Tempered Fractional Calculus

Transcription

1 Tempered Fractional Calculus International Symposium on Fractional PDEs: Theory, Numerics and Applications 3 5 June 2013 Mark M. Meerschaert Department of Statistics and Probability Michigan State University mcubed@stt.msu.edu Partially supported by NSF grant DMS and NIH grant R01-EB

2 Abstract Fractional Calculus has a close relation with Probability. Random walks with heavy tails converge to stable stochastic processes, whose probability densities solve space-fractional diffusion equations. Continuous time random walks, with heavy tailed waiting times between particle jumps, converge to non-markovian stochastic limits, whose probability densities solve time-fractional diffusion equations. Time-fractional derivatives and integrals of Brownian motion produce fractional Brownian motion, a useful model in many applications. Fractional derivatives and integrals are convolutions with a power law. Including an exponential term leads to tempered fractional derivatives and integrals. Tempered stable processes are the limits of random walk models where the power law probability of long jumps is tempered by an exponential factor. These random walks converge to tempered stable stochastic process limits, whose probability densities solve tempered fractional diffusion equations. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. Applying this idea to Brownian motion leads to tempered fractional Brownian motion, a new stochastic process that can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data.

3 Acknowledgments Farzad Sabzikar, Statistics and Probability, Michigan State Boris Baeumer, Maths & Stats, University of Otago, New Zealand Anna K. Panorska, Mathematics and Statistics, U of Nevada Parthanil Roy, Indian Statistical Institute, Kolkata Hans-Peter Scheffler, Math, Universität Siegen, Germany Qin Shao, Mathematics, University of Toledo, Ohio Alla Sikorskii, Statistics and Probability, Michigan State Yong Zhang, Desert Research Institute, Las Vegas Nevada

4 New Books Stochastic Models for Fractional Calculus Mark M. Meerschaert and Alla Sikorskii De Gruyter Studies in Mathematics 43, 2012 Advanced graduate textbook ISBN 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

5 Continuous time random walks R d X 2 X 3 0 X 1 W 1 W 2 W 3 W 4 W 5 X 4 X 5 t A random particle arrives at location S(n) = X X n at time T n = W 1 + +W n. After N t = max{n 0 : T n t} jumps, particle location is S(N t ).

6 CTRW limit theory If P(X n > x) x α and E(X n ) = 0 then n 1/α S(nu) A(u). The α-stable limit A(u) has pdf p(x,u) with Fourier transform ˆp(k,u) = e uψ A(k) and Fourier symbol ψ A (k) = (ik) α for 1 < α 2. If P(W n > t) t β then n 1/β T nu D u. The β-stable limit has pdf g(t,u) with Laplace transform g(s,u) = e uψ D(s) and Laplace symbol ψ D (s) = s β for 0 < β < 1. Inverse process: n β N nt E t = inf{u > 0 : D u > t}, and then n β/α S(N nt ) = n β/α S(n β n β N nt ) (n β ) 1/α S(n β E t ) A(E t ). Since E t has a pdf h(u,t) with LT h(u,s) = s 1 ψ D (s)e uψd(s), the CTRW limit A(E t ) has pdf q(x,t) = 0 p(x,u)h(u,t)du u P(A(u) = x E t = u)p(e t = u).

7 Probability and fractional calculus CTRW limit pdf has Fourier-Laplace transform q(k,s) = = 0 ˆp(k,u) h(u,s)du 0 eu(ik)α s β 1 e usβ du = sβ 1 s β (ik) α Then s β q(k,s) s β 1 = (ik) α q(k,s) which inverts to β t q(x,t) = Dα x q(x,t), using Caputo on the LHS and Riemann-Liouville on the RHS. Note β t codes power law waiting times, D α x power law jumps. Reduces to traditional diffusion for α = 2 and β = 1.

8 Anomalous diffusion processes Here α = 2.0,1.7 (left/right) and β = 1.0,0.9 (top/bottom). At At t t AEt AEt t t

9 Tempered fractional calculus Tempered stable D t has pdf with LT g(t,u) = e uψ D(s) where the Laplace symbol ψ D (s) = (λ+s) β λ β for some λ > 0 (small). Tempered stable A(u) has pdf with FT ˆp(k,u) = e uψ A(k) where the Fourier symbol ψ A (k) = (λ+ik) α λ α ikαλ α 1 with λ > 0. Again E t has pdf with LT h(u,s) = s 1 ψ D (s)e uψ D(s), and again A(E t ) has pdf q(x,t) = 0 p(x,u)h(u,t)du with FLT q(k,s) = s 1 ψ D (s) ψ D (s) ψ A (k) Then ψ D (s) q(k,s) s 1 ψ D (s) = ψ A (k) q(k,s). Invert to get β,λ t q(x,t) = D α,λ x q(x,t). Tempered jumps P(X n > x) x α e λx and waiting times.

