Analysis of clusters formed by the moving average of a long-range correlated time series

A. Carbone; G. Castelli; H. E. Stanley

doi:10.1103/PhysRevE.69.026105

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Analysis of clusters formed by the moving average of a long-range correlated time series

A. Carbone¹, G. Castelli¹, and H. E. Stanley²

¹Dipartimento di Fisica and INFM, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
²Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA

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Phys. Rev. E 69, 026105 – Published 19 February, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.026105

Abstract

We analyze the stochastic function $C_{n} (i) \equiv y (i) - y_{n} (i),$ where $y (i)$ is a long-range correlated time series of length $N_{\max}$ and $y_{n} (i) \equiv (1 / n) \sum_{k = 0}^{n - 1} y (i - k)$ is the moving average with window n. We argue that $C_{n} (i)$ generates a stationary sequence of self-affine clusters $C$ with length $l,$ lifetime $τ,$ and area s. The length and the area are related to the lifetime by the relationships $l \sim τ^{ψ_{l}}$ and $s \sim τ^{ψ_{s}},$ where $ψ_{l} = 1$ and $ψ_{s} = 1 + H .$ We also find that $l,$ $τ,$ and s are power law distributed with exponents depending on H: $P (l) \sim l^{- α},$ $P (τ) \sim τ^{- β},$ and $P (s) \sim s^{- γ},$ with $α = β = 2 - H$ and $γ = 2 / (1 + H) .$ These predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned Hurst exponent H (ranging from 0.05 to 0.95) of length up to $N_{\max} {= 2}^{21}$ and n up to $2^{13} .$

References (19)

D. Ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000).
M.S. Taqqu, V. Teverosky, and W. Willinger, Fractals 3, 785 (1995).
P.Ch. Ivanov, L.A.N. Amaral, A.L. Goldberger, S. Havlin, M.G. Rosenblum, R. Zbigniew, and H.E. Stanley, Nature (London) 410, 242 (1999).
Z. Chen, P.Ch. Ivanov, K. Hu, and H.E. Stanley, Phys. Rev. E 65, 041107 (2002).
K. Hu, P.Ch. Ivanov, Z. Chen, P. Carpena, and H.E. Stanley, Phys. Rev. E 64, 011114 (2001).
Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H.E. Stanley, Phys. Rev. E 60, 1390 (1999).
The Science of Disasters: Climate Disruptions, Heart Attacks and Market Crashes, edited by A. Bunde, H. J. Schellnhuber, and J. Kropp (Springer, Berlin, 2002).
N. Vandewalle and M. Ausloos, Phys. Rev. E 58, 6832 (1998).
Y. Ashkenazy, M. Lewkowicz, J. Levitan, S. Havlin, K. Saermark, H. Moelgaard, and P.E.B. Thomsen, Fractals 7, 85 (1999).
G. Rangarajan and M. Ding, Phys. Rev. E 61, 04991 (2000).
E. Alessio, A. Carbone, G. Castelli, and V. Frappietro, Eur. Phys. J. B 27, 197 (2002).
A. Carbone and G. Castelli, Proc. SPIE 5114, 406 (2003).
The moving average filter dynamically detrends the series: as the discrete index i increases by unity, the box window n switches its position by the same amount.
C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, and A.L. Goldberger, Phys. Rev. E 49, 1685 (1994); C.K. Peng, S. Ha, H.E. Stanley, and A.L. Goldberger, Chaos 5, 82 (1995).
Equation (8) follows if the constitutive relation of the fractional Brownian motion $Δ y (i) \sim Δ i^{H}$ is taken into account to calculate the length of the $y (i)$ segments limited by $i_{c} (j)$ and $i_{c} (j + 1) .$ Equation (9) follows if the relationship $y_{n} (i) - y (i) \approx D_{H} \frac{Δ i}{Δ n^{2}} \nabla^{2} y (i)$ is taken into account, where $D_{H}$ is the generalized diffusion coefficient for the fractional brownian motion and the other quantities have the usual meaning. The term on the right side of Eq. (16) is proportional to the average displacement of the random walker during a time interval $Δ i$ and thus varies as $Δ i^{H} .$ Using Eq. (16) to calculate the sum of terms $\tilde{y} (i) - y_{n} (i)$ over the time interval $τ_{j} \equiv i_{c} (j + 1) - i_{c} (j)$ [Eq. (7)], the behavior of s as $τ^{H + 1}$ is justified. It is worth mentioning that Eq. (16) has been worked out in J.R. Weimar and J.-P. Boon, Phys. Rev. E 49, 1749 (1994).
Equation (11) can be derived from the relation $〈 τ 〉 \equiv N_{\max} {/ N}_{\times}$ between the mean time interval $〈 τ 〉$ and the total number of crossing points $N_{\times} .$ To express $〈 τ 〉$ in terms of $N_{\max},$ we write $〈 τ 〉 \equiv \int_{1}^{N_{\max}} τ P (τ) d τ \sim N_{\max}^{2 - β} .$ Similarly, we can express $N_{\times}$ in terms of $N_{\max}$ as $N_{\times} \sim N_{\max}^{1 - H},$ which follows from the fact that the fractal dimension of the set of crossing points is $1 - H$ (the codimension of the set of crossing points $d - d_{\times}$ is equal to the sum of the codimension $2 - H$ of the signal and the codimension $2 - 1$ of the moving average). Thus $2 - β = 1 - (1 - H),$ from which Eq. (11) follows. A similar reasoning gives the value of the exponent $α = 2 - H$ for the cluster length probability distribution function [Eqs. (12) and (14)].
M. Ding and W. Yang, Phys. Rev. E 52, 207 (1995).
S. Maslov, M. Paczuski, and P. Bak, Phys. Rev. Lett. 73, 2162 (1994).
P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).

