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The expression 'long-range correlation' specifically refers to the slow decay of the (temporal or spatial) [[:Wikipedia:Correlation|correlation]] function. | The expression 'long-range correlation' specifically refers to the slow decay of the (temporal or spatial) [[:Wikipedia:Correlation|correlation]] function, defined as | ||
:<math>\gamma_{XY}(\Delta) = \langle X(t) Y(t+\Delta)\rangle.</math> | |||
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. | |||
Ignoring coherent states ('modes', to which the concept does not apply), the correlation function typically decays exponentially and can be characterized by a 'decorrelation time' (or length), calculated as the distance at which the correlation has dropped from its maximum value by | Ignoring coherent states (regular oscillations or 'modes', to which the concept does not apply), the correlation function typically decays exponentially as a function of Δ and can be characterized by a 'decorrelation time' (or length), calculated as the distance at which the correlation has dropped from its maximum value by a factor ''1/e''. | ||
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). | ||
This unusual, slow decay of the correlation function has important consequences, implying that the system exhibits 'memory effects' or 'non-local behaviour', which can be understood in the framework of [[Self-Organised Criticality]] | This unusual, slow decay of the correlation function has important consequences, implying that the system exhibits 'memory effects' or 'non-local behaviour' (self-similarity), which can be understood in the framework of [[Self-Organised Criticality]]. Also see [[Non-diffusive transport]] and [[Continuous Time Random Walk]]. | ||
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. | |||
== References == | == References == | ||
<references /> | <references /> | ||