Generating L©vy Random Fields

OutlineIntroduction

Constructing Levy Random FieldsInference

Discussion & Future Work

Generating Levy Random Fields

Robert L Wolperthttp://www.stat.duke.edu/∼rlw/

Department of Statistical Scienceand

Nicholas School of the EnvironmentDuke University

Durham, NC USA

Arhus University2009 May 21–22

R L Wolpert Generating Levy Random Fields

OutlineIntroduction



Outline

IntroductionMotivation: Moving AveragesContinuous TimeLevy-Khinchine

Constructing Levy Random FieldsBare-Hands ConstructionElegant ConstructionMusielak-Orlicz spaces for Poisson IntegralsMultivariate Random FieldsInverse Levy Measure ConstructionExplicit Examples: St(α, β, γ, δ)

Inference



OutlineIntroduction



Motivation: Moving AveragesContinuous TimeLevy-Khinchine

Moving Averages

A common, flexible way to construct stationary time series(discrete-time stochastic processes):Begin w/ i.i.d. sequence ζi : i ∈ N (or maybe i ∈ Z), set

Xi :=∑

j

bjζi−j

Mean and covariance of Xi easy to compute;

OLS forecasting also easy (at least if b(z) :=∑q

j=0 bj zq−j is apolynomial with all its roots in the unit disk)


OutlineIntroduction




Continuous Time

The obvious analog for continuous time would be to construct astochastic integral

Xt :=

∫ t

−∞b(t − θ) ζ(dθ)

for some random measure ζ(dθ) which is “i.i.d.” in that it:

assigns independent random variables to disjoint sets, and

assigns the same distribution to all translates ζ(t + B)


OutlineIntroduction




Examples

Brownian Motion ζ(Bj)ind∼ No(δ|Bj |, σ

2 |Bj |)

Poisson RF ζ(Bj)ind∼ Po(λ |Bj |)

Gamma RF ζ(Bj)ind∼ Ga(a |Bj |, b)

Cauchy RF ζ(Bj)ind∼ Ca(0, γ|Bj |)

α-Stable RF ζ(Bj)ind∼ St(α, β, γ|Bj |, δ|Bj |)

Note these all make sense for Bj ⊂ Rd ...

Or on a topological group, with Haar measure |Bj |... Or on anyspace S with σ-finite measures a(Bj), γ(Bj), δ(Bj ), λ(Bj), σ2(Bj).


OutlineIntroduction




Examples


2 |Bj |)








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Examples


2 |Bj |)








OutlineIntroduction




Examples

Brownian Motion ζ(Bj)ind∼ No(δ(Bj ), σ

2 (Bj))

Poisson RF ζ(Bj)ind∼ Po(λ (Bj ))

Gamma RF ζ(Bj)ind∼ Ga(a (Bj), b)

Cauchy RF ζ(Bj)ind∼ Ca(0, γ(Bj ))

α-Stable RF ζ(Bj)ind∼ St(α, β, γ(Bj ), δ(Bj ))




OutlineIntroduction




Levy-Khinchine

The two requirements

ζ(A) ⊥⊥ ζ(B), ζ(t + B) ∼ ζ(B)

force strict limits on the probability distribution for ζ(B):The SP ζt := ζ

(

(0, t])

must have stationary independentincrements from an Infinitely Divisible (ID) distribution, soLevy-Khinchine ⇒

E[

e iω ζt

]

= exp

itδω−tω2σ2/2 + t

∫

[

e iωu − 1]

ν(du)

for some drift δ ∈ R, diffusion σ2 ∈ R+, andLevy measure ν satisfying

∫

R

(1 ∧ u2) ν(du) < ∞.


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Examples of Levy Measures

Familiar examples with no Gaussian component (i.e., σ2 = 0):

Po(λ |Bj |) ν(du) = λδ1(du)

Ga(α |Bj |, β) ν(du) = αu−1e−βu1u>0 du

Ca(0, γ|Bj |) ν(du) = γπu−2 du

SαS(α, γ|Bj |) ν(du) = αγπ

Γ(α) sin πα2 |u|−α−1 du

St(α, β, γ|Bj |, δ|Bj |) ν(du) = αγπ

Γ(α) sin πα2 |u|−α−1(1 + β sgn u) du


OutlineIntroduction




Poisson Representation

Let H(du ds) ∼ Po(

ν(du) ds)

on R × S for S = R or Rd or. . . ,

and let δ(ds) be a signed measure on S:

ζ(B) = δ(B) +

∫∫

R×B

u H(du ds)

= δ(B) +∑

σn∈B

υn

∫

Sf (s)ζ(ds) =

∫

Sf (s) δ(ds) +

∑

f (σn) un,

where (υn, σn) = spt(

H(du ds))

(for details, see Wolpert+Ickstadt: 1998 Bka; 1998 Dey et al.,PN&SBS ; 2004 Inv Probs).