[c1] D. Ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000).

[c2] M.S. Taqqu, V. Teverosky, and W. Willinger, Fractals 3, 785 (1995).

[c3] P.Ch. Ivanov, L.A.N. Amaral, A.L. Goldberger, S. Havlin, M.G. Rosenblum, R. Zbigniew, and H.E. Stanley, Nature (London) 410, 242 (1999).

[c4] Z. Chen, P.Ch. Ivanov, K. Hu, and H.E. Stanley, Phys. Rev. E 65, 041107 (2002).

[c5] K. Hu, P.Ch. Ivanov, Z. Chen, P. Carpena, and H.E. Stanley, Phys. Rev. E 64, 011114 (2001).

[c6] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H.E. Stanley, Phys. Rev. E 60, 1390 (1999).

[c7] The Science of Disasters: Climate Disruptions, Heart Attacks and Market Crashes, edited by A. Bunde, H. J. Schellnhuber, and J. Kropp (Springer, Berlin, 2002).

[c8] N. Vandewalle and M. Ausloos, Phys. Rev. E 58, 6832 (1998).

[c9] Y. Ashkenazy, M. Lewkowicz, J. Levitan, S. Havlin, K. Saermark, H. Moelgaard, and P.E.B. Thomsen, Fractals 7, 85 (1999).

[c10] G. Rangarajan and M. Ding, Phys. Rev. E 61, 04991 (2000).

[c11] E. Alessio, A. Carbone, G. Castelli, and V. Frappietro, Eur. Phys. J. B 27, 197 (2002).

[c12] A. Carbone and G. Castelli, Proc. SPIE 5114, 406 (2003).

[c13] The moving average filter dynamically detrends the series: as the discrete index i increases by unity, the box window n switches its position by the same amount.

[c14] C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, and A.L. Goldberger, Phys. Rev. E 49, 1685 (1994); C.K. Peng, S. Ha, H.E. Stanley, and A.L. Goldberger, Chaos 5, 82 (1995).

[c15] Equation (8) follows if the constitutive relation of the fractional Brownian motion $Δ y (i) \sim Δ i^{H}$ is taken into account to calculate the length of the $y (i)$ segments limited by $i_{c} (j)$ and $i_{c} (j + 1) .$ Equation (9) follows if the relationship $y_{n} (i) - y (i) \approx D_{H} \frac{Δ i}{Δ n^{2}} \nabla^{2} y (i)$ is taken into account, where $D_{H}$ is the generalized diffusion coefficient for the fractional brownian motion and the other quantities have the usual meaning. The term on the right side of Eq. (16) is proportional to the average displacement of the random walker during a time interval $Δ i$ and thus varies as $Δ i^{H} .$ Using Eq. (16) to calculate the sum of terms $\tilde{y} (i) - y_{n} (i)$ over the time interval $τ_{j} \equiv i_{c} (j + 1) - i_{c} (j)$ [Eq. (7)], the behavior of s as $τ^{H + 1}$ is justified. It is worth mentioning that Eq. (16) has been worked out in J.R. Weimar and J.-P. Boon, Phys. Rev. E 49, 1749 (1994).

[c16] Equation (11) can be derived from the relation $〈 τ 〉 \equiv N_{\max} {/ N}_{\times}$ between the mean time interval $〈 τ 〉$ and the total number of crossing points $N_{\times} .$ To express $〈 τ 〉$ in terms of $N_{\max},$ we write $〈 τ 〉 \equiv \int_{1}^{N_{\max}} τ P (τ) d τ \sim N_{\max}^{2 - β} .$ Similarly, we can express $N_{\times}$ in terms of $N_{\max}$ as $N_{\times} \sim N_{\max}^{1 - H},$ which follows from the fact that the fractal dimension of the set of crossing points is $1 - H$ (the codimension of the set of crossing points $d - d_{\times}$ is equal to the sum of the codimension $2 - H$ of the signal and the codimension $2 - 1$ of the moving average). Thus $2 - β = 1 - (1 - H),$ from which Eq. (11) follows. A similar reasoning gives the value of the exponent $α = 2 - H$ for the cluster length probability distribution function [Eqs. (12) and (14)].

[c17] M. Ding and W. Yang, Phys. Rev. E 52, 207 (1995).

[c18] S. Maslov, M. Paczuski, and P. Bak, Phys. Rev. Lett. 73, 2162 (1994).

[c19] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).

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Analysis of clusters formed by the moving average of a long-range correlated time series

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