Tempered Fractional Calculus

2014 International Conference on

Fractional Differentiation and Its Applications

University of Catania, Italy

23–25 June 2014

Mark M. Meerschaert

Department of Statistics and Probability

Michigan State University

mcubed@stt.msu.edu

http://www.stt.msu.edu/users/mcubed

Abstract

Fractional derivatives and integrals are convolutions with a powerlaw. Including an exponential term leads to tempered fractionalderivatives and integrals. Tempered fractional Brownian motion,the tempered fractional integral or derivative of a Brownian mo-tion, is a new stochastic process whose increments can exhibitsemi-long range dependence. A tempered Grunwald-Letnikovformula provides the basis for finite difference methods to solvetempered fractional diffusion equations. The tempered finitedifference operator is also useful in time series analysis, whereit provides a useful new stochastic model for turbulent velocitydata. Tempered stable processes are the limits of random walkmodels, where the power law probability of long jumps is tem-pered by an exponential factor. These random walks convergeto tempered stable stochastic process limits, whose probabilitydensities solve tempered fractional diffusion equations. Tem-pered power law waiting times lead to tempered fractional timederivatives. Applications include geophysics and finance.

Acknowledgments

Farzad Sabzikar, Statistics and Probability, Michigan State

Inmaculada Aban, Biostatistics, Univ Alabama Birmingham

Boris Baeumer, Maths & Stats, University of Otago, New Zealand

Jinghua Chen, School of Sciences, Jimei University, China

Peter Franz Kern, Math, Universitat Dusseldorf, Germany

Anna K. Panorska, Mathematics and Statistics, U of Nevada

M.S. Phanikumar, Civil & Environ. Eng., Michigan State

Parthanil Roy, Indian Statistical Institute, Kolkata

Hans-Peter Scheffler, Math, Universitat Siegen, Germany

Qin Shao, Mathematics, University of Toledo, Ohio

Alla Sikorskii, Statistics and Probability, Michigan State

Aklilu Zeleke, Lyman Briggs School, Michigan State

Yong Zhang, Department of Geological Sciences, U Alabama

Recent Books

Stochastic Models for Fractional Calculus

Mark M. Meerschaert and Alla Sikorskii

De Gruyter Studies in Mathematics 43, 2012

Advanced graduate textbook

ISBN 978-3-11-025869-1

Mathematical Modeling, 4th Edition

Mark M. Meerschaert

Academic Press, Elsevier, 2013

Advanced undergraduate / beginning graduate textbook

(new sections on particle tracking and anomalous diffusion)

ISBN 978-0-12-386912-8

Fourier transform definition

Define the Fourier transform f(k) =∫ ∞

−∞e−ikxf(x) dx.

Positive fractional derivatives and integrals of order α > 0 satisfy

Dαxf(x) ⇐⇒ (ik)αf(k) and I

αxf(x) ⇐⇒ (ik)−αf(k)

Negative fractional derivatives and integrals of order α > 0 satisfy

Dα−xf(x) ⇐⇒ (−ik)αf(k) and I

α−xf(x) ⇐⇒ (−ik)−αf(k)

Tempered fractional derivatives with α, λ > 0 satisfy

Dα,λx f(x) ⇐⇒ (λ+ ik)αf(k) and I

α,λx f(x) ⇐⇒ (λ+ ik)−αf(k)

Dα,λ−x f(x) ⇐⇒ (λ− ik)αf(k) and I

α,λ−x f(x) ⇐⇒ (λ− ik)−αf(k)

Note the Fourier symbol (λ± ik)−α no longer blows up at k = 0.

Riemann-Liouville definition

Tempered fractional integrals can also be defined by

Iα,λx f(x) =

1

Γ(α)

∫ x

−∞f(u)(x− u)α−1e−λ(x−u)du

Iα,λ−x f(x) =

1

Γ(α)

∫ ∞

xf(u)(u− x)α−1e−λ(u−x)du

for any α, λ > 0. For λ = 0, these are Riemann-Liouville frac-

tional integrals.

