Self-similar process
Self-similar processes are stochastic processes satisfying a mathematically precise version of the self-similarity property. Several related properties have this name, and some are defined here.
A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension. Because stochastic processes are random variables with a time and a space component, their self-similarity properties are defined in terms of how a scaling in time relates to a scaling in space.
Distributional self-similarity
Definition
A continuous-time stochastic process is called self-similar with parameter if for all , the processes and have the same law.[1]
Examples
- The Wiener process (or Brownian motion) is self-similar with .[2]
- The fractional Brownian motion is a generalisation of Brownian motion that preserves self-similarity; it can be self-similar for any .[3]
- The class of self-similar Lévy processes are called stable processes. They can be self-similar for any .[4]
Second-order self-similarity
Definition
A wide-sense stationary process is called exactly second-order self-similar with parameter if the following hold:
- (i) , where for each ,
- (ii) for all , the autocorrelation functions and of and are equal.
If instead of (ii), the weaker condition
- (iii) pointwise as
holds, then is called asymptotically second-order self-similar.[5]
Connection to long-range dependence
In the case , asymptotic self-similarity is equivalent to long-range dependence.[1] Self-similar and long-range dependent characteristics in computer networks present a fundamentally different set of problems to people doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.[6]
Long-range dependence is closely connected to the theory of heavy-tailed distributions.[7] A distribution is said to have a heavy tail if
One example of a heavy-tailed distribution is the Pareto distribution. Examples of processes that can be described using heavy-tailed distributions include traffic processes, such as packet inter-arrival times and burst lengths.[8]
Examples
- The Tweedie convergence theorem can be used to explain the origin of the variance to mean power law, 1/f noise and multifractality, features associated with self-similar processes.[9]
- Ethernet traffic data is often self-similar.[5] Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic.[8]
References
- ^ a b §1.4.1 of Park, Willinger (2000)
- ^ Chapter 2: Lemma 9.4 of Ioannis Karatzas; Steven E. Shreve (1991), Brownian Motion and Stochastic Calculus (second ed.), Springer Verlag, doi:10.1007/978-1-4612-0949-2, ISBN 978-0-387-97655-6
- ^ Gennady Samorodnitsky; Murad S. Taqqu (1994), "Chapter 7: "Self-similar processes"", Stable Non-Gaussian Random Processes, Chapman & Hall, ISBN 0-412-05171-0
- ^ Theorem 3.2 of Andreas E. Kyprianou; Juan Carlos Pardo (2022), Stable Lévy Processes via Lamperti-Type Representations, New York, NY: Cambridge University Press, doi:10.1017/9781108648318, ISBN 978-1-108-48029-1
- ^ a b Will E. Leland; Murad S. Taqqu; Walter Willinger; Daniel V. Wilson (February 1994), "On the Self-similar Nature of Ethernet Traffic (Extended Version)", IEEE/ACM Transactions on Networking, 2 (1), IEEE: 1–15, doi:10.1109/90.282603
- ^ "The Self-Similarity and Long Range Dependence in Networks Web site". Cs.bu.edu. Archived from the original on 2019-08-22. Retrieved 2012-06-25.
- ^ §1.4.2 of Park, Willinger (2000)
- ^ a b Park, Willinger (2000)
- ^ Kendal, Wayne S.; Jørgensen, Bent (2011-12-27). "Tweedie convergence: A mathematical basis for Taylor's power law, 1/f noise, and multifractality". Physical Review E. 84 (6). American Physical Society (APS): 066120. Bibcode:2011PhRvE..84f6120K. doi:10.1103/physreve.84.066120. ISSN 1539-3755. PMID 22304168.
Sources
- Kihong Park; Walter Willinger (2000), Self-Similar Network Traffic and Performance Evaluation, New York, NY, USA: John Wiley & Sons, Inc., doi:10.1002/047120644X, ISBN 0471319740
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