Exit-times and ϵ-entropy for dynamical systems, stochastic processes, and turbulence

Abel, Markus; Biferale, Luca; Cencini, Massimo; Falcioni, M.; Vergni, Davide; Vulpiani, Angelo

doi:10.1016/s0167-2789(00)00147-0

Cited by 26 publications

(28 citation statements)

References 43 publications

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“…Therefore, to extract interesting information on the statistics of smooth signals, new statistical tools are needed. Recent contributions have shown that laminar events are optimally characterized in terms of their exit-distance statistics, also known as inverse statistics [12][13][14][15][16]. In a nutshell, in such approach one "inverts" the usual way of looking at signals.…”

Section: Introductionmentioning

confidence: 99%

Inverse velocity statistics in two-dimensional turbulence

et al. 2003

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We present a numerical study of two-dimensional turbulent flows in the enstrophy cascade regime, with different large-scale forcings and energy sinks. In particular, we study the statistics of morethan-differentiable velocity fluctuations by means of two recently introduced sets of statistical estimators, namely inverse statistics and second order differences. We show that the 2D turbulent velocity field, u, cannot be simply characterized by its spectrum behavior, E(k) ∝ k −α . There exists a whole set of exponents associated to the non-trivial smooth fluctuations of the velocity field at all scales. We also present a numerical investigation of the temporal properties of u measured in different spatial locations.

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Section: Introductionmentioning

confidence: 99%

Inverse velocity statistics in two-dimensional turbulence

et al. 2003

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“…Recently, a new stylized fact have been unveiled dealing with the inverse statistics of the exit time in the Dow Jones Industrial Average [1,2,3] and in the foreign exchange markets [11]. Interestingly, this concept of inverse statistics was also borrowed from turbulence [12] and applied in turbulence extensively [13,14,15,16,17,18].For a given series of log prices {s i } where i corresponds to trading days, the exit time (or first passage time) τ at time i for a given return threshold ρ > 0 is defined as the minimal time span needed for the difference of log prices exceeds ρ for the first time. In other words, one says mathematically…”

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Inverse statistics in stock markets: Universality and idiosyncracy

Zhou

Yuan

2005

Physica A: Statistical Mechanics and its Applications

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Investigations of inverse statistics (a concept borrowed from turbulence) in stock markets, exemplified with filtered Dow Jones Industrial Average, S&P 500, and NASDAQ, have uncovered a novel stylized fact that the distribution of exit time follows a power law p(τ ρ ) ∼ τ −α ρ with α ≈ 1.5 at large τ ρ and the optimal investment horizon τ * ρ scales as ρ γ [1,2,3]. We have performed an extensive analysis based on unfiltered daily indices and stock prices and high-frequency (5-min) records as well in the markets all over the world. Our analysis confirms that the power-law distribution of the exit time with an exponent of about α = 1.5 is universal for all the data sets analyzed. In addition, all data sets show that the power-law scaling in the optimal investment horizon holds, but with idiosyncratic exponent. Specifically, γ ≈ 1.5 for the daily data in most of the developed stock markets and the five-minute highfrequency data, while the γ values of the daily indexes and stock prices in emerging markets are significantly less than 1.5. We show that there is of little chance that this discrepancy in γ stems from the difference of record sizes in the two kinds of stock markets. IntroductionEconophysics is an interdisciplinary science which applies statistical physics and complex system theories to economics [4,5,6,7]. More than ten stylized facts of asset returns have been discovered or re-discovered in the community [8], some of which are inspired originally by the analogy between finance markets and turbulence [9,10]. In these works, the asset return, as the counterpart of velocity difference in turbulence, plays a central role, which is defined as the difference of logarithmic prices at a given time lag. Recently, a new stylized fact have been unveiled dealing with the inverse statistics of the exit time in the Dow Jones Industrial Average [1,2,3] and in the foreign exchange markets [11]. Interestingly, this concept of inverse statistics was also borrowed from turbulence [12] and applied in turbulence extensively [13,14,15,16,17,18].For a given series of log prices {s i } where i corresponds to trading days, the exit time (or first passage time) τ at time i for a given return threshold ρ > 0 is defined as the minimal time span needed for the difference of log prices exceeds ρ for the first time. In other words, one says mathematicallyWe see that τ ρ ≥ 1 is integer. If the stock price rises, τ ρ = 1. It is argued that more small τ ρ in a period implies a bullish market while large τ ρ indicates a lasting bearish market. For fractional Brownian motion of Hurst exponent H, Ding and Yang [19] have found that the distribution density p(τ ρ ) scales aswith α = 2 − H, when τ ρ is large. For Brownian motion, the exit time distribution has been solved analytically [20,21,22] where K is the generalized diffusion constant. The most probable exit time τ * ρ (also called the optimal investment horizon in Finance) is thus scaled asThe first passage time was studied in physics, biology and engineering [20,21,22, and r...

