continuous time random walks in the continuum …thiele.au.dk/fileadmin/ · continuous time random...

Continuous Time Random Walks

in the Continuum Limit

Simulation of Atmospheric Wind Speeds

David Kleinhans, Joachim Peinke, Rudolf Friedrich

[email protected]

FORWIND

Zentrum fur Windenergieforschung

D-26129 Oldenburg, Germany

Institut fur Physik

Carl-von-Ossietzky-Universit at Oldenburg

D-26111 Oldenburg, Germany

Institut fur Theoretische Physik

Westf alische Wilhelms-Universit at Munster

D-48149 Munster, Germany

Version vom 31. Januar 2008

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 1

Turbulence and Finance:

Instationary Processes


Motivation

[Böttcher, Bath & Peinke 2007]

–0.04 0.00 0.04–4

0

4

8

y(τ )

log

p(y(

τ))

[Nawroth & Peinke 2003]

u

Lo

g(P

(u))

[a

rb.

un

its

]

[Friedrich 2003]

[Mordant, Metz, Michel & Pinton 2001]


Motivation


–0.04 0.00 0.04–4

0

4

8

y(τ )

log

p(y(

τ))


u

Lo

g(P

(u))

[a

rb.

un

its

]

[Friedrich 2003]


CTRWs are potential generators of Lagrangian tracer dynamics

Similar structure of increment statistis in Finance and Atmospheric turbulence


Motivation


–0.04 0.00 0.04–4

0

4

8

y(τ )

log

p(y(

τ))


u

Lo

g(P

(u))

[a

rb.

un

its

]

[Friedrich 2003]


Outline:

Introduction to Continuous Time Random Walks (CTRWs)

Initial definition

Continuous sample paths, application to finance

CTRW model for atmospheric turbulence


Continuous time random walks

(CTRWs)


Discrete Random Walks

[Metzler & Klafter 2000]

Continuous time random walk:

xi+1 = xi + ηi

ti+1 = ti + τi

PDFs of jumping times are continuous , paths are discontinuous


Diffusion Limit of CTRWs

Evolution of CTRWs

Sums of random variables ⇒ Fourier / Laplace-Representation

Montroll-Weiss equation:

ˆP (k, u) =1 − Pt(u)

uh

1 − Px(k)Pt(u)i . (1)



Evolution of CTRWs

Sums of random variables ⇒ Fourier / Laplace-Representation

Montroll-Weiss equation:

ˆP (k, u) =1 − Pt(u)

uh

1 − Px(k)Pt(u)i . (2)

Diffusion Limit (⇒ Long Time)

Assumptions:

Jump PDF has finite variance

Asymptotical Power-Law decay ∼ x−(1+α) of waiting time PDF (heavy tailed)

Fractional Diffusion equation

∂

∂tW (x, t) = δ(k)δ(t) +

σ2

T α 0D1−α

t

X

i,j

∂2

∂xi∂xjW (x, t) (3)



With 0D1−αt f(t), 0 < α ≤ 1: Riemann-Liouville integro-differential fractional operator

0D1−αt f(t) :=

1

Γ(α)

∂

∂t

∞Z

0

dt′f(t)

(t − t′)1−α(4)

Connection to integer PDEs: Memory Kernel [Metzler & Klafter 2000, Barkai 2001]

W (x, t) =

∞Z

0

s A(s|t)W1(x, (Kα/K1)1/α s) (5)

with

A(s|t) =1

α

t

s1+1/αLα,1

„

t

s1/α

«

(6)


Long Time Limit: t ≫ 1

Waiting time PDF P (τ) =

8

>

<

>

:

τ ≥ 0 :q

2

πσ2exp

„

τ2

2σ2

«

τ < 0 : 0

8

<

:

τ ≤ 100 : N (0.8, 100)L0.8,1(τ)

τ > 100 : 0

L0.8,1(τ)

