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Detrended Fluctuation Analysis
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 obtained exponent is similar to the Hurst exponent, except that DFA may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are non-stationary (changing with time). It is related to measures based upon spectral techniques such as autocorrelation and Fourier transform. Peng et al. introduced DFA in 1994 in a paper that has been cited over 3,000 times as of 2022 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities. Calculation Consider a bounded time series x_t of length N, where t \in \mathbb, and let its mean value be denoted \langle x\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting. When the problem has substantial uncertainties in the independent variable (the ''x'' variable), then simple regression and least-squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regressio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Brownian Motion
In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process ''BH''(''t'') on , ''T'' that starts at zero, has expectation zero for all ''t'' in , ''T'' and has the following covariance function: :E _H(t) B_H (s)\tfrac (, t, ^+, s, ^-, t-s, ^), where ''H'' is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by . The value of ''H'' determines what kind of process the ''fBm'' is: * if ''H'' = 1/2 then the process is in fact a Brownian motion or Wiener process; * if ''H'' > 1/2 then the increments of the process are positively correlated; * if ''H'' < 1/2 then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Color Of Noise
In audio engineering, electronics, physics, and many other fields, the color of noise or noise spectrum refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantly different properties. For example, as audio signals they will sound different to human ears, and as images they will have a visibly different texture. Therefore, each application typically requires noise of a specific color. This sense of 'color' for noise signals is similar to the concept of timbre in music (which is also called "tone color"; however, the latter is almost always used for sound, and may consider very detailed features of the spectrum). The practice of naming kinds of noise after colors started with white noise, a signal whose spectrum has equal power within any equal interval of frequencies. That name was given by analogy with white light, which was (incorrectly) assumed to have such a flat power spectrum over the visibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Khinchin Theorem
In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process. History Norbert Wiener proved this theorem for the case of a deterministic function in 1930; Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published that probabilistic analogue in 1934. Albert Einstein explained, without proofs, the idea in a brief two-page memo in 1914. The case of a continuous-time process For continuous time, the Wiener–Khinchin theorem says that if x is a wide-sense stochastic process whose autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value, r_(\tau) = \mathbb\big (t)^*x(t - \tau)\big/math> (the asterisk denotes complex co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of Power (physics), power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including Noise (electronics), noise) as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over ''all'' time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations. Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations. Correlation functions used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MF-DFA
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 obtained exponent is similar to the Hurst exponent, except that DFA may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are non-stationary (changing with time). It is related to measures based upon spectral techniques such as autocorrelation and Fourier transform. Peng et al. introduced DFA in 1994 in a paper that has been cited over 3,000 times as of 2022 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities. Calculation Consider a bounded time series x_t of length N, where t \in \mathbb, and let its mean value be denoted \langle x\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multifractal
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. Multifractal systems are common in nature. They include the length of coastlines, mountain topography, fully developed turbulence, real-world scenes, heartbeat dynamics, human gait and activity, human brain activity, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in fluid dynamics to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more. The origin of multifractality in sequential (time series) data has been attributed to mathematical convergence effects related to the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie exponential dispersion models, as well as the geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rescaled Range
The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst (1880–1978). Its purpose is to provide an assessment of how the apparent variability of a series changes with the length of the time-period being considered. The rescaled range of time series is calculated from dividing the range of its mean adjusted cumulative deviate series (see the Calculation section below) by the standard deviation of the time series itself. For example, consider a time series , which has a mean m = 2 and standard deviation S = 1.79. Subtracting m from each value of the series gives mean adjusted series . To calculate cumulative deviate series we take the first value -1, then sum of the first two values -1+1=0, then sum of the first three values and so on to get , range of which is R = 3, so the rescaled range is R/S = 1.68. If we consider the same time series, but increase the number of observations of it, the rescaled ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Noise
] In science, Brownian noise, also known as Brown noise or red noise, is the type of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" does not come from the color, but after Robert Brown, who documented the erratic motion for multiple types of inanimate particles in water. The term "red noise" comes from the "white noise"/"white light" analogy; red noise is strong in longer wavelengths, similar to the red end of the visible spectrum. Explanation The graphic representation of the sound signal mimics a Brownian pattern. Its spectral density is inversely proportional to ''f'' 2, meaning it has higher intensity at lower frequencies, even more so than pink noise. It decreases in intensity by 6 dB per octave (20 dB per decade) and, when heard, has a "damped" or "soft" quality compared to white and pink noise. The sound is a low roar resembling a waterfall or heavy rainfall. See also violet noise, which is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pink Noise
Pink noise or noise is a signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave interval (halving or doubling in frequency) carries an equal amount of noise energy. Pink noise sounds like a waterfall. It is often used to tune loudspeaker systems in professional audio. Pink noise is one of the most commonly observed signals in biological systems. The name arises from the pink appearance of visible light with this power spectrum. This is in contrast with white noise which has equal intensity per frequency interval. Definition Within the scientific literature, the term 1/f noise is sometimes used loosely to refer to any noise with a power spectral density of the form S(f) \propto \frac, where ''f'' is frequency, and 0 < α < 2, with exponent α usually close to 1. One-dimensional signals with α = 1 are usually called pink noise. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
