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Long-range dependency
Long-range dependency (LRD), also called long memory or long-range persistence, is a phenomenon that may arise in the analysis of spatial or time series data. It relates to the rate of decay of statistical dependence, with the implication that this decays more slowly than an exponential decay, typically a power-like decay. LRD is often related to self-similar processes or fields. LRD has been used in various fields such as internet traffic modelling, econometrics, hydrology, linguistics and the earth sciences. Different mathematical definitions of LRD are used for different contexts and purposes.
==Short-range dependence versus long-range dependence==
One way of characterising long-range and short-range dependent stationary process is in terms of their autocovariance functions. For a short-range dependent process, the coupling between values at different times decreases rapidly as the time difference increases. Either the autocovariance drops to zero after a certain time-lag, or it eventually has an exponential decay. In the case of LRD, there is much stronger coupling. The decay of the autocovariance function is power-like and so decays slower than exponentially.
A second way of characterizing long- and short-range dependence is in terms of the variance of partial sum of consecutive values. For short-range dependence, the variance grows typically proportionally to the number of terms. As for LRD, the variance of the partial sum increases more rapidly which is often a power function with the exponent greater than 1. A way of examining this behavior uses the rescaled range. This aspect of long-range dependence is important in the design of dams on rivers for water resources, where the summations correspond to the total inflow to the dam over an extended period.〔
*Hurst, H.E., Black, R.P., Simaika, Y.M. (1965) ''Long-term storage: an experimental study'' Constable, London.〕
The above two ways are mathematically related to each other, but they are not the only ways to define LRD. In the case where the autocovariance of the process does not exist (heavy tails), one has to find other ways to define what LRD means, and this is often done with the help of self-similar processes.
The Hurst parameter ''H'' is a measure of the extent of long-range dependence in a time series (while it has another meaning in the context of self-similar processes). ''H'' takes on values from 0 to 1. A value of 0.5 indicates the absence of long-range dependence.〔Beran (1994) page 34〕 The closer ''H'' is to 1, the greater the degree of persistence or long-range dependence. ''H'' less that 0.5 corresponds to anti-persistency, which as the opposite of LRD indicates strong negative correlation so that the process fluctuates violently.
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