-6

-4

-2

0

2

4

6

8

0 25 50 75 100 125 150

x 1(t

)

t

-6

-4

-2

0

2

4

6

8

0 25 50 75 100 125 150

x 2(t

)

t

-6

-4

-2

0

2

4

6

8

0 25 50 75 100 125 150

x 3(t

)

t

-40-20

0 20 40

60 80

100 120 140 160

0 2500 5000 7500 10000 12500 15000

x 1(t

)

t

-40-20

0 20 40 60 80

100 120 140 160

0 2500 5000 7500 10000 12500 15000

x 2(t

)

t

-40-20

0 20 40

60 80

100 120 140 160

0 2500 5000 7500 10000 12500 15000x 3

(t)

t


Continuous Trajectories at Finite Time

Adequate scaling of processes: Continuous (fractional) trajectories at finite time

⇒ Monte-Carlo simulation of fractional PDEs





Initial work:

Heinsalu, Patriarca, Goychuk, Schmid & Hänggi 2006,

Fractional Fokker-Planck dynamics: Numerical algorithm and simulations





Initial work:

Heinsalu, Patriarca, Goychuk, Schmid & Hänggi 2006,

Fractional Fokker-Planck dynamics: Numerical algorithm and simulations

Recent advancements:

Magdziarz & Weron 2007,

Fractional FP dynamics: Stochastic representation and computer simulation

Gorenflo, Mainardi & Vivoli 2007,

Continuous-time random walk and parametric subordination in fractional diffusion

Kleinhans & Friedrich 2007,

Continuous time random walks: Simulation of continuous trajectories


Continuum limit [Fogedby 1994]

According to Fogedby: Continuum limit of discrete equations,

xi+1 = xi + ηi

ti+1 = ti + τi

9

>

=

>

;

i→s⇒

8

>

<

>

:

∂∂s

x(s) = η(s)

∂∂s

t(s) = τ(s)




xi+1 = xi + ηi

ti+1 = ti + τi

9

>

=

>

;

i→s⇒

8

>

<

>

:

∂∂s

x(s) = η(s)

∂∂s

t(s) = τ(s)

Notation:

dxs = dWs

dts = dLαs




xi+1 = xi + ηi

ti+1 = ti + τi

9

>

=

>

;

i→s⇒

8

>

<

>

:

∂∂s

x(s) = η(s)

∂∂s

t(s) = τ(s)




xi+1 = xi + ηi

ti+1 = ti + τi

9

>

=

>

;

i→s⇒

8

>

<

>

:

∂∂s

x(s) = η(s)

∂∂s

t(s) = τ(s)

Here: η(s) and τ(s) have to obey stable distribution

Lα,1(x) =1

πRe

8

<

:

∞Z

0

dk exph

−ikx − |k|α exp“

−iπα

2

”i

9

=

;

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2

L α(x

)x

α=0.9α=0.8α=0.7α=0.6α=0.5α=0.4

Result: Fractional dynamics at finite time


Intrinsic Time in Finance [Müller 1993]


Numerical algorithm

Numerical Integration Scheme: (Itô)

x(s + ∆s) = x(s) + ∆sF (x(s)) + (∆s)1/2 η(s)

t(s + ∆s) = t(s) + (∆s)1/α τα(s) .


Numerical algorithm


x(s + ∆s) = x(s) + ∆sF (x(s)) + (∆s)1/2 η(s)

t(s + ∆s) = t(s) + (∆s)1/α τα(s) .

Discontinuous character of t(s):

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3

t(s)

s


Numerical algorithm


x(s + ∆s) = x(s) + ∆sF (x(s)) + (∆s)1/2 η(s)

t(s + ∆s) = t(s) + (∆s)1/α τα(s) .

Algorithm for simulation of x(t):

Initialisation of xs(0) and ts(0), set s = 0

for every j = 0 to N :

1. while (ts(s) < tj):

(a) calculate xs(s + ∆s) and ts(s + ∆s) from discrete equations (see above)

(b) increase s by ∆s

2. set x(tj) := xs(s)


Numerical algorithm


x(s + ∆s) = x(s) + ∆sF (x(s)) + (∆s)1/2 η(s)

t(s + ∆s) = t(s) + (∆s)1/α τα(s) .

