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	~~Irwin Butts~~ is ~~what my spouse loves~~ to ~~call me although~~ I ~~don't truly like being called like that~~. ~~Bookkeeping~~ is ~~her day job now~~. The ~~thing she adores most~~ is ~~physique building~~ and ~~now she~~ is ~~trying~~ to ~~earn money~~ with it. ~~California~~ is exactly ~~where I've always been residing~~ and ~~I adore every working day residing here~~.<br><br>~~Look~~ at ~~my page [~~http://~~xyz~~.ms/~~diettogo30380 diet meal delivery~~]		'''Long-range dependency''' (abbreviated as LRD, also addressed as ''long memory'' or ''long-range persistence'') is a phenomenon that may arise in the analysis of [[Spatial analysis\|spatial]] or [[time series analysis\|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 process]]es 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. Some references are <ref>{{cite book\|last=Beran\|first=Jan\|title=Statistics for Long-Memory Processes\|year=1994\|publisher=CRC Press}}</ref> <ref>{{cite book\|last=Doukhan et al.\|title=Theory and Applications of Long-Range Dependence\|year=2003\|publisher=Birkhäuser}}</ref> <ref>{{cite journal \|last1=Malamud\|first1=Bruce D.\|last2= Turcotte\|first2=Donald L.\|title=Self-Affine Time Series: I. Generation and Analyses \|journal=Advances in Geophysics \|year=1999 \|volume=40 \|pages=1–90 \|doi=10.1016/S0065-2687(08)60293-9}}</ref> <ref>{{cite book\|last=Samorodnitsky\|first=Gennady\|title=Long range dependence\|year=2007\|publisher=Foundations and Trends® in Stochastic Systems}}</ref> <ref>{{cite book\|last=Beran et al.\|title=Long memory processes: probabilistic properties and statistical methods\|year=2013\|publisher=Springer}}</ref> <ref>{{cite journal\|last1=Witt\|first1=Annette\|last2=Malamud\|first2=Bruce D.\|title=Quantification of Long-Range Persistence in Geophysical Time Series: Conventional and Benchmark-Based Improvement Techniques\|journal=Surveys in Geophysics\|date=September 2013\|volume=34\|issue=5\|pages=541-651\|doi=10.1007/s10712-012-9217-8\|url=http://link.springer.com/article/10.1007/s10712-012-9217-8\|publisher=Springer}}</ref>

			==Short-range dependence vs. 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 [[dam]]s on rivers for [[water resources]], where the summations correspond to the total inflow to the dam over an extended period.<ref>*Hurst, H.E., Black, R.P., Simaika, Y.M. (1965) ''Long-term storage: an experimental study'' Constable, London.</ref>

			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 process]]es.

			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 process]]es). ''H'' takes on values from 0 to 1. A value of 0.5 indicates the absence of long-range dependence.<ref>Beran (1994) page 34</ref> 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.

			<!--
			==Exactly second-order self-similar vs. asymptotically second-order self-similar==
			A process ''X'' is called exactly second-order self-similar, if, for all <math>m=1,2,3,\cdots</math>

			:<math>Var(X^{(m)}) =\sigma^2 m^{-(2-2H)}</math> and <math>r^{(m)}(k) = r(k).</math>

			On the other hand, ''X'' is called asymptotically second-order self-similar with self-similarity parameter ''H'', if, for all ''k'' large enough,

			:<math>r^{(m)}(k) \to r(k) </math> as <math>m \to \infty.</math>
			-->
			==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 [[self-similar process]] with stationary increments 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]]s that are be used for long-range dependence, some popular ones are [[autoregressive fractionally integrated moving average]] models, which are defined for discrete-time processes, while continuous-time models might start from [[fractional Brownian motion]].

			==See also==
			* [[Long-tail traffic]]
			* [[Traffic generation model]]
			* [[Detrended fluctuation analysis]]
			* [[Tweedie distributions]]

			==Notes==
			{{Reflist}}

			==References==
			:*Beran, J. (1994) ''Statistics for Long-Memory Processes'', Chapman & Hall. ISBN 0-412-04901-5.

			==Further reading==
			:*{{cite journal \|author1=Granger, C. W. J.\|author2=Joyeux, R. \|authorlink1=Granger, C.
			\|title=An introduction to long-memory time series models and fractional differencing \|journal=Journal of Time Series Analysis \|year=1980 \|volume=1 \|pages=15–30 \|doi=10.1111/j.1467-9892.1980.tb00297.x}}
			:*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.
			:*Witt, A. and Malamud, B. D. (2013) "Quantification of long-range persistence in geophysical time series: Conventional and benchmark-based improvement techniques" ''Surveys in Geophysics'', 34(5), 541–651. Available online at: http://link.springer.com/article/10.1007/s10712-012-9217-8

			[[Category:Statistical dependence]]
			[[Category:Long-memory processes]]
			[[Category:Time series analysis]]
			[[Category:Teletraffic]]

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Irwin Butts is what my spouse loves to call me although I don't truly like being called like that. Bookkeeping is her day job now. The thing she adores most is physique building and now she is trying to earn money with it. California is exactly where I've always been residing and I adore every working day residing here.<br><br>Look at my page [http://xyz.ms/diettogo30380 diet meal delivery]

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en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q971651

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