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 data. It relates to the rate of decay of statistical dependence 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, 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.[1][2][3][4][5][6]
Short-range dependence versus long-range dependenceedit
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 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 dams on rivers for water resources, where the summations correspond to the total inflow to the dam over an extended period.[7]
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 parameterH 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.[8] 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 Parameteredit
Slowly decaying variances, LRD, and a spectral density obeying a power-law are different manifestations of the property of the underlying covariance of a 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 processesedit
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, the most typical one being fractional Brownian motion. 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).
^Beran, Jan (1994). Statistics for Long-Memory Processes. CRC Press.
^Doukhan; et al. (2003). Theory and Applications of Long-Range Dependence. Birkhäuser.
^Malamud, Bruce D.; Turcotte, Donald L. (1999). Self-Affine Time Series: I. Generation and Analyses. Vol. 40. pp. 1–90. Bibcode:1999AdGeo..40....1M. doi:10.1016/S0065-2687(08)60293-9. ISBN9780120188406. {{cite book}}: |journal= ignored (help)
^Samorodnitsky, Gennady (2007). Long range dependence. Foundations and Trends in Stochastic Systems.
^Beran; et al. (2013). Long memory processes: probabilistic properties and statistical methods. Springer.
^Witt, Annette; Malamud, Bruce D. (September 2013). "Quantification of Long-Range Persistence in Geophysical Time Series: Conventional and Benchmark-Based Improvement Techniques". Surveys in Geophysics. 34 (5): 541–651. Bibcode:2013SGeo...34..541W. doi:10.1007/s10712-012-9217-8.
^*Hurst, H.E., Black, R.P., Simaika, Y.M. (1965) Long-term storage: an experimental study Constable, London.
Bariviera, A.F. (2011). "The influence of liquidity on informational efficiency: The case of the Thai Stock Market". Physica A: Statistical Mechanics and Its Applications. 390 (23): 4426–4432. Bibcode:2011PhyA..390.4426B. doi:10.1016/j.physa.2011.07.032. S2CID 120377241.
Bariviera, A.F.; Guercio, M.B.; Martinez, L.B. (2012). "A comparative analysis of the informational efficiency of the fixed income market in seven European countries". Economics Letters. 116 (3): 426–428. doi:10.1016/j.econlet.2012.04.047. hdl:11336/66311. S2CID 153323583.
Brockwell, A.E. (2006). "Likelihood-based analysis of a class of generalized long-memory time series models". Journal of Time Series Analysis. 28 (3): 386–407. doi:10.1111/j.1467-9892.2006.00515.x. S2CID 122206112.
Granger, C. W. J.; Joyeux, R. (1980). "An introduction to long-memory time series models and fractional differencing". Journal of Time Series Analysis. 1: 15–30. doi:10.1111/j.1467-9892.1980.tb00297.x.
Witt, A.; 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. Bibcode:2013SGeo...34..541W. doi:10.1007/s10712-012-9217-8.
Cohn, T. A.; Lins, H. F. (2005). "Nature's style: Naturally trendy". Geophysical Research Letters. 32 (23). doi:10.1029/2005GL024476.
April 11, 2024
long, range, dependence, also, called, long, memory, long, range, persistence, phenomenon, that, arise, analysis, spatial, time, series, data, relates, rate, decay, statistical, dependence, points, with, increasing, time, interval, spatial, distance, between, . 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 data It relates to the rate of decay of statistical dependence 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 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 1 2 3 4 5 6 Contents 1 Short range dependence versus long range dependence 2 Estimation of the Hurst Parameter 3 Relation to self similar processes 4 Models 5 See also 6 Notes 7 Further readingShort range dependence versus long range dependence editOne 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 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 dams on rivers for water resources where the summations correspond to the total inflow to the dam over an extended period 7 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 8 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 editSlowly decaying variances LRD and a spectral density obeying a power law are different manifestations of the property of the underlying covariance of a 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 analysisRelation to self similar processes editGiven 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 the most typical one being fractional Brownian motion In the converse given a self similar process with stationary increments with Hurst index H gt 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 Models editAmong stochastic models that are 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 editLong tail traffic Traffic generation model Detrended fluctuation analysis Tweedie distributions Fractal dimension Hurst exponentNotes edit Beran Jan 1994 Statistics for Long Memory Processes CRC Press Doukhan et al 2003 Theory and Applications of Long Range Dependence Birkhauser Malamud Bruce D Turcotte Donald L 1999 Self Affine Time Series I Generation and Analyses Vol 40 pp 1 90 Bibcode 1999AdGeo 40 1M doi 10 1016 S0065 2687 08 60293 9 ISBN 9780120188406 a href Template Cite book html title Template Cite book cite book a journal ignored help Samorodnitsky Gennady 2007 Long range dependence Foundations and Trends in Stochastic Systems Beran et al 2013 Long memory processes probabilistic properties and statistical methods Springer Witt Annette Malamud Bruce D September 2013 Quantification of Long Range Persistence in Geophysical Time Series Conventional and Benchmark Based Improvement Techniques Surveys in Geophysics 34 5 541 651 Bibcode 2013SGeo 34 541W doi 10 1007 s10712 012 9217 8 Hurst H E Black R P Simaika Y M 1965 Long term storage an experimental study Constable London Beran 1994 page 34Further reading editBariviera A F 2011 The influence of liquidity on informational efficiency The case of the Thai Stock Market Physica A Statistical Mechanics and Its Applications 390 23 4426 4432 Bibcode 2011PhyA 390 4426B doi 10 1016 j physa 2011 07 032 S2CID 120377241 Bariviera A F Guercio M B Martinez L B 2012 A comparative analysis of the informational efficiency of the fixed income market in seven European countries Economics Letters 116 3 426 428 doi 10 1016 j econlet 2012 04 047 hdl 11336 66311 S2CID 153323583 Brockwell A E 2006 Likelihood based analysis of a class of generalized long memory time series models Journal of Time Series Analysis 28 3 386 407 doi 10 1111 j 1467 9892 2006 00515 x S2CID 122206112 Granger C W J Joyeux R 1980 An introduction to long memory time series models and fractional differencing Journal of Time Series Analysis 1 15 30 doi 10 1111 j 1467 9892 1980 tb00297 x Schennach S M 2018 Long Memory via Networking Econometrica 86 6 2221 2248 doi 10 3982 ECTA11930 hdl 10419 189779 Witt A 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 Bibcode 2013SGeo 34 541W doi 10 1007 s10712 012 9217 8 Cohn T A Lins H F 2005 Nature s style Naturally trendy Geophysical Research Letters 32 23 doi 10 1029 2005GL024476 Retrieved from https en wikipedia org w index php title Long range dependence amp oldid 1193823249, wikipedia, wiki, book, books, library,