Appropriate ∆s can be determined:

10-2

10-1

100

10-5 10-4 10-3 10-2 10-1 100

∆s

√⟨∆

x2 ⟩


Numerical results

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 0

2

4

6

8

10

x(t)

s(t)

t

α=1.00

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 0

2

4

6

8

10

x(t)

s(t)

t

α=0.95

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 0

2

4

6

8

10

x(t)

s(t)

t

α=0.90

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 0

2

4

6

8

10

x(t)

s(t)

t

α=0.85


Validation: Fractional statistics

0

0.05

0.1

0.15

0.2

0.25

-4 -2 0 2 4 6

P(x

)

x


Validation: Fractional statistics

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

<x(

t 1=

1)x(

t 2)>

t2

[Kleinhans & Friedrich 2007, Baule & Friedrich 2007]David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 13

Simulation of

Atmospheric Wind Fields


Atmospheric Wind Fields

[NASA visible earth]

0

20

40

60

80

100

0 2 4 6 8 10wind speed (m/s)

heig

ht (

m)

logarithmic profile: atmosphere unstable / neutral

atmosphere stable

[Berg (2004)] [R. Gasch, Windkraftanlagen,

Teubner, 1993.]


Turbulence intensity

Engineer: Character of wind field fully specified by (here: T = 600s)

Mean wind speed 〈u(t)〉T

Turbulence intensity TI :=〈u2〉

T−[〈u(t)〉T ]2

〈u(t)〉T


Turbulence intensity

Engineer: Character of wind field fully specified by (here: T = 600s)

Mean wind speed 〈u(t)〉T

Turbulence intensity TI :=〈u2〉

T−[〈u(t)〉T ]2

〈u(t)〉T

Example:

Temporal drift of wind velocity u(t) = u0 + at + u′(t) with˙

u′(t)¸

=˙

tu′(t)¸

= 0D

u′2(t)E

= σ2

TI = 100q

a2 T2

12+ σ2/u0

⇒ Turbulence intensity very weak parameter

7 8 9

10 11 12 13

-300 -150 0 150 300

u(t)

[m/s

]

t [s]

7 8 9

10 11 12 13

-300 -150 0 150 300

u(t)

[m/s

]

t [s]

Two sample time series with turbulence intensity TI= 10%


Windfield models: Overview

DNS, Large Eddy Simulation (LES), Reynolds Averaged Navier Stokes (RANS)

Spectral Models (IEC compliant)

Veers: Three-dimensional wind fiels simulation (1984)

Mann: Wind field simulation (1998)

some further works e.g. by Bierbooms, Nielsen, Larsen, Hansen . . .

(Fung: Kinematic Simulation (KS) (1992))

Fractal models

Schertzer, Lovejoy: . . . anisotropic scaling multiplicative processes (1987)

Cleve: Fractional Brownian motion (2005)

Stochastic / Probabilistic

Nawroth: Reconstruction of processes with Markov properties in scale

Schmiegel / Barndorff-Nielsen: Delay kernel

Cleve: Cascade model


Spectral surrogate data (1D) [Shinozuka, Deodatis 1991]

Aim: Simulation of u′(t) with 〈u′〉 = 0 and proper energy spectrum .

’Fourier-Stieltjes-Integral’ : u′(t) =∞R

0

[cos(ωt)du(ω) + sin(ωt)dv(ω)]

Properties of the coefficients (for ω, ω′ ≥ 0):

Mean 〈du(ω)〉 = 〈dv(ω)〉 = 0

Autocorrelation 〈du(ω)du(ω′)〉 = 2δ(ω′ − ω)S(ω)dω

〈dv(ω)dv(ω′)〉 = 2δ(ω′ − ω)S(ω)dω

Crosscorrelation 〈du(ω)dv(ω′)〉 = 0


Intermittent velocity increments

Atmospheric measurement: Spectral surrogates:

[Shinozuka, Deodatis 1991; Veers 1988;

Mann 1998]

-4

-2

0

2

4

6

8

10

12

14

-4 -3 -2 -1 0 1 2 3 4 5Lo

g[P

(∆x)

]∆x [σ m/s]

Spectral surrogates do not reproduce intermittent statistics

How could they?

Advanced methods for fast simulation of atmospheric winds required


Motivation: Starting point

Requirements

Intermittent statistics of atmospheric wind

Simulation of a 2D (3D?) field in time

Adaptability to measured wind data

Recent work:

[Friedrich (2003)]

Motivation of the applicability of CTRW’s for Lagrangian tracers

[Baule (2005)]

Several tools for the application of CTRW processes.

