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The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum. The Fourier transform of C 3 (t 1,t 2) (third-order cumulant) is called bispectrum or bispectral density. They fall in the category of Higher Order Spectra, or Polyspectra and provide supplementary information to the power ...
The (potentially time-dependent) autocorrelation matrix (also called second moment) of a (potentially time-dependent) random vector = (, …,) is an matrix containing as elements the autocorrelations of all pairs of elements of the random vector .
Suppose be a weakly stationary (2nd-order stationary) process with mean , variance , and autocorrelation function ().Assume that the autocorrelation function () has the form () as , where < < and () is a slowly varying function at infinity, that is () = for all >.
In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series that appear to be long-memory processes (diverging correlation time, e.g. power-law decaying autocorrelation function) or 1/f noise.
The second-order logic without these restrictions is sometimes called full second-order logic to distinguish it from the monadic version. Monadic second-order logic is particularly used in the context of Courcelle's theorem, an algorithmic meta-theorem in graph theory. The MSO theory of the complete infinite binary tree is decidable.
The notation () indicates an autoregressive model of order p.The AR(p) model is defined as = = + where , …, are the parameters of the model, and is white noise. [1] [2] This can be equivalently written using the backshift operator B as
A plot showing 100 random numbers with a "hidden" sine function, and an autocorrelation (correlogram) of the series on the bottom. In the analysis of data, a correlogram is a chart of correlation statistics.
In 1930 O. Perron constructed an example of a second-order system, where the first approximation has negative Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of the original nonlinear system is Lyapunov unstable. Furthermore, in a certain neighborhood of this zero solution almost all ...