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Confession

Experts—

I lied a little about the representation, to make it look simpler.The expressions above will only converge if ν(du) satisfies

∫

R

(

1 ∧ |u|)

ν(du) < ∞; (1)

this excludes the Cauchy and (for α ≥ 1) α-Stable cases.We can still construct ζ(B) under the weaker condition

∫

R

(

1 ∧ u2)

ν(du) < ∞ (2)

once we introduce a “compensator function” h(u) below.


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Confession

Experts—


∫

R

(

1 ∧ |u|)

ν(du) < ∞; (1)


∫

R

(

1 ∧ u2)

ν(du) < ∞ (2)



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Confession

Experts—


∫

R

(

1 ∧ |u|)

ν(du) < ∞; (1)


∫

R

(

1 ∧ u2)

ν(du) < ∞ (2)



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Levy-Khinchine Revisited

Let’s construct a stochastic process (and random field)

ζt := ζ(

(0, t])

, ζ[f ] :=

∫

Sf (s)ζ(ds)

with ch.f.

E[

e iω ζt

]

= exp

itδω + t

∫

[

e iωu − 1 − iωh(u)]

ν(du)

for a bounded Borel compensator h(u) = u + O(u2) andLevy measure ν satisfying

∫

R

(1 ∧ u2) ν(du) < ∞.

Note[


≤ cω(1 ∧ u2), so the ch.f. is OK.


OutlineIntroduction




Levy-Khinchine Revisited

Let’s construct a stochastic process (and random field)

ζt := ζ(

(0, t])

, ζ[f ] :=

∫

Sf (s)ζ(ds)

with ch.f.

E[

e iω ζt

]

= exp

itδω + t

∫

[


ν(du)

for a bounded Borel compensator h(u) = u + O(u2) andLevy measure ν satisfying

∫

R

(1 ∧ u2) ν(du) < ∞.

Note[


≤ cω(1 ∧ u2), so the ch.f. is OK.


OutlineIntroduction



Bare HandsLess InelegantMusielak-Orlicz spaces for Poisson IntegralsMultivariate Random FieldsInverse Levy Measure (ILM) ConstructionExplicit Examples: St(α, β, γ, δ)

Bare-Hands ConstructionFor “small” ǫ > 0,

ζǫt := δt +

∫∫

(−ǫ,ǫ)c×(0,t]u H(du ds) −

∫∫

(−ǫ,ǫ)c×(0,t]h(u) ν(du)ds

:= δǫt +

∫∫

(−ǫ,ǫ)c×(0,t]u H(du ds), δǫ := δ −

∫

(−ǫ,ǫ)ch(u) ν(du)

ζt := limǫ→0

ζǫt = lim

ǫ→0

[

δǫt +∑

uj : |uj | > ǫ, sj ≤ t]

(3)

ujiid∼ νǫ(du)/νǫ

+ ⊥⊥ sjiid∼ Un(T), νǫ(du) := 1|u|>ǫν(du)

Note:

If∫ (

1 ∧ |u|)

ν(du) = ∞ then∑

uj doesn’t converge absolutely &typically δǫ → ±∞ as ǫ → 0... but limit (3) still OK


OutlineIntroduction





ζǫt := δt +

∫∫


∫∫


:= δǫt +

∫∫

(−ǫ,ǫ)c×(0,t]u H(du ds), δǫ := δ −

∫


ζt := limǫ→0

ζǫt = lim

ǫ→0

[

δǫt +∑

uj : |uj | > ǫ, sj ≤ t]

(3)



Note:

If∫ (

1 ∧ |u|)




OutlineIntroduction





ζǫt := δt +

∫∫


∫∫


:= δǫt +

∫∫

(−ǫ,ǫ)c×(0,t]u H(du ds), δǫ := δ −

∫


ζt := limǫ→0

ζǫt = lim

ǫ→0

[

δǫt +∑

uj : |uj | > ǫ, sj ≤ t]

