Log-amplitude statistics of non-Gaussian fluctuations

1. THIC&APFA7 1. Motivation 　□ Characterization of intermittent fluctuations 　□ Coarse-grained time series and its probability density function 　□ Castaing's turbulence statistics and Beck and Cohen's superstatistics 2. Log-amplitude statistics 　□ Superstatistics and log-amplitude fluctuation 　□ Logarithmic absolute moments and log-amplitude moments 　□ Log-amplitude moments of the superstatistics 　□ The extreme value statistics in the superstatistics 3. Non-Gaussian properties of Nikkei 225 futures 　□ Power-law tails and Gumbel-distributed log-amplitude 　□ Slow convergence to a Gaussian 4. Summary Log-amplitude statistics of non-Gaussian fluctuations Ken Kiyono College of Engineering, Nihon University, JAPAN

2. [Y. Gagne, Thèse de Docteur-Ingénieur, Univ. Grenoble (1980)] Motivation ■ Characterization of intermittent fluctuations Variance heterogeneity has been observed in a wide range of complex systems. Intermittency problem in hydrodynamic turbulence, Volatility clustering in financial time series, Intermittent heart rate fluctuations, Intermittent chaos near the bifurcation point, etc. Non-multifractal intemittency Breakdown of the Kolmogorov similarity hypothesis in developed turbulence [B. Chabaud et al. Transition toward developed turbulence, Phys Rev Lett, 73 (1994) 3227]

3. Coarse-grained time series and its probability density function Partial sum process {DsZi} Fine resolution where Deformation of PDFs across scales Convergence to a Gaussian Coarse resolution

4. Characterization of non-Gaussian distributions ■ Model selection and parameter estimation Castaing’s model [Castaing, Gagne & Hopfinger, Physica D, 46, 177 (1990)] This model was proposed in the framework of turbulent cascade picture. Superstatistics[Beck & Cohen, Physica A, 322, 267 (2003)] Superstatistics considers an inhomogeneous driven nonequilibrium system that consists of many subsystems with different values of some intensive parameter b (the inverse effective temperature). Heavy tailed distributions ◇ Cauchy distribution, P(x) ~ |x|-2 for large |x| ◇ Student’s t-distribution, P(x) ~ |x|-(n+1) (n > 0) for large |x| ◇ Pearson type VII distribution (generalized Cauchy distribution) ◇ symmetric Levy stable distribution, P(x) ~ |x|-(a+1) (a> 0) for large |x| ◇ bilateral exponential distribution , P(x) ∝ exp(-b|x|a) PL(x): PDF at integral scale L G(s): fluctuations through energy cascade PL(x): local equilibrium distribution f(b): fluctuations of intensive parameter

5. Characterization of non-Gaussian distributions ■ Model selection and parameter estimation Castaing’s model [Castaing, Gagne & Hopfinger, Physica D, 46, 177 (1990)] This model was proposed in the framework of turbulent cascade picture. Superstatistics[Beck & Cohen, Physica A, 322, 267 (2003)] Superstatistics considers an inhomogeneous driven nonequilibrium system that consists of many subsystems with different values of some intensive parameter b (the inverse effective temperature). Heavy tailed distributions ◇ Cauchy distribution, P(x) ~ |x|-2 for large |x| ◇ Student’s t-distribution, P(x) ~ |x|-(n+1) (n > 0) for large |x| ◇ Pearson type VII distribution (generalized Cauchy distribution) ◇ symmetric Levy stable distribution, P(x) ~ |x|-(a+1) (a> 0) for large |x| ◇ bilateral exponential distribution , P(x) ∝ exp(-b|x|a) Is it possible to determine the form of G from observed time series? PL(x): PDF at integral scale L G(s): fluctuations through energy cascade Is it possible to determine the form of f from observed time series? PL(x): local equilibrium distribution f(b): fluctuations of intensive parameter How to measure the deviation from a Gaussian distribution?