For 0 < α < 1 and λ > 0 we also have

Dα,λx f(x) = λαf(x) +

α

Γ(1− α)

∫ x

−∞

f(x)− f(u)

(x− u)α+1e−λ(x−u)du

Dα,λ−x f(x) = λαf(x) +

α

Γ(1− α)

∫ +∞

x

f(x)− f(u)

(u− x)α+1e−λ(u−x)du

For λ = 0, these are Marchaud fractional derivatives.

Grunwald-Letnikov definition

For λ > 0 and 0 < α < 1 we have

limh→0

h−α∆α,λh f(x) = D

α,λx f(x)

where the tempered fractional difference

∆α,λh f(x) =

∞∑

j=0

(

αj

)

(−1)je−λjhf(x− jh)− (1− e−λh)αf(x).

using the fractional binomial coefficients(

αj

)

=Γ(α+1)

j!Γ(α− j +1)

Can use to construct finite difference codes [special session].

These codes are mass-preserving since (by the Binomial formula)

∞∑

j=0

(

αj

)

(−1)je−λjh = (1− e−λh)α

Tempered fractional Brownian motion

Yaglom noise is defined by

Yα,λ(t) =∫ t

−∞(t− u)−αe−λ(t−u) W (u) du = I

1−α,λt W (t)

where W (u) is a white noise.

Tempered fractional Brownian motion (TFBM) is defined by

Bα,λ(t) = Yα,λ(t)− Yα,λ(0)

=∫ ∞

−∞

[

e−λ(t−u)+(t− u)−α+ − e−λ(0−u)+(0− u)−α

+

]

W (u) du

where (x)+ = x for x > 0 and (x)+ = 0 for x ≤ 0.

When λ = 0, Yaglom noise is undefined, and TFBM = FBM.

FBM and tempered FBM

FBM (thin line) and TFBM (thick line) with same W (u) for

H = 0.3, λ = 0.03 (left) and H = 0.7, λ = 0.01 (right).

0 100 200 300 400 500

−5

05

10

t

X(t

)

0 100 200 300 400 500

−20

010

20t

X(t

)

Sample paths are Holder continuous of order H = 1/2− α.

Scaling: Bα,λ(ct)d= cHBα,cλ(t) .

Tempered fractional time series

The ARMA(p, q) time series model is

Yt −p∑

j=1

φjYt−j = Zt +q∑

i=1

θiZt−i

where Zt ∼ IID(0, σ2).

We say that Xt follows an ARTFIMA(p, α, λ, q) model if

Yt = ∆α,λ1 Xt =

∞∑

j=0

(

αj

)

(−1)je−λjXt−j

follows an ARMA(p, q) model.

Then Xt = ∆−α,λ1 Yt, a kind of tempered fractional integration.

Spectral approach

Kolmogorov argued that the spectral density of turbulence data

follows h(k) ≈ k−5/3 in the inertial range of frequency k.

Tempered fractional Gaussian noise (TFGN) is defined by

Xj = Bα,λ(j)−Bα,λ(j − 1) = Yα,λ(j)− Yα,λ(j − 1).

Its spectral density (FT of the autocovariance function) is

h(k) = C|e−ik − 1|2

(λ2 + k2)H+1/2

≈ Ck1−2H

for k small and λ ≈ 0. Set H = 4/3 for Kolmogorov scaling.

The ARTFIMA(0, α, λ,0) model has spectral density

h(k) = C∣

∣

∣1− e−(λ+ik)∣

∣

∣

−2α≈ Ck−2α

for k small and λ ≈ 0. Set α = 5/6 for Kolmogorov scaling.

Spectral density of TFGN and ARTFIMA

log

h(k

)

log k

Log-log plot of spectral density for TFGN (solid), ARTFIMA

(dotted), and Kolmogorov scaling (dashed). Here λ = 0.03.