[Castaing (1990), Beck (2003), Böttcher (2005)]

Analysis of turbulent data sets:

Generation of intermittency by superposition of gaussian p rocesses .


Model

Coupled Langevin equations

∂

∂txref (s) = −γref

ˆ

xref (s) − x0

˜

+q

DrefΓref (s)

∂

∂sxi(s) = −γ

ˆ

xi(s) − αixref (s)˜

+X

j

p

DijΓj(s) ∀ i

∂

∂st(s) = τ(s)

Properties

x0: Mean wind speed at reference height

Reference process xref : Models wind variations on larger timescales (γref ≪ γ)


Model

Coupled Langevin equations

∂

∂txref (s) = −γref

ˆ

xref (s) − x0

˜

+q

DrefΓref (s)

∂

∂sxi(s) = −γ

ˆ

xi(s) − αixref (s)˜

+X

j

p

DijΓj(s) ∀ i

∂

∂st(s) = τ(s)

Properties

x0: Mean wind speed at reference height

Reference process xref : Models wind variations on larger timescales (γref ≪ γ)

αi incorporate (logarithmic) wind profile

Interaction of fluctuations decays exponentially, Dij ∼ exp(−˛

˛ri − rj

˛

˛)

Kramers-like equations (11a,11a) can be treates analytically

Stochastic process t(s) naturally introduces intermittency


Fokker-Planck Equation

Evolution equation for PDF:

∂

∂sP (x, xref , s) =

h

∇xγˆ

x + αxref

˜

+ ∇x∇Tx D (13)

+∂

∂xrefγ

ˆ

xref + x0

˜

+∂2

∂x2ref

Dref

#

P (x, xref , s)

Time evolution of expectation values

Estimation of stationary results

Here: Mean and Variance:

〈xi〉 = αix0 (14)

D

(xi − 〈xi〉)2

E

=D

γ+ α2

i

γ

γ + γref

Dref

γref(15)


Spectrum

Spectrum of (hidden) x(s):

(s ≫ γ, γref und i ≡ k)

f(ω) =

D + α2i

γ3

(γ + γref )(γ − γref )

Dref

γref

ff

1

γ2 + ω2

+α2i

γ2

(γ + γref )(γ − γref )Dref

1

γ2ref + ω2

-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4-0.2

-10 -8 -6 -4 -2 0 2 4d

d(l

og[ω

])lo

g{

f(l

og[ω

])}

log[ω]David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 23

Comparisson: Measurement ↔ Model

7

8

9

10

11

12

13

0 100 200 300 400 500 600

x(t)

t [s]

-5

0

5

10

-6 -4 -2 0 2 4 6

Log[

P(∆

x)]

∆x [σ m/s]

4 6 8

10 12 14 16 18

0 100 200 300 400 500 600

x(t)

t [s]

-4-2 0 2 4 6 8

10 12 14

-4 -3 -2 -1 0 1 2 3 4 5Lo

g[P

(∆x)

]∆x [σ m/s]

2 4 6 8

10 12 14 16 18

0 100 200 300 400 500 600

x(t)

t [s]

-5

0

5

10

-6 -4 -2 0 2 4 6

Log[

P(∆

x)]

∆x [σ m/s]


Results: Loads on Wind Turbine Blade

[H. Gontier, A.P. Schaffarczyk, Fachhochschule Kiel, Germany]


Conclusion


Conclusion

CTRWs

Powerfull tool for numerical simulation of fractional processes

Simulation of continuous sample paths

⇒ Accurate reproduction of fractional dynamics at finite time

Multivariate joint statistics?

Wind field model

Modells currently applied: Poor reproduction of intermittent statistics

Stochastic approach based on CTRW

⇒ Intermittency can be controlled

Infinite waiting times unphysical: Truncation of power laws ()

Multiplicative character of atmospheric turbulence?


Conclusion

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 5 10 15 20 25 30 35

sqrt

[<(u

-<u>

)2 >] [

m/s

]

u [m/s]

Fit: 0.027 u + 0.009Analysis of FINO data


Thank you

for your attention!


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