Long-range dependency
Encyclopedia
Long-range dependency 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. Some self-similar process
es may exhibit long-range dependence, but not all processes having long-range dependence are self-similar. LRD has been used in various field such as video traffic modelling, econometrics
, hydrology
and linguistics
Different definitions of LRD are used for different applications: a review is given by Samorodnitsky.
functions. In short-range dependent processes, 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 long-range processes there is much stronger coupling. The decay of the autocovariance function is power-like and so decays slower than exponentially.
A second way of characterising long- and short-range dependence is in terms of the properties of sums of consecutive values and, in particular, how the properties change as the number of terms in the summation increases. In long-range dependent processes the variance and range of the run-sums are larger and increase more rapidly, compared to properties of the marginal distribution
, than for short-range dependence or independent processes. One way of examining this behaviour uses the rescaled range
. This aspect of long-range dependence is important in the design of dam
s on rivers for water resources
, where the summations correspond to the total inflow to the dam over an extended period.
A formal statement of difference between SRD and LRD is given as follows:
"All short-range dependent processes are characterized by an autocorrelation
function which decays exponentially fast; processes with long-range dependence exhibit a much slower decay of the correlations - their autocorrelation functions typically obey some power law
."
The Hurst parameter H is a measure of the extent of long-range dependence in a time series. H takes on values from 0.5 to 1. A value of 0.5 indicates the absence of long-range dependence. The closer H is to 1, the greater the degree of persistence or long-range dependence.
that exhibit long-range dependence are often represented by starting from models that have been devised to be either exactly self similar (a self-similar process
), or approximately so. Additional components of dependence can then be added in the same way that short-term dependence models are derived from independence
or white noise
.
Among stochastic models that can be used for long-range dependence behaviour are autoregressive fractionally integrated moving average
models, which are defined for discrete-time processes, while continuous-time models might start from fractional Brownian motion.
Spatial analysis
Spatial analysis or spatial statistics includes any of the formal techniques which study entities using their topological, geometric, or geographic properties...
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. Some self-similar process
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 . Self-similar processes can sometimes be described using...
es may exhibit long-range dependence, but not all processes having long-range dependence are self-similar. LRD has been used in various field such as video traffic modelling, econometrics
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...
, hydrology
Hydrology
Hydrology is the study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability...
and linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
Different definitions of LRD are used for different applications: a review is given by Samorodnitsky.
Short-range dependence vs. long-range dependence
One way of characterising long-range and short-range dependent processes is in terms of their autocovarianceAutocovariance
In statistics, given a real stochastic process X, the autocovariance is the covariance of the variable with itself, i.e. the variance of the variable against a time-shifted version of itself...
functions. In short-range dependent processes, 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 long-range processes there is much stronger coupling. The decay of the autocovariance function is power-like and so decays slower than exponentially.
A second way of characterising long- and short-range dependence is in terms of the properties of sums of consecutive values and, in particular, how the properties change as the number of terms in the summation increases. In long-range dependent processes the variance and range of the run-sums are larger and increase more rapidly, compared to properties of the marginal distribution
Marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. The term marginal variable is used to refer to those variables in the subset of variables being retained...
, than for short-range dependence or independent processes. One way of examining this behaviour uses the rescaled range
Rescaled range
The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst...
. This aspect of long-range dependence is important in the design of dam
Dam
A dam is a barrier that impounds water or underground streams. Dams generally serve the primary purpose of retaining water, while other structures such as floodgates or levees are used to manage or prevent water flow into specific land regions. Hydropower and pumped-storage hydroelectricity are...
s on rivers for water resources
Water resources
Water resources are sources of water that are useful or potentially useful. Uses of water include agricultural, industrial, household, recreational and environmental activities. Virtually all of these human uses require fresh water....
, where the summations correspond to the total inflow to the dam over an extended period.
A formal statement of difference between SRD and LRD is given as follows:
"All short-range dependent processes are characterized by an autocorrelation
Autocorrelation
Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...
function which decays exponentially fast; processes with long-range dependence exhibit a much slower decay of the correlations - their autocorrelation functions typically obey some power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
."
The Hurst parameter H is a measure of the extent of long-range dependence in a time series. H takes on values from 0.5 to 1. A value of 0.5 indicates the absence of long-range dependence. The closer H is to 1, the greater the degree of persistence or long-range dependence.
Models
Stochastic processStochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
that exhibit long-range dependence are often represented by starting from models that have been devised to be either exactly self similar (a self-similar process
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 . Self-similar processes can sometimes be described using...
), or approximately so. Additional components of dependence can then be added in the same way that short-term dependence models are derived from independence
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
or white noise
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
.
Among stochastic models that can be used for long-range dependence behaviour are autoregressive fractionally integrated moving average
Autoregressive fractionally integrated moving average
In statistics, autoregressive fractionally integrated moving average models are time series models that generalize ARIMA models by allowing non-integer values of the differencing parameter and are useful in modeling time series with long memory...
models, which are defined for discrete-time processes, while continuous-time models might start from fractional Brownian motion.
Further reading
- Ledesma S. and Liu D. (2000) "Synthesis of fractional Gaussian noise using linear approximation for generating self-similar network traffic", Computer Communication Review, 30, 4–17.
- Ledesma S., Liu D. and Hernandez D. (2007) "Two Approximation Methods to Synthesize the Power Spectrum of Fractional Gaussian Noise",Computational Statistics and Data Analysis Journal, 52 (2), 1047–1062.