Lake Huron water velocity spectrum

-8

-6

-4

-2

0

2

4

6

8

10

12

14

-10 -8 -6 -4 -2 0

ln(spectrum)

ln(frequency)

Spectrum of Saginaw Bay velocity data (symbols), fitted

ARTFIMA model (thick line) with α = 5/6 and λ = 0.006, and

Kolmogorov model (thin line).

Davenport spectrum for wind gusts

Spectral density of wind gustiness from Davenport (1961). The

Davenport spectrum is the same as TFGN with H = 5/6.

Fractional diffusion and probability

The space-time fractional diffusion equation

Dβt p(x, t) = aDα

xp(x, t) + bDα−xp(x, t) + δ(x)

t−β

Γ(1− β)

governs a stochastic process Xt with power law jumps in space

P(|∆Yu| > x) ≈ Ax−α

and power law waiting times

P(∆Du > t) ≈ Bt−β

for 0 < α < 2 and 0 < β < 1.

Graph of sample paths (particle traces): (t,Xt) = (Du, Yu) .

Hence Dβt codes power law waiting times, D

α±x power law jumps.

Continuous time random walks

-

6

J1

J2J3

J4J5

u

u

u

u

u

u

Rd

0tW1 W2 W3 W4 W5

For particle jumps P(J > x) ≈ Ax−α, J1 + · · · + J[nu] ≈ n1/αYu.

For waiting times P(W > t) ≈ Bx−β, W1 + · · ·+W[nu] ≈ n1/βDu.

Fractional diffusion particle tracking

Here α = 2.0,1.7 (left/right) and β = 1.0,0.9 (top/bottom).

0 2000 4000 6000 8000 10000

−50

050

100

200

t

At

0 2000 4000 6000 8000 10000

−60

0−

400

−20

00

t

At

0 500000 1000000 1500000

050

100

150

200

250

t

AE

t

0 500000 1000000 1500000

−30

0−

100

010

020

0

t

AE

t

Tempered fractional diffusion

The tempered space-fractional diffusion equation

Dtp(x, t) = a∂α,λx p(x, t) + b∂α,λ−x p(x, t)

governs a stochastic process Xt with tempered power law jumps

P(|∆Xu| > x) ≈ Ax−αe−λx

for 0 < α < 2 and λ > 0.

This normalized tempered fractional derivative is defined by

∂α,λ±x f(x) ⇐⇒ [(λ± ik)α − λα]f(k) for 0 < α < 1, and

∂α,λ±x f(x) ⇐⇒ [(λ± ik)α − λα ∓ ikαλα−1]f(k) for 1 < α < 2.

The λα term makes p(x, t) a PDF.

The ikαλα−1 term makes p(x, t) have zero mean.

Tempered fractional diffusion particle tracking

Tempered stable process with α = 1.2 transitions from

Brownian motion to stable Levy motion as λ decreases.

0 5000 10000−150

−100

−50

0

λ = 0.1

0 5000 10000−200

0

200λ = 0.01

0 5000 10000−500

0

500λ = 0.001

0 5000 10000

−500

0

500

1000λ = 0.0001

Tempered power laws in finance

AMZN stock price changes fit a tempered power law model

P (X > x) ≈ x−0.6e−0.3x for x large

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooo ooooooo oooooooooooooooooo o o o ooo o ooo

ooo

o

o

o

o

1.5 2.0 2.5 3.0

-8-7

-6-5

-4-3

-2

1.5 2.0 2.5 3.0

-8-7

-6-5

-4-3

-2

ln(x)

ln(P

(X>

x))

1.5 2.0 2.5 3.0

-8-7

-6-5

-4-3

-2

Tempered power laws in hydrology

Tempered power law model P (X > x) ≈ x−0.6e−5.2x for incre-

ments in hydraulic conductivity at the MADE site.

