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Pink noise. Within the scientific literature the term pink noise is sometimes used a little more loosely to refer to any noise with a power spectral density of the form where f is frequency and 0 < α < 2, with exponent α usually close to 1. These pink -like noises occur widely in nature and are a source of considerable interest in many fields. The distinction between the noises with α near 1 and those with a broad range of α approximately corresponds to a much more basic distinction. The former (narrow sense) generally come from condensed matter systems in quasi-equilibrium, as discussed below.[2] The latter (broader sense) generally correspond to wide range of non-equilibrium driven dynamical systems.

The term flicker noise is sometimes used to refer to pink noise, although this is more properly applied only to its occurrence in electronic devices due to a direct current. Description[edit] Web Site for Perfectly Random Sampling with Markov Chains: There is a chapter on ``Coupling from the past,'' by James G. Propp and David B. Wilson, Chapter 22 of the textbook ''Markov Chains and Mixing Times,'' by David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, to be published by the American Mathematical Society, 2008. Introduction and Scope Random sampling has found numerous applications in physics, statistics, and computer science. In most cases one simply does not know how many Markov chain steps are needed to get a sufficiently random state. In the past decade there is been much research on obtaining rigorous bounds of how many Markov chain steps are needed to generate a random sample.

In recent years there have been a large number of algorithms developed for sampling from the steady state distribution of suitably well-structured Markov chains, which require no a priori knowledge of how long the Markov chains take to get mixed. Is placed next to those articles that contain simulation results or give sample outputs. Lévy flight.

The term "Lévy flight" was coined by Benoît Mandelbrot,[1] who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution,[2] and Rayleigh flight for when the distribution is a normal distribution[3] (which is not an example of a heavy-tailed probability distribution). Later researchers have extended the use of the term "Lévy flight" to include cases where the random walk takes place on a discrete grid rather than on a continuous space.[4][5] A Lévy flight is a random walk in which the steps are defined in terms of the step-lengths, which have a certain probability distribution, with the directions of the steps being isotropic and random.

The particular case for which Mandelbrot used the term "Lévy flight"[1] is defined by the survivor function (commonly known as the survival function) of the distribution of step-sizes, U, being[6] for some k satisfying 1 < k < 3. John Nolan's Stable Distribution Page. Autoregressive conditional heteroskedasticity. ARCH(q) model Specification[edit] Suppose one wishes to model a time series using an ARCH process.

Autoregressive conditional heteroskedasticity

Let denote the error terms (return residuals, with respect to a mean process) i.e. the series terms. These are split into a stochastic piece and a time-dependent standard deviation characterizing the typical size of the terms so that The random variable is a strong White noise process. Is modelled by where and An ARCH(q) model can be estimated using ordinary least squares. Estimate the best fitting autoregressive model AR(q) .Obtain the squares of the error and regress them on a constant and q lagged values: where q is the length of ARCH lags.The null hypothesis is that, in the absence of ARCH components, we have for all .

GARCH[edit] Nakagami distribution. Credit Scoring, Data Mining, Predictive Analytics, Statistics, StatSoft Electronic Textbook. "Thank you and thank you again for providing a complete, well-structured, and easy-to-understand online resource.

Credit Scoring, Data Mining, Predictive Analytics, Statistics, StatSoft Electronic Textbook

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