Long-term correlation

Long-term correlations , also known as long-term persistence , maintenance tendency or memory effect , are correlations with diverging correlation lengths .

In the case of positive correlations, a high value is more likely to be followed by another high value and a low value by a lower one; In the case of long-term correlations , due to the slowly falling correlation function, this also applies to extended high or low ranges, which are then correlated with one another in the same way as the individual values. This leads to a pronounced mountain and valley structure, which manifests itself in the fact that long-term correlated sequences are difficult to distinguish from trends .

Long-term correlations are self-affine structures that show self-similarity only under anisotropic length transformation. So z. For example, if a long-term correlated series of random numbers is similar to itself, the abscissa and ordinate must be stretched or compressed with different factors.

An extension of the description of long-term correlations, the multifractality is where different moments are long-term correlates different, which is especially strong in runoff time series occurs.

Occur

Long-term correlations have so far mainly been investigated in the case of autocorrelations , but can in principle also occur in the case of cross-correlations and generally in the multivariate case. They have been found in a wide variety of areas, e.g. B. in

• Runoff time series
• long weather records
• DNA sequences
• Fluctuating heartbeat
• Fluctuations in neuronal action potentials
• the human walk .

The effect of long-term correlations was described for the first time in 1951 by HE Hurst when he examined the long-term Nile series. He investigated which level fluctuations of the Nile a dam must contain without overflowing or drying out, which led to his R / S analysis with the Hurst exponent (related to , see below). In the course of chaos research , the topic was taken up and is now the subject of research in many areas. ${\displaystyle H}$ ${\displaystyle \alpha }$

Mathematical description

In the case of long-term correlations , the integral over the correlation function has no finite value: ${\displaystyle C(s)}$

${\displaystyle s_{\times }=\int _{0}^{\infty }\!\!C(s)\cdot {\rm {d}}s\quad \rightarrow \quad \infty \,.}$

This is especially true for a correlation function that decreases like a power law :

${\displaystyle C(s)\sim s^{-\gamma }}$

with a correlation exponent (in the one-dimensional case). ${\displaystyle 0<\gamma <1}$

Such correlations can be quantified using various methods :

• the numerically calculated correlation function provides the above correlation exponent . ${\displaystyle \gamma }$
• the power spectrum decreases with the exponent . ${\displaystyle \beta }$
• the fluctuation analysis shows the fluctuation exponent . ${\displaystyle \alpha }$
• and others, e.g. B. Wavelets .

The following relationships apply between the three exponents:

${\displaystyle {\begin{alignedat}{2}&\alpha +\gamma /2&&=1\\&\beta +\gamma &&=1\,,\end{alignedat}}}$

the latter can be shown using the Wiener-Chinchin theorem .

In contrast to long-term correlations, short-term correlations , e.g. B. emerge from an autoregressive process , a finite correlation length, e.g. B.

${\displaystyle C(s)={\rm {e}}^{-s/s_{\times }}}$.

literature

• Harold Edwin Hurst : Long-term storage capacity of reservoirs . In: Transactions of the American Society of Civil Engineers , Vol. 116 (1951), Issue 2447, pp. 770-808, ISSN  0066-0604
• Jens Feder: Fractals (Physics of solids and liquids). Plenum Press, New York 1988, ISBN 0-306-42851-2 .
• Armin Bunde , Shlomo Havlin (Ed.): Fractals and Disordered Systems . 2nd Edition. Springer, Berlin 1996, ISBN 3-540-56219-2 .
• Armin Bunde, Jan W. Kantelhardt: Long-term correlations in nature: of climate, genome and heart rhythm . (PDF; 896 kB). In: Physikalische Blätter , Volume 57, 2001, pp. 49-54, ISSN  1617-9439 .