−3.0 −2.5 −2.0 −1.5 −1.0 −0.5

−8

−7

−6

−5

−4

−3

ln(x)

ln(P

(X>

x))

Tempered power laws in atmospheric science

Tempered power law model P (X > x) ≈ x−0.2e−0.01x for daily

precipitation data at Tombstone AZ.

4.0 4.5 5.0 5.5 6.0 6.5

−8

−6

−4

−2

ln(x)

ln(P

(X>

x))

Tempered stable pdf in macroeconomics

data

Den

sity

−0.2 0.0 0.2 0.4

02

46

810

12

One-step BARMA forecast errors for annual inflation rates fit a

symmetric tempered stable with α = 1.1 and λ = 12.

Tempered time-fractional diffusion model

Fitted concentration data from a 3-D supercomputer simulation.

ADE fit uses α = 2, β = 1. Without cutoff uses λ = 0.

Summary

• Tempered fractional derivatives

• Tempered fractional Brownian motion

• Applications to turbulence

• Tempered fractional diffusion

• Numerical methods [special session]

• Applications to finance and geophysics

References1. I.B. Aban, M.M. Meerschaert, and A.K. Panorska (2006) Parameter Estimation for the

Truncated Pareto Distribution. Journal of the American Statistical Association: Theory

and Methods. 101(473), 270–277.

2. B. Baeumer and M.M. Meerschaert (2010) Tempered stable Levy motion and transientsuper-diffusion. Journal of Computational and Applied Mathematics 233, 2438–2448.

3. A. Cartea and D. del Castillo-Negrete (2007) Fluid limit of the continuous-time randomwalk with general Levy jump distribution functions. Phys. Rev. E 76, 041105.

4. A. Chakrabarty and M. M. Meerschaert (2011) Tempered stable laws as random walklimits. Statistics and Probability Letters 81(8), 989–997.

5. A. V. Chechkin, V. Yu. Gonchar, J. Klafter and R. Metzler (2005) Natural cutoff inLevy flights caused by dissipative nonlinearity. Phys. Rev. E 72, 010101.

6. S. Cohen and J. Rosinski (2007) Gaussian approximation of multivariate Levy processeswith applications to simulation of tempered stable processes, Bernoulli 13, 195-210.

7. A. G. Davenport (1961) The spectrum of horizontal gustiness near the ground in highwinds. Quarterly Journal of the Royal Meteorological Society 87, 194–211.

8. I. Koponen (1995) Analytic approach to the problem of convergence of truncated Levyflights towards the Gaussian stochastic process, Phys. Rev. E 52, 1197–1199.

9. M.M. Meerschaert and H.P. Scheffler (2001) Limit Theorems for Sums of Independent

Random Vectors: Heavy Tails in Theory and Practice. Wiley Interscience, New York.

10. M.M. Meerschaert and H.P. Scheffler (2004) Limit theorems for continuous-time ran-dom walks with infinite mean waiting times. J. Appl. Probab. 41(3), 623–638.

11. Meerschaert, M.M., Scheffler, H.P. (2008) Triangular array limits for continuous timerandom walks. Stoch. Proc. Appl. 118(9), 1606-1633.

12. Meerschaert, M.M., Y. Zhang and B. Baeumer (2008) Tempered anomalous diffusionin heterogeneous systems. Geophys. Res. Lett. 35, L17403.

13. M.M. Meerschaert and A. Sikorskii (2012) Stochastic Models For Fractional Calculus.De Gruyter, Berlin/Boston.

14. Meerschaert, M.M., P. Roy and Q. Shao (2012) Parameter estimation for temperedpower law distributions. Communications in Statistics Theory and Methods 41(10),1839–1856.

15. M.M. Meerschaert (2013) Mathematical Modeling. 4th Edition, Academic Press, Boston.

16. M.M. Meerschaert and F. Sabzikar (2013) Tempered fractional Brownian motion.Statist. Prob. Lett. 83 (2013), 2269–2275.