(3)



Note:

If∫ (

1 ∧ |u|)




OutlineIntroduction




More Elegant Construction

Set H(du ds) := H(du ds) − ν(du ds) (Compensated Poisson) and:

ζt := δt +

∫

R×(0,t]

(

u − h(u))

H(du ds) +

∫

R×(0,t]h(u)H(du ds)

Well-defined because:

(

u−h(u))

1(0,t](s) ∈ Ψ0∧1 :=

f :

∫

(1 ∧ |f (u, s)|) ν(du ds) < ∞

h(u)1(0,t](s) ∈ Ψ1∧2 :=

f :

∫

(

|f (u, s)| ∧ |f (u, s)|2)

ν(du ds) < ∞

Musielak-Orlicz spaces


OutlineIntroduction






ζt := δt +

∫

R×(0,t]

(

u − h(u))

H(du ds) +

∫



(

u−h(u))

1(0,t](s) ∈ Ψ0∧1 :=

f :

∫

(1 ∧ |f (u, s)|) ν(du ds) < ∞

h(u)1(0,t](s) ∈ Ψ1∧2 :=

f :

∫

(

|f (u, s)| ∧ |f (u, s)|2)

ν(du ds) < ∞



OutlineIntroduction






ζt := δt +

∫

R×(0,t]

(

u − h(u))

H(du ds) +

∫



(

u−h(u))

1(0,t](s) ∈ Ψ0∧1 :=

f :

∫

(1 ∧ |f (u, s)|) ν(du ds) < ∞

h(u)1(0,t](s) ∈ Ψ1∧2 :=

f :

∫

(

|f (u, s)| ∧ |f (u, s)|2)

ν(du ds) < ∞



OutlineIntroduction




Musielak-Orlicz for (uncompensated) H(dx):

H(dx) ∼ Po(

ν(dx))

on X ⇒

E[

e iωH(A)]

= e(e iω−1)ν(A); for simple f =∑

aj1Aj ∈ L1,

H[f ] :=

∫

Xf (x)H(dx) =

∑

ajH(Aj )

log E[

e iωH[f ]]

=∑

j

(e iωaj − 1)ν(Aj )

=

∫

X

(

e iωf (x) − 1)

ν(dx), well-defined

∀ f ∈ Ψ0∧1:=

f :

∫

X(1 ∧ |f (x)|) ν(dx) < ∞


OutlineIntroduction





H(dx) ∼ Po(

ν(dx))

on X ⇒

E[

e iωH(A)]


aj1Aj ∈ L1,

H[f ] :=

∫

Xf (x)H(dx) =

∑

ajH(Aj )

log E[

e iωH[f ]]

=∑

j


=

∫

X

(

e iωf (x) − 1)


∀ f ∈ Ψ0∧1:=

f :

∫

X(1 ∧ |f (x)|) ν(dx) < ∞


OutlineIntroduction





H(dx) ∼ Po(

ν(dx))

on X ⇒

E[

e iωH(A)]


aj1Aj ∈ L1,

H[f ] :=

∫

Xf (x)H(dx) =

∑

ajH(Aj )

log E[

e iωH[f ]]

=∑

j


=

∫

X

(

e iωf (x) − 1)


∀ f ∈ Ψ0∧1:=

f :

∫

X(1 ∧ |f (x)|) ν(dx) < ∞


OutlineIntroduction




Musielak-Orlicz for (fully compensated) H(dx):

H(dx) = H(dx)−ν(dx), H(dx) ∼ Po(

ν(dx))

on X ⇒

E[

e iωH(A)]

= e(e iω−1−iω)ν(A); for simple f =∑

aj1Aj ∈ L2,

H[f ] :=

∫

Xf (x) H(dx) =

∑

ajH(Aj)

log E[

e iωH[f ]]

=∑

j

(e iωaj − 1 − iωaj)ν(Aj)

=

∫

X

(

e iωf (x) − 1 − iωf (x))


∀ f ∈ Ψ2:=

f :

∫

X

(

|f (x)||f (x)|2)

ν(dx) < ∞

H[f ] ∈ Lp

(

(Ω,F ,P))

, .


