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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.
Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. [2] Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to ...
Self-similar solutions appear whenever the problem lacks a characteristic length or time scale (for example, the Blasius boundary layer of an infinite plate, but not of a finite-length plate). These include, for example, the Blasius boundary layer or the Sedov–Taylor shell .
The increment process X(t) is known as fractional Gaussian noise. There is also a generalization of fractional Brownian motion: n-th order fractional Brownian motion, abbreviated as n-fBm. [1] n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order n are stationary. For n = 1, n-fBm is classical fBm.
A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process. In mathematics, the Wiener process (or Brownian motion, due to its historical connection with the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener.
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.
A self-similar process is one way of modeling network data dynamics with such a long range correlation. This article defines and describes network data transfer dynamics in the context of a self-similar process.
Let {} be a random process, and be any point in time (may be an integer for a discrete-time process or a real number for a continuous-time process). Then X t {\displaystyle X_{t}} is the value (or realization ) produced by a given run of the process at time t {\displaystyle t} .