6. Characterization of non-Gaussian distributions ■ Model selection and parameter estimation Castaing’s model [Castaing, Gagne & Hopfinger, Physica D, 46, 177 (1990)] This model was proposed in the framework of turbulent cascade picture. Superstatistics[Beck & Cohen, Physica A, 322, 267 (2003)] Superstatistics considers an inhomogeneous driven nonequilibrium system that consists of many subsystems with different values of some intensive parameter b (the inverse effective temperature). Heavy tailed distributions ◇ Cauchy distribution, P(x) ~ |x|-2 for large |x| ◇ Student’s t-distribution, P(x) ~ |x|-(n+1) (n > 0) for large |x| ◇ Pearson type VII distribution (generalized Cauchy distribution) ◇ symmetric Levy stable distribution, P(x) ~ |x|-(a+1) (a> 0) for large |x| ◇ bilateral exponential distribution , P(x) ∝ exp(-|x|) Multiplicative stochastic process{Xt} X = WexpY where X ~ P(x), W ~ PL(w) and Y ~ G(y) X = W/Z1/2 where X ~ P(x), W ~ PL(w) and Z ~ f(z) E(ln|X|)n(n = 1, 2, 3, ・・・) All order moments of ln|X| are finite

7. Superstatistics and log-amplitude fluctuation ■ Universality classes of superstatistics Superstatistical fluctuations can be described by log-amplitude fluctuations {Yt } P(x): Student’s t-distribution When n = 2, extreme value statistics appears! P(x) ∝ exp(-b|x|), when n = 2 The extreme value statistics is linked to P(x) ~ |x|-3orP(x) ~ exp(-b|x|).

8. Logarithmic absolute moments of heavy tailed distributions ■ Logarithmic absolute moments E{ln|X|- E(ln|X|)}n(n = 1, 2, 3, ・・・) The logarithmic absolute moments of f(x) are finite, if there exist C, M, andx0 > 0 such that f(x) < C|x|-1for |x| > x0and f(x) ≦M for |x| < x0. Example. Student’s t-distribution var(X) = n/(n-2) for n > 2, otherwise NOT defined. for n > 0

9. Log-amplitude moments of superstatistics ■ Definition of log-amplitude variance and higher moments m2= 0 ⇔ “Xis a Gaussian” (Cumulants of Y are also useful.)

10. Log-amplitude moments of superstatistics ■ m3 vs m2 Log-amplitude moments of the superstatistics can be calculated analytically.

11. Log-amplitude moments of superstatistics ■ m3 vs m2 Log-amplitude statistics is applicable to a wide variety of non-Gaussian distributions.

12. Examples of multiplicative stochastic processes log-amplitude statistics Time series {Xi} Multiplicative log-normal model: Multiplicative log-Poisson model: Superstatistical (c2) model:

13. Estimation of log-amplitude moments ■ Numerical test of the estimation Theoretical values are estimated well from the observed time series. log-normal log-Poisson superstatistics

14. Non-Gaussian properties of Nikkei 225 futures ■ Power-law tails and Gumbel-distributed log-amplitude PDFs of log-returns of Nikkei 225 future (29 June 2006~1 August 2008). Y = max{V1, V2, ・・・, Vn} obeys the Gumbel distribution, where V are independent random variables having a Gaussian, Log-normal, Gamma, logistic, Weibulle, or negative Frechet distribution. WhenY ~ Gumbel distribution, the PDF of X is t-distribution with 2 degrees of freedom (dashed lines). Log-volatility depends on the instantaneous maximum value of something in the stock market?

15. Non-Gaussian properties of Nikkei 225 futures ■ Slow convergence to a Gaussian Probability density functions of log-returns of Nikkei 225 future (29 June 2006~1 August 2008). multiplicative log-normal models (dashed lines) ← multiplicative log-normal model

16. Summary ■ Log-amplitude statistics as a measure of non-Gaussian fluctuations Log-amplitude moments of superstatistics are obtained. ■ Gumbel-distributed log-amplitude fluctuations The Gumbel (maximum) distribution is linked to the t-distribution with 2 degrees of freedom, The Gumbel (minimum) distribution is linked to the bilateral exponential distribution, ■ Non-Gaussian properties of Nikkei 225 futures PDFs of log-returns at short-time scales (< 10 min) is described well by the t-distribution with 2 degrees of freedom, which suggests that the log-volatility fluctuations obey the Gumbel distribution.