Reliability of self-affine measurements

Schmittbuhl, Jean; Vilotte, J. P.; Roux, Stéphane

doi:10.1103/physreve.51.131

Cited by 252 publications

(254 citation statements)

References 11 publications

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“…Another method related to dispersional analysis is "Fano factor" analysis (25), also known as the variable bandwidth method (24). In this method one averages the variances over short records and over successively longer records.…”

Section: Discussionmentioning

confidence: 99%

“…This technique has been evaluated for the effects of record length, but not for its dependence on the value of the mean, or the effects of noise, so such studies should be undertaken before Fano factor or variable bandwidth analysis is either accepted or rejected. Schmittbuhl et al (24) shows that it is a stronger technique than either box analysis or caliper (or divider) analysis (22), neither of which is suited to self-affine signals but which are suited to self-similar functions.…”

Section: Discussionmentioning

confidence: 99%

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Evaluation of the dispersional analysis method for fractal time series

1995

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Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is <0.5 for rough anticorrelated series, >0.5 for positively correlated series, and =0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 − D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (>0.8) and low H (<0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H.Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Evaluation of the dispersional analysis method for fractal time series

1995

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“…We have measured the roughness of the cracks in the model with both the variable bandwidth max-min method of Eq. (1) and the variable bandwidth RMS method [13]. In order to avoid strong finite size effects we have used the results of Ref.…”

Section: Resultsmentioning

confidence: 99%

Random and correlated roughening in slow fracture by damage nucleation

2006

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We address the role of the nature of material disorder in determining the roughness of cracks which grow by damage nucleation and coalescence ahead of the crack tip. We highlight the role of quenched and annealed disorders in relation to the length scales d and ξc associated with the disorder and the damage nucleation respectively. In two related models, one with quenched disorder in which d ≃ ξc, the other with annealed disorder in which d ≪ ξc, we find qualitatively different roughening properties for the resulting cracks in 2-dimensions. The first model results in random cracks with an asymptotic roughening exponent ζ ≈ 0.5. The second model shows correlated roughening with ζ ≈ 0.66. The reasons for the qualitative difference are rationalized and explained.

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“…Compared to existing techniques, quantum probes offers the advantage of requiring measurements performed at a fixed single (optimized) instant of time, without the need of observing the system for a long time in order to collect a time series, and thus avoiding any issue related to poor sampling [24,25,26]. As we will see, quantum probes may be effectively employed to characterize fractional Gaussian process when the the systemenvironment coupling is weak, provided that a long interaction time is achievable, or when the coupling is strong and the quantum probe may be observed shortly after that the interaction has been switched on.…”

Section: ] B Denotes Expectation Values Taken Over the Values Of Thementioning

confidence: 99%

Quantum probes for fractional Gaussian processes

Paris¹

2014

Physica A: Statistical Mechanics and its Applications

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We address the characterization of classical fractional random noise via quantum probes. In particular, we focus on estimation and discrimination problems involving the fractal dimension of the trajectories of a system subject to fractional Brownian noise. We assume that the classical degree of freedom exposed to the environmental noise is coupled to a quantum degree of freedom of the same system, e.g. its spin, and exploit quantum limited measurements on the spin part to characterize the classical fractional noise. More generally, our approach may be applied to any two-level system subject to dephasing perturbations described by fractional Brownian noise, in order to assess the precision of quantum limited measurements in the characterization of the external noise. In order to assess the performances of quantum probes we evaluate the Bures metric, as well as the Helstrom and the Chernoff bound, and optimize their values over the interaction time. We find that quantum probes may be successfully employed to obtain a reliable characterization of fractional Gaussian process when the coupling with the environment is weak or strong. In the first case decoherence is not much detrimental and for long interaction times the probe acquires information about the environmental parameters without being too much mixed. Conversely, for strong coupling information is quickly impinged on the quantum probe and can effectively retrieved by measurements performed in the early stage of the evolution. In the intermediate situation, none of the two above effects take place: information is flowing from the environment to the probe too slowly compared to decoherence, and no measurements can be effectively employed to extract it from the quantum probe. The two regimes of weak and strong coupling are defined in terms of a threshold value of the coupling, which itself increases with the fractional dimension.

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Reliability of self-affine measurements

Cited by 252 publications

References 11 publications

Evaluation of the dispersional analysis method for fractal time series

Evaluation of the dispersional analysis method for fractal time series

Random and correlated roughening in slow fracture by damage nucleation

Quantum probes for fractional Gaussian processes

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