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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. [1]

## Autocorrelation

Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay.

## Brownian noise

In science, Brownian noise, also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion, hence its alternative name of random walk noise.

## Chaos theory

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

## Chung-Kang Peng

Chung-Kang Peng, PhD, is the Director of the Center for Dynamical Biomarkers at / (BIDMC/HMS).

## Colors of noise

In audio engineering, electronics, physics, and many other fields, the color of noise refers to the power spectrum of a noise signal (a signal produced by a stochastic process).

## 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.

## Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

## Fractal dimension

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured.

## Fractional Brownian motion

In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion.

## Hausdorff dimension

Hausdorff dimension is a measure of roughness in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a space, taking into account the distance between its points.

## Hurst exponent

The Hurst exponent is used as a measure of long-term memory of time series.

## Independent and identically distributed random variables

In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d. or iid or IID) if each random variable has the same probability distribution as the others and all are mutually independent.

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Least squares

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns.

## Log–log plot

In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes.

## Long-range dependence

Long-range dependence (LRD), also called long memory or long-range persistence, is a phenomenon that may arise in the analysis of spatial or time series data.

## MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks.

## Minkowski–Bouligand dimension

Estimating the box-counting dimension of the coast of Great Britain In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, or more generally in a metric space (X, d).

## Multifractal system

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.

## Neurophysiological Biomarker Toolbox

The Neurophysiological Biomarker Toolbox (NBT) is an open source Matlab toolbox for the computation and integration of neurophysiological biomarkers (e.g., biomarkers based on EEG or MEG recordings).

## Pink noise

Pink noise or noise is a signal or process with a frequency spectrum such that the power spectral density (energy or power per frequency interval) is inversely proportional to the frequency of the signal.

## Random walk

A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.

## 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).

## Self-organized criticality

In physics, self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor.

## Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts).

## Spectral density

The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal.

## Stationary process

In mathematics and statistics, a stationary process (a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time.

## Stochastic process

--> In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables.

## Time series

A time series is a series of data points indexed (or listed or graphed) in time order.

## White noise

In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density.

## Wiener–Khinchin theorem

In applied mathematics, the Wiener–Khinchin theorem, also known as the Wiener–Khintchine theorem and sometimes 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.

## References

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