OutlineIntroduction






ν(dx))

on X ⇒

E[

e iωH(A)]


aj1Aj ∈ L2,

H[f ] :=

∫

Xf (x) H(dx) =

∑

ajH(Aj)

log E[

e iωH[f ]]

=∑

j


=

∫

X

(

e iωf (x) − 1 − iωf (x))


∀ f ∈ Ψ2:=

f :

∫

X

(

|f (x)||f (x)|2)

ν(dx) < ∞

H[f ] ∈ Lp

(

(Ω,F ,P))

, .


OutlineIntroduction






ν(dx))

on X ⇒

E[

e iωH(A)]


aj1Aj ∈ L2,

H[f ] :=

∫

Xf (x) H(dx) =

∑

ajH(Aj)

log E[

e iωH[f ]]

=∑

j


=

∫

X

(

e iωf (x) − 1 − iωf (x))


∀ f ∈ Ψ1∧2:=

f :

∫

X

(

|f (x)|1 ∧ |f (x)|2)

ν(dx) < ∞

H[f ] ∈ Lp

(

(Ω,F ,P))

, .


OutlineIntroduction






ν(dx))

on X ⇒

E[

e iωH(A)]


aj1Aj ∈ L2,

H[f ] :=

∫

Xf (x) H(dx) =

∑

ajH(Aj)

log E[

e iωH[f ]]

=∑

j


=

∫

X

(

e iωf (x) − 1 − iωf (x))


∀ f ∈ Ψp∧2:=

f :

∫

X

(

|f (x)|p ∧ |f (x)|2)

ν(dx) < ∞

H[f ] ∈ Lp

(

(Ω,F ,P))

, 1 ≤ p ≤ 2.


OutlineIntroduction






ν(dx))

on X ⇒

E[

e iωH(A)]


aj1Aj ∈ L2,

H[f ] :=

∫

Xf (x) H(dx) =

∑

ajH(Aj)

log E[

e iωH[f ]]

=∑

j


=

∫

X

(

e iωf (x) − 1 − iωf (x))


∀ f ∈ Ψp∨2:=

f :

∫

X

(

|f (x)|p ∨ |f (x)|2)

ν(dx) < ∞

H[f ] ∈ Lp

(

(Ω,F ,P))

, 2 ≤ p < ∞.


OutlineIntroduction




More Generally... Rd -valued SII Random Field on R

p:

For d , p ∈ N and bounded h(u) = u + O(

|u|2)

on Rd , and suitable

(e.g. cpt’ly-spt’d or, more generally, Musielak-Orlicz) φ : Rp → R,

H(du ds) ∼ Po(

ν(du ds))

on Rd×R

p,

ζ[φ] := δ[φ] +

∫

Rd+p

(

u − h(u)φ(s))

H(du ds) +

∫

Rd+ph(u)φ(s)H(du ds)

= δ[φ] + limǫ→0

[

δǫ[φ] +∑

ujφ(sj) : |uj | > ǫ]

,

δǫ := δ[φ] −

∫

B(ǫ)c×Rp

h(u)φ(s) ν(du ds)

May also replace Rp with any measurable space S


OutlineIntroduction




More Generally... Rd -valued SII Random Field on S:

For d ∈ N and bounded h(u) = u + O(

|u|2)

on Rd , and suitable

(e.g. cpt’ly-spt’d or, more generally, Musielak-Orlicz) φ : S → R,H(du ds) ∼ Po

(

ν(du ds))

on Rd×S,

ζ[φ] := δ[φ] +

∫

Rd×S

(

u − h(u)φ(s))

H(du ds) +

∫

Rd×Sh(u)φ(s)H(du ds)

= δ[φ] + limǫ→0

[

δǫ[φ] +∑

ujφ(sj) : |uj | > ǫ]

,

δǫ := δ[φ] −

∫

B(ǫ)c×Sh(u)φ(s) ν(du ds)

May also replace Rp with any measurable space S


OutlineIntroduction




ILM Construction for nonnegative random fields:

Let ν(du ds) = ν(du|s)Π(ds) for nice Π(ds); thenu 7→ t = T (u|s) := ν((u,∞)|s) has inverse t 7→ u = T ←(t|s).Let τn be std. Poisson event times and:

sniid∼ Π(ds), un|sn := T ←(τn|sn)

δn[φ] := δ[φ] −

∫

h(u)φ(s)1τn>T (u|s)ν(du ds)

then

ζ[φ] = limn→∞

δn[φ] +∑

j≤n

ujφ(sj )

The un are drawn in decreasing order! Very efficient.