10 Space fractional tempered stable Tempered stable Lévy motion with α = λ = λ = λ = λ =

11 Tempered power laws in finance AMZN stock price changes fit a tempered power law model P(X > x) x 0.6 e 0.3x for x large ln(p(x> x)) oo o o o ooooo ooooooooooo oooo oooooo oooo ooooo oooo o o o ooo o o o o o o o o o o ln(x)

12 Tempered power laws in hydrology Tempered power law model P(X > x) x 0.6 e 5.2x for increments in hydraulic conductivity at the MADE site. ln(p(x>x)) ln(x)

13 Tempered power laws in atmospheric science Tempered power law model P(X > x) x 0.2 e 0.01x for daily precipitation data at Tombstone AZ. ln(p(x>x)) ln(x)

14 Tempered stable pdf in macroeconomics Density data One-step BARMA forecast errors for annual inflation rates, and a tempered stable with α = 1.1 and λ = 12 (rough fit).

15 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.

16 Numerical codes for fractional diffusion For α > 0 we have D α x f(x) = lim h 0h α α f(x) where ( ) ( ) α α f(x) = ( 1) j α f(x jh), = Γ(α+1) j j j!γ(α j +1) j=0 Use to construct explicit/implicit finite difference codes. These codes are mass-preserving since j=0 ( ) α ( 1) j = 0. j Change jh to (j α )h for unconditional stability (implicit Euler). Use operator splitting for 2-d, 3-d, or reaction term. Finite element and finite volume methods also available.

17 Numerical example Exact solution p(x,t) = e t x 3 and numerical approximation to p(x, t) = d(x) 1.8 p(x,t) t x 1.8 +r(x,t) on 0 < x < 1 with p(x,0) = x 3, p(0,t) = 0, p(1,t) = e t, r(x,t) = (1+x)e t x 3, d(x) = Γ(2.2)x 2.8 / Exact CN

18 Tempered fractional diffusion codes For 0 < α < 1, lim h 0 h α α,λ f(x) = D α,λ x f(x) where ( ) α,λ α f(x) = ( 1) j e λjh f(x jh) (1 e λh ) α f(x). j j=0 Codes are mass-preserving since (by the Binomial formula) ( ) α ( 1) j e λjh = (1 e λh ) α j j=0 Stable, consistent implicit Euler codes: shift jh (j α )h. Crank-Nicolson codes are O( t 2 + x). Apply Richardson extrapolation to get O( t 2 + x 2 ). For 1 < α < 2, h α α,λ f(x) D α,λ x f(x)+αλ α 1 f (x).

19 Numerical example Exact solution u(x,t) = x β e λx t /Γ(1+β) and Crank-Nicolson solution (extrapolated) to t p(x,t) = c(x)d α,λ x p(x,t)+r(x,t) on 0 < x < 1 with α = 1,6, λ = 2, β = 2.8, p(x,0) = x β e λx /Γ(β +1), p(0,t) = 0, p(1,t) = e λ t /Γ(β +1), c(x) = x α Γ(1+β α)/γ(β +1), and r(x,t) = r 1 (x,t)e λx t Γ(1+β α)/γ(β +1) where r 1 (x,t) = (1 α)λα x α+β Γ(β +1) + αβλα 1 x α+β 1 Γ(β) 2x β Γ(1+β α). t x CN Max Error Rate CNX Max Error Rate 1/10 1/ /20 1/ /40 1/ /80 1/

20 Fractional derivatives: Integral forms In the simplest case 0 < α < 1 the generator form is D α x f(x) = α ( ) f(x) f(x y) y α 1 dy. Γ(1 α) 0 Check: Since f(x y) has FT e ikyˆf(k), then the RHS has FT α Γ(1 α) 0 ( 1 e iky ) ˆf(k)y α 1 dy. For λ > 0 it is not hard to compute [Prop in MS 2012] α Γ(1 α) 0 ( 1 e (λ+ik)y ) y α 1 dy = (λ+ik) α and then (let λ 0) D α x f(x) has FT (ik)αˆf(k). For 1 < α < 2 D α α(α 1) xf(x) = Γ(2 α) and again the RHS has FT (ik) αˆf(k). 0 ( f(x y) f(x)+yf (x) ) y α 1 dy

21 Tempered fractional derivatives In the simplest case 0 < α < 1 the generator form is D α,λ x f(x) = α ( ) f(x) f(x y) e λy y α 1 dy. Γ(1 α) 0 Check: The integral on the RHS has Fourier transform α ( 1 e iky ) ˆf(k)e λy y α 1 dy Γ(1 α) 0 α ( = (1 e (λ+ik)y ) (1 e λy ) ) ˆf(k)y α 1 dy Γ(1 α) 0 which reduces to [(λ+ik) α λ α ]ˆf(k). For 1 < α < 2 we have D α,λ α(α 1) x f(x) = Γ(2 α) 0 ( f(x y) f(x)+yf (x) ) e λy y α 1 dy and the RHS has FT [(λ+ik) α λ α ikαλ α 1 ]ˆf(k).

