Long-range dependency

Long -term correlations are correlations with divergent correlation length. That is, the integral of the correlation function has no finite value:

This is especially true for a potency- law -like sloping correlation function

With a correlation exponent ( in the one-dimensional case ). One speaks also of long-term persistence, preservation inclination or memory effect.

Basically, long-term correlations may in autocorrelations, cross-correlations and generally occur in the multivariate case, but have so far been studied mainly in the former.

Due to the slowly decaying correlation function, successive values ​​are far / long correlated. In positive correlations follows a rather high value, a further high and low at a low one. This applies for the above at Langzeikorrelationen Potenzgesetes well for extended high and low areas which are then correlated with each other in the same manner as the individual values. This leads to a pronounced mountain and valley structure, which is expressed about the fact that long time-correlated sequences can be difficult to distinguish trends.

Long -term correlations have been found in various areas, such as in runoff time series, long weather records, DNA sequences fluctuation of heartbeat fluctuations in neuronal action potentials in human gait, and many others.

Such correlations can be quantified by various methods:

• The numerically calculated correlation function gives the above correlation exponent.
• The power spectrum decreases with the exponent.
• The fluctuation analysis (see Trendbereinigende fluctuation analysis ) shows the fluctuation exponent.
• And others, such as wavelets.

Between the three exponents relations apply:

• .

The former can be shown by means of the Wiener- Khinchin theorem.

Long -term correlations are self- affine structures, which show self-similarity only under anisotropic length transformation. So say, for example, a long time-correlated series of random numbers is similar to itself, the abscissa and the ordinate must be stretched or compressed by the same factors.

In contrast to the long-term correlations have short term correlations, resulting from an autoregressive process eg, a finite length of correlation (for example ).

The effect of long-term correlations was in 1951 for the first time by HE Hurst described in the study of long-term Nilreihe. Hurst investigated which level fluctuation of the Nile, a dam must hold without overflowing or drying up, what ( related to ) led to his R / S analysis with the Hurst exponent. In the wake of chaos research the topic was picked up and is now in many areas the subject of research.

An extension of the description of long-term correlations represents the multifractality in which various moments are long-term correlation varies, which is particularly strong discharge occurs in the time series.