Long-range correlation: Difference between revisions
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Here, <math>\langle . \rangle</math> refers to an average over ''t'' and the observables ''X'' and ''Y'' depend on the time ''t'', but an analogous expression can be written down for spatial dependence. | Here, <math>\langle . \rangle</math> refers to an average over ''t'' and the observables ''X'' and ''Y'' depend on the time ''t'', but an analogous expression can be written down for spatial dependence. | ||
== Coherent states == | |||
Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting ''X = sin(ωt)'' and taking, e.g., ''Y=X''. | Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting ''X = sin(ωt)'' and taking, e.g., ''Y=X''. | ||
Note also that the correlation function is a convolution, hence its spectrum is the product of the spectra of ''X'' and ''Y'', so that γ<sub>XY</sub> 'inherits' the spectral properties of the original time series. | Note also that the correlation function is a convolution, hence its spectrum is the product of the spectra of ''X'' and ''Y'', so that γ<sub>XY</sub> 'inherits' the spectral properties of the original time series. | ||
== Turbulence == | |||
More interesting is the typical behaviour of the correlation function for turbulent states. | More interesting is the typical behaviour of the correlation function for turbulent states. | ||
In this case, the correlation function typically decays exponentially as a function of Δ and can be characterized by a single number: the 'decorrelation time' (or length) Δ<sub>corr</sub>, calculated as the distance at which the correlation has dropped from its maximum value by a factor ''1/e''. | In this case, the correlation function typically decays exponentially as a function of Δ and can be characterized by a single number: the 'decorrelation time' (or length) Δ<sub>corr</sub>, calculated as the distance at which the correlation has dropped from its maximum value by a factor ''1/e''. | ||
== Long range effects == | |||
When the correlation exhibits a slower decay for large values of the delay (or distance) Δ, namely an algebraic decay proportional to 1/Δ<sup>α</sup> (α > 0 but not too large, < 2), the correlations at large delay may be quite important to understand the global system behaviour (contrasting sharply with the exponential decay case, in which large values of Δ can be safely ignored). | When the correlation exhibits a slower decay for large values of the delay (or distance) Δ, namely an algebraic decay proportional to 1/Δ<sup>α</sup> (α > 0 but not too large, < 2), the correlations at large delay may be quite important to understand the global system behaviour (contrasting sharply with the exponential decay case, in which large values of Δ can be safely ignored). | ||
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These issues can be understood in the framework of [[Self-Organised Criticality]]. | These issues can be understood in the framework of [[Self-Organised Criticality]]. | ||
The mathematical modelling of such systems is based on the [[Continuous Time Random Walk]] and the Generalized Master Equation. | The mathematical modelling of such systems is based on the [[Continuous Time Random Walk]] and the Generalized Master Equation. | ||
=== Experimental determination === | === Experimental determination === | ||
It can be shown that determining the long-range behaviour of the correlation function directly from γ<sub>XY</sub> is not a good idea, due to its sensitivity to noise.<ref>[[doi:10.1063/1.873192|B.A. Carreras, D.E. Newman, B.Ph. van Milligen, and C. Hidalgo, ''Long-range time dependence in the cross-correlation function'', Phys. Plasmas '''6''' (1999) 485]]</ref> | It can be shown that determining the long-range behaviour of the correlation function directly from γ<sub>XY</sub> is not a good idea, due to its sensitivity to noise.<ref>[[doi:10.1063/1.873192|B.A. Carreras, D.E. Newman, B.Ph. van Milligen, and C. Hidalgo, ''Long-range time dependence in the cross-correlation function'', Phys. Plasmas '''6''' (1999) 485]]</ref> | ||
Rather, techniques such as the [[:Wikipedia:Rescaled range|Rescaled Range]], [[:Wikipedia:Hurst exponent|Hurst]] analysis, or Structure functions should be used to determine long-range correlations in data series. | Rather, techniques such as the [[:Wikipedia:Rescaled range|Rescaled Range]], [[:Wikipedia:Hurst exponent|Hurst]] analysis, or Structure functions should be used to determine long-range correlations in data series. | ||
== See also == | |||
* [[:Wikipedia:Long-range dependency|Long-range dependency]] | |||
* [[Non-diffusive transport]] | |||
== References == | == References == | ||
<references /> | <references /> | ||
Revision as of 11:42, 5 April 2012
The expression 'long-range correlation' specifically refers to the slow decay of the (temporal or spatial) correlation function (covariance), defined as
Here, refers to an average over t and the observables X and Y depend on the time t, but an analogous expression can be written down for spatial dependence.
Coherent states
Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting X = sin(ωt) and taking, e.g., Y=X. Note also that the correlation function is a convolution, hence its spectrum is the product of the spectra of X and Y, so that γXY 'inherits' the spectral properties of the original time series.
Turbulence
More interesting is the typical behaviour of the correlation function for turbulent states. In this case, the correlation function typically decays exponentially as a function of Δ and can be characterized by a single number: the 'decorrelation time' (or length) Δcorr, calculated as the distance at which the correlation has dropped from its maximum value by a factor 1/e.
Long range effects
When the correlation exhibits a slower decay for large values of the delay (or distance) Δ, namely an algebraic decay proportional to 1/Δα (α > 0 but not too large, < 2), the correlations at large delay may be quite important to understand the global system behaviour (contrasting sharply with the exponential decay case, in which large values of Δ can be safely ignored).
This unusual, slow decay of the correlation function has important consequences, implying that the system exhibits 'memory effects' or 'non-local behaviour' (self-similarity). A 'memory effect' refers to the fact that the evolution of the system is affected by previous system states over times (much) longer than the turbulence decorrelation time. An analogous interpretation is possible for 'non-local' behaviour, in which the system state at remote points affects the local evolution of the system. These issues can be understood in the framework of Self-Organised Criticality. The mathematical modelling of such systems is based on the Continuous Time Random Walk and the Generalized Master Equation.
Experimental determination
It can be shown that determining the long-range behaviour of the correlation function directly from γXY is not a good idea, due to its sensitivity to noise.[1] Rather, techniques such as the Rescaled Range, Hurst analysis, or Structure functions should be used to determine long-range correlations in data series.