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“…Indeed laminar fluctuations, corresponding to the events described by the peak of the probability distribution, posses non-trivial scaling properties. Recently, it has been shown that laminar fluctuations of rough and multiaffine fields are optimally characterized in terms of their exit-distance statistics, also known as inverse-statistics [3][4][5][6].The aim of this letter is twofold. First we want to extend the application of inverse statistics [3,4] to the case of smooth signals with a given power spectrum, E(k) ∼ k −α .…”

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Inverse Statistics of Smooth Signals: The Case of Two Dimensional Turbulence

et al. 2001

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The problem of inverse statistics (statistics of distances for which the signal fluctuations are larger than a certain threshold) in differentiable signals with power law spectrum, E(k) ∼ k −α , 3 ≤ α < 5, is discussed. We show that for these signals, with random phases, exit-distance moments follow a bi-fractal distribution. We also investigate two dimensional turbulent flows in the direct cascade regime, which display a more complex behavior. We give numerical evidences that the inverse statistics of 2d turbulent flows is described by a multi-fractal probability distribution, i.e. the statistics of laminar events is not simply captured by the exponent α characterizing the spectrum. PACS number(s) : 47.27.Eq, 05.45.-a Many phenomena in natural science posses complex behaviors over a wide range of spatial and temporal scales. Complexity is quantified by the non-Gaussian properties of probability distribution functions (pdf) of signal increments over a given range of scales. Whenever pdf's at different scales cannot be superposed by a simple rescaling procedure one speaks about intermittency [1]. Such non-trivial rescaling properties may be exhibited by pdf's tails or peaks, or both [2]. Strongly intermittent and rough signals, like those encountered in three dimensional turbulent flows, are the typical examples of systems with non-trivial, say multiaffine, scaling of pdf's tails. Other important natural phenomena develop simple pdf's tails but non-trivial pdf's peaks. This is the case of two dimensional turbulence as it will be shown in this letter. Indeed laminar fluctuations, corresponding to the events described by the peak of the probability distribution, posses non-trivial scaling properties. Recently, it has been shown that laminar fluctuations of rough and multiaffine fields are optimally characterized in terms of their exit-distance statistics, also known as inverse-statistics [3][4][5][6].The aim of this letter is twofold. First we want to extend the application of inverse statistics [3,4] to the case of smooth signals with a given power spectrum, E(k) ∼ k −α . In particular, we discuss signals only one time differentiable, i.e. 3 ≤ α < 5. For such signals, direct statistics, i.e. moments of signal increments over a given scale, do not bring any information: they are always dominated by the differentiable event, v(x + r) − v(x) ∼ r, which are always present when the spectrum has a slope α > 3. On the contrary, we will show that the exit-distance statistics is given by a bifractal distribution if only one kind of more than smooth fluctuation exists. With more than smooth fluctuations we mean events where the signal has a local scaling as v(x + r) − v(x) ∼ r h with h > 1. Second, we will apply the inverse statistics analysis to the case of two dimensional turbulence. Two-dimensional turbulence in the direct enstrophy cascade regime is of obvious importance both theoretically and practically to understand a variety of different natural processes, e.g. in geophysics and astrophysics [7,8]. By applying the e...

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Exit-times and ϵ-entropy for dynamical systems, stochastic processes, and turbulence

Cited by 26 publications

References 43 publications

Inverse velocity statistics in two-dimensional turbulence

Inverse velocity statistics in two-dimensional turbulence

Inverse statistics in stock markets: Universality and idiosyncracy

Inverse Statistics of Smooth Signals: The Case of Two Dimensional Turbulence

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