OutlineIntroduction




ILM Construction for nonnegative random fields:

Let ν(du ds) = ν(du|s)Π(ds) for nice Π(ds); thenu 7→ t = T (u|s) := ν((u,∞)|s) has inverse t 7→ u = T ←(t|s).Let τn be std. Poisson event times and:

sniid∼ Π(ds), un|sn := T ←(τn|sn)

δn[φ] := δ[φ] −

∫

h(u)φ(s)1τn>T (u|s)ν(du ds)

then

ζ[φ] = limn→∞

δn[φ] +∑

j≤n

ujφ(sj )

The un are drawn in decreasing order! Very efficient.


OutlineIntroduction




Example: α-Stable Process on Rp

ν(du ds) = γαΓ(α) sin πα

2

π|u|−α−1

(

1 + β sgn u)

du ds

h(u) = sin u

δǫ = −γβ ×

tan πα2 +

2αΓ(α) sin πα2

π(α−1) ǫ1−α α 6= 12π

(1 − γe − log ǫ) α = 1

→

−γβ tan πα2 0 < α < 1

−∞ sgnβ 1 ≤ α < 2as ǫ → 0

νǫ+ =

2γ|T|Γ(α) sin πα2

π ǫα, uj

iid∼ ±Pa(α, ǫ), sj

iid∼ Un(T)


OutlineIntroduction






2

π|u|−α−1

(

1 + β sgn u)

du ds

h(u) = sin u

δǫ = −γβ ×

tan πα2 +

2αΓ(α) sin πα2

π(α−1) ǫ1−α α 6= 12π

(1 − γe − log ǫ) α = 1

→

−γβ tan πα2 0 < α < 1

−∞ sgnβ 1 ≤ α < 2as ǫ → 0

νǫ+ =


π ǫα, uj


iid∼ Un(T)


OutlineIntroduction






2

π|u|−α−1

(

1 + β sgn u)

du ds

h(u) = sin u

δǫ = −γβ ×

tan πα2 +

2αΓ(α) sin πα2

π(α−1) ǫ1−α α 6= 12π

(1 − γe − log ǫ) α = 1

→

−γβ tan πα2 0 < α < 1

−∞ sgnβ 1 ≤ α < 2as ǫ → 0

νǫ+ =


π ǫα, uj


iid∼ Un(T)


OutlineIntroduction




Example: Fully-Skewed Cauchy (α=1, β=1, γ=1, δ=0)

With ǫ = 10−3 on t ∈ T = [0, 5],

ν(du ds) =2

πu−2 1u>0 du ds

νǫ+ = ν

(

(−ǫ, ǫ)c × T)

=2 |T|

π ǫ= 104/π ≈ 3183.1

δǫ = −2

π(1 − γe − log ǫ) ≈ −4.66677

Jǫ ∼ Po(νǫ+); (uj , sj)

iid∼ Pa(1, ǫ) ⊗ Un(T)

≈ (ǫ/Vj , 5Wj) for Vj ,Wjiid∼ Un(0, 1)

ζǫt = tδǫ +

Jǫ∑

j=1

uj1sj≤t


OutlineIntroduction




Example: Fully-Skewed Cauchy (α = 1, β = 1) Process

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

Time

Jum

ps

3256 Jumps for α−Stable St(α = 1, β = 1, γ = 1, δ = 0), w/ ε = 0.001

0 1 2 3 4 5

01

23

4

Time

Pro

cess

Sin−compensated α−Stable Process St(α = 1, β = 1, γ = 1, δ = 0), w/ ε = 0.001, δε = −4.67


OutlineIntroduction




Example: ILM for Fully-Skewed Cauchy

ν(du ds) =2

πu−2 1u>0 du ds

= ν(du|s)Π(ds) =2|T|

πu2du

1

|T|ds

T (u|s) := ν(

(u,∞)∣

∣s) =2|T|

πu, T ←(τn|sn) =

2|T|

πτn

δnt = −t

2

π

[

1 − γe + logπ

2|T|+ log τn

]

ζnt = δn

t +2|T|

π

n∑

j=1

1

τj

1sj≤t


OutlineIntroduction




Amusing side-light...

Let τj be event times of standard Poisson process andlet σj = ±1 with equal probabilities; then the limits

X := (2/π) limn→∞

n∑

j=1

1

τj

− log n

∼ St(1, 1, 1, δ), δ =2

πlog

πe

2+ γe

Y := (2/π) limn→∞

n∑

j=1

σj

τj∼ St(1, 0, 1, 0) = Ca(0, 1)

exist even though neither sum converges absolutely!


