Long-range dependence (LRD), also called long memory or long-range persistence, is a phenomenon that may arise in the analysis of spatial or

time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

data. It relates to the rate of decay of

statistical dependence Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...

of two points with increasing time interval or spatial distance between the points. A phenomenon is usually considered to have long-range dependence if the dependence decays more slowly than an

exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...

, typically a power-like decay. LRD is often related to

self-similar process Self-similar processes are types of stochastic processes that exhibit the phenomenon of self-similarity. A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension (space or time ...

es or fields. LRD has been used in various fields such as internet traffic modelling,

econometrics Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...

hydrology Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...

linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...

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 In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process ...

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

. In the case of LRD, there is much stronger coupling. The decay of the autocovariance function is power-like and so is slower than exponential. 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

dam A dam is a barrier that stops or restricts the flow of surface water or underground streams. Reservoirs created by dams not only suppress floods but also provide water for activities such as irrigation, human consumption, industrial use, a ...

s on rivers for

water resources Water resources are natural resources of water that are potentially useful for humans, for example as a source of drinking water supply or irrigation water. 97% of the water on the Earth is salt water and only three percent is fresh water; slight ...

, where the summations correspond to the total inflow to the dam over an extended period. 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 In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...

), one has to find other ways to define what LRD means, and this is often done with the help of

es. The

Hurst parameter The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst expone ...

''H'' is a measure of the extent of long-range dependence in a time series (while it has another meaning in the context of

es). ''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 than 0.5 corresponds to anti-persistency, which as the opposite of LRD indicates strong negative correlation so that the process fluctuates violently.

Estimation of the Hurst Parameter

Slowly decaying variances, LRD, and a spectral density obeying a power-law are different manifestations of the property of the underlying covariance stationary process X. Therefore, it is possible to approach the problem of estimating the Hurst parameter from three difference angles: *Variance-time plot: based on the analysis of the variances of the aggregate processes *R/S statistics: based on the time-domain analysis of the rescaled adjusted range *Periodogram: based on a frequency-domain analysis

Relation to self-similar processes

Given a stationary LRD sequence, the partial sum if viewed as a process indexed by the number of terms after a proper scaling, is a

with

stationary increments In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on the time when the observation was started. Many large families of stochastic processes have stat ...

asymptotically. In the converse, given a self-similar process with stationary increments with Hurst index ''H'' > 0.5, its increments (consecutive differences of the process) is a stationary LRD sequence. This also holds true if the sequence is short-range dependent, but in this case the self-similar process resulting from the partial sum can only be Brownian motion (''H'' = 0.5), while in the LRD case the self-similar process is a self-similar process with ''H'' > 0.5, the most typical one being fractional Brownian motion.

Models

Among

stochastic model In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

s that are used for long-range dependence, some popular ones are

autoregressive fractionally integrated moving average In statistics, autoregressive fractionally integrated moving average models are time series models that generalize ARIMA (''autoregressive integrated moving average'') models by allowing non-integer values of the differencing parameter. These model ...

models, which are defined for discrete-time processes, while continuous-time models might start from fractional Brownian motion.

Short-range dependence versus long-range dependence

Estimation of the Hurst Parameter

Relation to self-similar processes

Models

See also

Notes

Further reading