, X I
t =
- e
(0,1)
ftdMα and X N
t = N
- i=1
ft(i)Zi, where Zi = Mα{i} are i.i.d. standard α−Fr´ echet, independent of X I
t .
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Outline Representations of Max–Stable Processes Structure and Classification Examples
- {X I
t } is the continuous and {X N t }t∈T the discrete spectral components
- f X.
Fine print: One of the compponents vanishes if N = 0 or I = 0.
- The continous–discerete decomposition does not depend on the choice
- f the representation.
Thm (Wang & S., 2009) Let {gt}t∈T ⊂ Lα
+(SI ′,N′, λI ′,N′) be another
minimal rep of X with standardized support, then (I, N) ≡ (I ′, N′), {X I
t } d
= {X I ′
t }t∈T
and {X N
t } d
= {X N′
t }t∈T,
where X I ′
t :=
- e
SI,N∩(0,1) gtdMα and X N′ t
=
- e
SI,N∩N dtdMα.
- Moreover, for the discrete component, we have that:
{X N
t }t∈T d
= N
- i=1
φt(i)Zi
- t∈T,
for some unique set of functions {φt(i), 1 ≤ i ≤ N}.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Discrete Principal Components Consider the spectrally discrete component of the process {Xt}t∈T: X N
t = N
- i=1
φt(i)Zi, (t ∈ T), for i.i.d. standard α−Fr´ echet Zi’s.
- The functions t → φt(i), 1 ≤ i ≤ N are unique up to permutation of
the indices.
- The φt(i)’s are the discrete principal components of X.
- Not all sequences of non–negative of functions can be discrete principal
components. Fine print: Prop: (Wang & S., 2009) A countable set of functions φ := {φt(i) ≥ 0, 1 ≤ i ≤ N} can be
discrete principal components of an α−Fr´ echet process if and only if, φ is a minimal representation. Namely, if (i) PN
i=1 φα t (i) < ∞ and (ii) ρ(φt(·), t ∈ T) = 2{1,··· ,N}.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Discrete Principal Components: Applications and Implications Applications: Given a spectrally discrete statistical model, estimate:
- The order N, if finite.
- The (unique) principal component functions t → φt(i), for 1 ≤ i ≤ N.
Thm (Wang & S., 2009) Let {Xt}t∈R be a measurable and stationary α−Fr´ echet process. Then, the spectrally discrete component of X is either zero or trivial, i.e. {X N
t }t∈R d
= {Z}t∈R, for some random variable Z.
- Certainly, with i.i.d. α−Fr´
echet Zi’s: Xt :=
- i∈Z
f (t − i)Zi, (t ∈ Z), is a non–trivial stationary and spectrally discrete processes.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Co–spectral functions Let now T be a separable metric space, equipped with a Borel measure λ. Consider the α−Fr´ echet process X = {Xt}t∈T: Xt :=
- e
U
f (t, u)Mα(du), (t ∈ T), where (t, u) → f (t, u) is measurable.
- Focus on the co–spectral functions:
t → f (t, u) ∈ L0
+(T, λ),
for fixed u ∈ U.
- Can show that the co–spectral functions of X do not depend on the
representation (up to rescaling)! Fine print: If (U, µ) is standard Lebesgue, then X is measurable, if and only if, (t, u) → f (t, u) has a
measurable modification.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Co–spectral functions: Classification Let P be a positive (measurable) cone in L0
+(T, λ), i.e. cP ⊂ P.
Consider the partition of U = A ∪ B into disjoint components: A := {u ∈ U : f (·, u) ∈ P} and B := U \ A = {u ∈ U : f (·, u) ∈ P}. This yields the decomposition: {Xt}t∈T
d
= {X A
t ∨ X B t }t∈T,
(1) where X A
t :=
- e
A
f (t, u)Mα(du) and X B
t :=
- e
B
f (t, u)Mα(du) are two independent processes.
- The decomposition (1) does not depend on the choice of the
measurable rep {f (t, u)}(t,u)∈T×U. Idea of proof: WLOG suppose that {f (t, u)}t∈T is minimal and let {g(t, v)}t∈T ⊂ Lα
+(V , ν) is another measurable rep of X = {Xt}t∈T.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Then, by Thm 1: g(t, v) = h(v)f (t, φ(v)), where h(v) ≥ 0. Since P is a cone, g(·, v) ∈ P ⇔ f (·, φ(v)) ∈ P, which shows that the corresponding partition of V is: V = A ∪ B := φ−1(A) ∪ φ−1(B) A change of variables, yields:
e A
ftdMα
- t∈T
d
=
e e A
gtd Mα
- t∈T,
completing the proof.
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Applications Corollary: Let {
- e
U
f (t, u)Mα(du)}
d
= {
- e
V
g(t, v) Mα(dv)}. Then, given a cone P ⊂ L0
+(T),
f (·, u) ∈ P, a.e. if and only if g(·, v) ∈ P, a.e.
- Let (U, µ) ≡ (Rd, dx) and f , g ∈ Lα
+(Rd). Consider the moving
maxima random fields Xt :=
- e
Rd f (t − u)Mα(du)
and Yt :=
- e
Rd g(t − u)Mα(du).
Then, {Xt}t∈Rd
d
= {Yt}t∈Rd, if and only if, for some τ ∈ Rd g(·) = f (· + τ).
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Outline Representations of Max–Stable Processes Structure and Classification Examples
Key References Brown, B. M. & Resnick, S. I. (1977), ‘Extreme values of independent stochastic processes’, J. Appl. Probability 14(4), 732–739. Cambanis, S., Hardin, Jr., C. D. & Weron, A. (1987), ‘Ergodic properties
- f stationary stable processes’, Stoch. Proc. Appl. 24, 1–18.
Davis, R. & Resnick, S.I. (1993), ‘Prediction of stationary max–stable processes’, Ann. Appl. Probab., 3(2), 497–525. de Haan, L. (1978), ‘A characterization of multidimensional extreme-value distributions’, Sankhy¯ a (Statistics). The Indian Journal of
- Statistics. Series A 40(1), 85–88.
de Haan, L. (1984), ‘A spectral representation for max–stable processes’,
- Ann. Probab. 12(4), 1194–1204.