OutlineIntroduction



InferenceThe likelihood function upon observing ζǫ

t : t ∈ T

(or, equivalently, (uj , sj) : |uj | > ǫ) for a somewhat uncertainνθ(du ds) = νθ(u, s) du ds is:

L(θ) = e−νθ

(

(−ǫ,ǫ)c×T

) Jǫ∏

j=1

νθ(uj , sj)

For a Subordinator St(α, 1, γ), the negative log likelihood for shapeα and jump rate λǫ := νǫ

+/|T| = (2γ/π)Γ(α) sin πα2 ǫ−α is given by:

ℓ(θ) := − log L(θ) = λǫ|T| − Jǫ log(αλǫ) + α

Jǫ∑

j=1

log(uj/ǫ)

Minimize ⇒ α = Jǫ/∑

log(uj/ǫ), λǫ = Jǫ/|T|


OutlineIntroduction



InferenceThe likelihood function upon observing ζǫ

t : t ∈ T

(or, equivalently, (uj , sj) : |uj | > ǫ) for a somewhat uncertainνθ(du ds) = νθ(u, s) du ds is:

L(θ) = e−νθ

(

(−ǫ,ǫ)c×T

) Jǫ∏

j=1

νθ(uj , sj)

For a Subordinator St(α, 1, γ), the negative log likelihood for shapeα and jump rate λǫ := νǫ

+/|T| = (2γ/π)Γ(α) sin πα2 ǫ−α is given by:

ℓ(θ) := − log L(θ) = λǫ|T| − Jǫ log(αλǫ) + α

Jǫ∑

j=1

log(uj/ǫ)

Minimize ⇒ α = Jǫ/∑

log(uj/ǫ), γ = πJǫǫα/2|T|Γ(α) sin πα

2


OutlineIntroduction



How much approximation error?

For h(u) = u1|u|<1, set Aǫt := (−ǫ, ǫ) × (0, t];

ζt − ζǫt =

∫∫

Aǫt

u H(du ds) −

∫∫

Aǫt

h(u) ν(du)ds =

∫∫

Aǫt

uH(du ds)

is an L2-martingale with QV

〈ζ − ζǫ〉t =

∫∫

Aǫt

u2 ν(du ds) [ζ − ζǫ]t =

∫∫

Aǫt

u2 H(du ds)

or, for the α-stable St(α, β, γ, δ),

〈ζ − ζǫ〉t =2tαγ

π(2 − α)Γ(α) sin(

πα

2) ǫ2−α, or =

2tǫ

πfor Cauchy.

Doob ⇒ P[

|ζs − ζǫs | > c for any 0 < s ≤ t

]

≤2 t ǫ

π c2


OutlineIntroduction



How much approximation error?

For h(u) = u1|u|<1, set Aǫt := (−ǫ, ǫ) × (0, t];

ζt − ζǫt =

∫∫

Aǫt

u H(du ds) −

∫∫

Aǫt

h(u) ν(du)ds =

∫∫

Aǫt

uH(du ds)

is an L2-martingale with QV

〈ζ − ζǫ〉t =

∫∫

Aǫt

u2 ν(du ds) [ζ − ζǫ]t =

∫∫

Aǫt

u2 H(du ds)

or, for the α-stable St(α, β, γ, δ),

〈ζ − ζǫ〉t =2tαγ

π(2 − α)Γ(α) sin(

πα

2) ǫ2−α, or =

2tǫ

πfor Cauchy.

Doob ⇒ P[

|ζs − ζǫs | > c for any 0 < s ≤ t

]

≤2 t ǫ

π c2


OutlineIntroduction



Example: PF Volumes for Sufriere Hills Volcano


OutlineIntroduction



Soufriere Hills Volcano

View from MVO at sunset, 18 April 2007


OutlineIntroduction



Where is Montserrat?


OutlineIntroduction



What’s there?


OutlineIntroduction



What’s been happening?

12

510

2050

100

200

PF volume v (m3)

Num

ber

Vj≥

v

2 5 1052 5 106

2 5 1072 5 108

2 5

Seems kind of linear, on log-log scale... remember that.