22 Laplace transform approach Since e λt f(t) has Laplace transform f(s λ) and since D α t LT s α f(s), we see that [ e λt f(t) ] dt = s α f(s λ). 0 e st D α t Using the shift property one more time, we see that 0 e st e λt D α [ t e λt f(t) ] dt = (s+λ) α f(s). Then for 0 < α 1 we have f(t) has D α,λ t f(t) = e λt D α t and for 1 < α 2 we have [ e λt f(t) ] λ α f(t) D α,λ x f(x) = e λx D α x[ e λx f(x) ] λ α f(x) αλ α 1 f (x).

23 Fractional derivatives (other integral forms) Recall that in the simplest case 0 < α < 1 the generator form D α x f(x) = 1 Γ(1 α) 0 ( f(x) f(x y) ) αy α 1 dy. Integrate by parts udv = uv vdu with u = f(x) f(x y) and dv = αy α 1 dy to get the Caputo form 1 Γ(1 α) 0 f (x y)y α dy = 1 Γ(1 α) f (u)(x u) α + du where (x u) + = (x u) for x > u and (x u) + = 0 otherwise. Move the derivative outside to get the Riemann-Liouville form 1 d Γ(1 α) dx Caputo is I 1 α x 0 f(x y)y α dy = 1 d Γ(1 α) dx f(u)(x u) α + dy. D 1 x f(x) and Riemann-Liouville is D1 x I1 α f(x) where I α x f(x) = 1 Γ(α) f(u)(x u)α 1 + du. x

24 Fractional Brownian motion Given Z n iid with mean zero and finite variance, the time series ( ) X t = α α Z n = ( 1) j Z j t j j=0 is long range dependent: E(X t X t+j ) j 2H 2 with H = 1/2 α. Then n H (X 1 + +X [nt] ) B H (t) fractional Brownian motion. Here B H (t) = α t B(t) α t B(0) where B(t) is a Brownian motion: B H (t) = 1 Γ(1 α) + [ ] (t u) α + (0 u) α + B(du) with B(du) = B (u)du in the distributional sense (Caputo). Hence FBM is the fractional integral of the white noise B(du).

25 Tempered fractional Brownian motion Tempered fractional Brownian motion with 1/2 < α < 1/2: B α,λ (t) = + [ e λ(t u) + (t u) α + e λ(0 u) +(0 u) α ] + B(du). TFBM is the tempered fractional integral of white noise B(du): I α,λ t f(t) = 1 Γ(α) f(u)e λ(t u) +(t u) α 1 + du. Tempered fractional Gaussian noise: X t = B α,λ (t) B α,λ (t 1). Semi-long range dependence: E(X t X t+j ) j 2H 2 for j small to moderate, then E(X t X t+j ) j 2 as j, where H = 1/2 α. For any Z n iid with mean zero and finite variance, the time series X t = α,λ Z n also exhibits semi-long range dependence.

26 FBM and tempered FBM FBM (thin line) and TFBM (thick line) with same B(du) for H = 0.3,λ = 0.03 (left) and H = 0.7,λ = 0.01 (right). X(t) X(t) t t Sample paths are Hölder continuous of order H. Scaling: B H (ct) d = c H B H (t) and B α,λ (ct) d = c H B α,cλ (t).

27 Semi-long range dependence The covariance functions E(X t X t+j ) for FGN and TFGN with H = 0.7 (and λ = 0.001) are quite similar until j gets large lag

28 Spectral density FGN spectral density blows up at low frequencies for H > 1/2: f H (ω) = 1 2π j= e iωj E(X t X t+j ) ω 1 2H as ω 0. For TFGN with λ 0 we get a similar result f α,λ (ω) ω 2 [λ 2 +ω 2 ] H+1/2 as ω 0. Hence f α,λ (ω) ω 1 2H for moderate frequencies, but remains bounded for very low frequencies (Davenport spectrum).

29 Spectral density comparison Spectral density for FGN and TFGN with H = 0.7,λ = frequency

30 Davenport spectrum Kolmogorov invented FBM to model turbulence in the inertial subrange. The Davenport spectrum f(ω) ω 2 /[1+ω 2 ] H+1/2 extends the model to include production and dissipation. Since TFBM has the same spectral density, it provides a comprehensive stochastic model for turbulence. Figure reproduced from Beaupuits et al. (2004).