17. M.M. Meerschaert and F. Sabzikar (2014) Stochastic integration for tempered fractionalBrownian motion. Stoch. Proc. Appl. 124, 2363–2387.

18. J. Rosinski (2007), Tempering stable processes. Stoch. Proc. Appl. 117, 677–707.

19. F. Sabzikar, M.M. Meerschaert, and Jinghua Chen (2014) Tempered Fractional Calcu-lus. J. Comput. Phys., to appear in the Special Issue on Fractional Partial DifferentialEquations, preprint at www.stt.msu.edu/users/mcubed/TFC.pdf

Simulating tempered stable laws (JCAP 2010)

Simulation codes for stable random variates are widely available.

If X > 0 has stable density density f(x), TS density is

fλ(x) =e−λxf(x)

∫ ∞

0e−λuf(u) du

Take Y ∼ exp(λ) independent of X, (Xi, Yi) IID with (X,Y ).

Let N = min{n : Xn ≤ Yn}. Then XN ∼ fλ(x).

Proof: Compute P (XN ≤ x) = P (X ≤ x|X ≤ Y ) by conditioning,

then take d/dx to verify.

Triangular array scheme (SPL 2011)

Take P (J > x) ≈ Cx−α with 1 < α < 2. Triangular array limit

[nt]∑

k=1

n−1/αJk − b(n)t ⇒ Yt

is stable. Define tempering variables:

P (Z > u) = uα∫ ∞

ur−α−1e−λrdr

Replace n−1/αXk by Zk if n−1/αXk > Zk.

Triangular array limit is tempered stable.

Exponential tempering: sum of α and α− 1 tempered stables.

Tail estimation (CIS 2012)

Hill-type estimator: Assume P (X > x) ≈ Cx−αe−λx for x large,

use order statistics X(1) ≤ X(2) ≤ · · · ≤ X(n).

Conditional MLE given X(n−k+1) > L ≥ X(n−k):

T1 : =k∑

i=1

(logX(n−i+1) − logL)

T2 : =k∑

i=1

(X(n−i+1) − L)

1 =k∑

i=1

x(n−i+1)

kx(n−i+1) + α(T2 − T1x(n−i+1))

λ = (k − αT1)/T2

C =k

nLαeλL

R code available at www.stt.msu.edu/users/mcubed/TempParetoR.zip

Testing for pure power law tail (JASA 06)

Null hypthesis H0 : P (X > x) = Cx−α Pareto for x > L.

Test based on extreme value theory rejects H0 if

X(1) <

(

nC

− ln q

)1/α

where α,C can be estimated using Hill’s estimator

αH =

k−1k∑

i=1

{lnX(n−i+1) − lnX(n−k)}

−1

CH = (k/n)(X(n−k))αH

Simple p-value formula p = exp{−nC X−α(n)

}.

Tempered fractional calculus operators (SPA 2014)

Iα,λ±x f(x) ⇐⇒ (λ±ik)−αf(k) for f ∈ L2(R) = {f :

∫

|f(x)|2dx < ∞}.

Define the fractional Sobolev space

Wα,2(R) := {f ∈ L2(R) :∫

(λ2 + k2)α|f(k)|2 dk < ∞}.

Then Dα,λ±x f(x) ⇐⇒ (λ± ik)αf(k) for f ∈ Wα,2(R).

Semigroup property: Iα,λ± I

β,λ± f = I

α+β,λ± f for f ∈ L2(R), and

Dα,λ± D

β,λ± f = D

α+β,λ± f for f ∈ Wα+β,2(R).

Adjoint property: 〈f, Iα,λ+ g〉

2= 〈I

α,λ− f, g〉

2for f, g ∈ L2(R), and

〈f,Dα,λ+ g〉

2= 〈D

α,λ− f, g〉

2for f, g ∈ Wα,2(R), using the L2 inner

product 〈f, g〉2 =∫

f(x)g(x) dx