OutlineIntroduction



Pyroclastic Flow Volumes

PF’s exceeding 104 m3: Daily, 0–1.6 km runout

PF’s exceeding 105 m3: Weekly, 2–3.0 km runout

PF’s exceeding 106 m3: Quarterly, 4–6.0 km runout

PF’s exceeding 107 m3: Yearly, 4–6.0 km runout

PF’s exceeding 108 m3: Just one, unknown runout

PF’s exceeding 109 m3: None YET, but ...

Let’s make inference and predictions.


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OutlineIntroduction












OutlineIntroduction



Modeling Pyroclastic Flow

Cumulative Flow Volume: Zt ∼ St(α, β = 1, γt, δ = 0); Individual Flows exceeding ǫ: Vj ∼ Pa(α, ǫ); Initiation Angles: ϕj ∼ Un[0, 2π).

Likelihood Function:

L(α, γ) ∝ (α λǫǫα)Jǫ exp

[

− λǫT − α∑

j≤Jǫ

log Vj

]

= (α λǫ)Jǫ e−λǫT−αSǫ , where

Sǫ :=∑

j≤Jǫ

log(Vj/ǫ) and λǫ :=2 γ

π ǫαΓ(α) sin

πα

2


OutlineIntroduction







[

− λǫT − α∑

j≤Jǫ

log Vj

]


Sǫ :=∑

j≤Jǫ


π ǫαΓ(α) sin

πα

2


OutlineIntroduction







[

− λǫT − α∑

j≤Jǫ

log Vj

]


Sǫ :=∑

j≤Jǫ


π ǫαΓ(α) sin

πα

2


OutlineIntroduction







[

− λǫT − α∑

j≤Jǫ

log Vj

]


Sǫ :=∑

j≤Jǫ


π ǫαΓ(α) sin

πα

2


OutlineIntroduction



Likelihood Contours for Sufriere Hills Volcano Data

α

γ

0.50 0.55 0.60 0.65 0.70 0.75 0.80

1500

015

500

1600

016

500

1700

017

500

+

Log Likelihood for Subordinator St(α, β = 1, γ)

MLE: α = 0.65, γ = 16530.50


OutlineIntroduction



Contours from the Computer Flow Model

8.6V=10

8.7V=10

8.8V=10

8.9V=10

9V=10

9.1V=10

Bramble Airport Ψ(⋅)↓

← Plymouth Ψ(⋅)

Angle/Volume combinations (ϕ,V ) outside loops cause disasters


OutlineIntroduction



Risk Computation

P(t) = P[At least one PF > Ψ(ϕ) in t yrs | data]

= 1 −

∫∫

R2+

exp

[

−t λ

2π

∫ 2π

0Ψ(ϕ)−α dϕ

]

π∗(α, λ) dα dλ

= 1 −Sǫ

Jǫ

Γ(Jǫ)

∫

R+

[

1 + (t/T )Iǫ(α)]−Jǫ−1

αJǫ−1e−αSǫ dα

where

Iǫ(α) :=1

2π

∮

[Ψ(ϕ)/ǫ]−α dϕ

Approximate integration (by simulation) leads to:


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Estimated Risk of Catastrophic Flows

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

← P(t) at Bramble Airport

P(t) at Plymouth →

Time t (years)

P(t)

= P[

Cat

astro

phe

in t

yrs

| Dat

a ]

Probability of disaster in next t yearsR L Wolpert Generating Levy Random Fields

OutlineIntroduction



Ongoing Work & Extensions

Related papers on spatial & spat/temp Levy-based BNPReg’n models (w/ N Best, M Clyde, E ter Horst, L House,K Ickstadt, S Mukherjee, N Pillai, C Tu) and upcomingpapers (w/M Huber, M OConnell, Z Ouyang, D Woodard)

Posterior distributions in Bayesian models (RJ-MCMC)

Elicitation issues (w/ K Mengersen)

Consistency properties (w/ N Pillai)

Spatial Maximal fields (w/ D Nychka)

Perfect simulation of Levy (w/G Roberts)

Spatial Risk Models (Volcanoes! w/ J Berger, S Bayarri, etal.)

Need better/more output measures (movies, etc.— help!)


OutlineIntroduction



Thanks!

to the Department of Mathematical Sciences at Alborg University,esp. Jesper Møller.

More details (references, this talk in .pdf, related work) areavailable from website

http://www.stat.duke.edu/∼rlw/or on request from

[email protected].

Thanks too to NSF, SAMSI, DSCR, MSO.


Documents

Generating L©vy Random Fields