31 Davenport spectrum for wind gusts Spectral density of wind gustiness from Davenport (1961). TFBM provides a stochastic model for the Davenport spectrum.

32 Wind power study Spectral density of wind speed at the Chajnantor radio telescope site in Chile.

33 Summary Tempered power laws Tempered fractional derivatives Numerical methods Tempered fractional Brownian motion Davenport model for wind speed

34 References 1. 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), P. Abry, P. Goncalves and P. Flandrin (1995) Wavelets, spectrum analysis and 1/f processes. In Wavelets and statistics, Springer New York. 3. B. Baeumer and M.M. Meerschaert (2010) Tempered stable Levy motion and transient super-diffusion. Journal of Computational and Applied Mathematics 233, J. P. Pérez Beaupuits, A. Otárola, F. T. Rantakyrö, R. C. Rivera, S. J. E. Radford, and L.-Å. Nyman (2004) Analysis of wind data gathered at Chajnantor. ALMA Memo Á. Cartea and D. del Castillo-Negrete (2007) Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76, A. Chakrabarty and M. M. Meerschaert (2011) Tempered stable laws as random walk limits. Statistics and Probability Letters 81(8), A. V. Chechkin, V. Yu. Gonchar, J. Klafter and R. Metzler (2005) Natural cutoff in Lévy flights caused by dissipative nonlinearity. Phys. Rev. E 72, S. Cohen and J. Rosiński (2007) Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered stable processes, Bernoulli 13, A. G. Davenport (1961) The spectrum of horizontal gustiness near the ground in high winds. Quarterly Journal of the Royal Meteorological Society 87, P. Flandrin (1989) On the spectrum of fractional Brownian motions. IEEE Trans. on Info. Theory IT-35,

35 11. R. N. Mantegna and H. E. Stanley (1994) Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. Phys. Rev. Lett. 73, 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. 13. M.M. Meerschaert and H.P. Scheffler (2004) Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41(3), Meerschaert, M.M., Scheffler, H.P. (2008) Triangular array limits for continuous time random walks. Stoch. Proc. Appl. 118(9), Meerschaert, M.M., Y. Zhang and B. Baeumer (2008) Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, L M.M. Meerschaert and A. Sikorskii (2012) Stochastic Models For Fractional Calculus. De Gruyter, Berlin/Boston. 17. Meerschaert, M.M., P. Roy and Q. Shao (2012) Parameter estimation for tempered power law distributions. Communications in Statistics Theory and Methods 41(10), M.M. Meerschaert (2013) Mathematical Modeling. 4th Edition, Academic Press, Boston. 19. M.M. Meerschaert and F. Sabzikar (2013) Tempered fractional Brownian motion. Preprint at Rosiński, J. (2007), Tempering stable processes. Stoch. Proc. Appl. 117, C. Tadjeran, M.M. Meerschaert, H.P. Scheffler (2006) A second order accurate numerical approximation for the fractional diffusion equation. Journal of Computational Physics 213(1),

36 Simulating tempered stable laws Simulation codes for stable random variates are widely available. If X > 0 has stable density density f(x), TS density is e λx f(x) f λ (x) = 0 e λu f(u)du Take Y exp(λ) independent of X, (X i,y i ) IID with (X,Y). Let N = min{n : X n Y n }. Then X N f λ (x). Proof: Compute P(X N x) = P(X x X Y) by conditioning, then take d/dx to verify.

37 Triangular array scheme (SPL 2011) Take P(X > x) Cx α with 1 < α < 2. Triangular array limit [nt] k=1 n 1/α X k b (n) t is stable. Define tempering variables: P(Z > u) = u α A(t) u r α 1 e λr dr Replace n 1/α X k by Z k if n 1/α X k > Z k. Triangular array limit is tempered stable. Exponential tempering: sum of α and α 1 tempered stables.

38 Tail estimation (CIS 2012) Hill-type estimator: Assume P(X > x) Cx α e λx for x large, use order statistics X (1) X (2) X (n). Conditional MLE given X (n k+1) > L X (n k) : T 1 : = T 2 : = 1 = k i=1 k i=1 k i=1 (logx (n i+1) logl) (X (n i+1) L) x (n i+1) kx (n i+1) + ˆα(T 2 T 1 x (n i+1) ) ˆλ = (k ˆαT 1 )/T 2 Ĉ = k n Lˆα eˆλl R code available at

39 Testing for pure power law tail (JASA 06) Null hypthesis H 0 : P(X > x) = Cx α Pareto for x > L. Test based on extreme value theory rejects H 0 if X (1) < ( nc lnq ) 1/α where α, C can be estimated using Hill s estimator ˆα H = k 1 k {lnx (n i+1) lnx (n k) } i=1 Ĉ H = (k/n)(x (n k) )ˆα H 1 Simple p-value formula p = exp{ ncx α (n) }.