A plot showing 100 random numbers with a "hidden" sine function, and an autocorrelation of the series on the bottom.

Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. Informally, it is a measure of how well a signal matches a time-shifted version of itself, as a function of the amount of time shift. More precisely, it is the cross-correlation of a signal with itself. Autocorrelation is useful for finding repeating patterns in a signal, such as determining the presence of a periodic signal which has been buried under noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. Image File history File links Acf. ... Image File history File links Acf. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar... A missing fundamental is a missing fundamental frequency which higher frequencies refer to. ... In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. ...

Definitions

Different definitions of autocorrelation are in use depending on the field of study which is being considered and not all of them are equivalent. In some fields, the term is used interchangeably with autocovariance. Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ...

Statistics

In statistics, the autocorrelation function (ACF) of a random process describes the correlation between the process at different points in time. Let X_t be the value of the process at time t (where t may be an integer for a discrete-time process or a real number for a continuous-time process). If X_t has mean μ and variance σ² then the definition of the ACF is A graph of a normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... Positive linear correlations between 1000 pairs of numbers. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

where E is the expected value operator. Note that this expression is not well-defined for all time series or processes, since the variance σ² may be zero (for a constant process) or infinite. If the function R is well-defined its value must lie in the range [−1, 1], with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

If X_t is second-order stationary then the ACF depends only on the difference between t and s and can be expressed as a function of a single variable. This gives the more familiar form In the mathematical sciences, a stationary process (or strict(ly) stationary process) is a stochastic process in which the probability density function of some random variable X does not change over time or position. ...

where k is the lag, | t − s |. It is common practice in many disciplines to drop the normalization by σ² and use the term autocorrelation interchangeably with autocovariance.

For a discrete time series of length n {X₁, X₂, … X_n} with known mean and variance, an estimate of the autocorrelation may be obtained as

for any positive integer k < n. When the true mean μ is known, this estimate is unbiased. However, if the true mean and variance of the process are not known, and μ and σ² are replaced by the standard formulae for sample mean and sample variance, then this is a biased estimate. As an alternative, a periodogram based estimate replaces n-k in the above formula with n. This estimate is always biased; however, it usually has a smaller mean square error.^[1][2]

Signal processing

In signal processing, the above definition is often used without the normalization, that is, without subtracting the mean and dividing by the variance. When the autocorrelation function is normalized by mean and variance, it is sometimes referred to as the autocorrelation coefficient.^[3] Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ...

Given a signal f(t), the continuous autocorrelation R_ff(τ) is most often defined as the continuous cross-correlation integral of f(t) with itself, at lag τ.

where represents the complex conjugate and * represents convolution. For a real function, . In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

The discrete autocorrelation R at lag j for a discrete signal x_n is

The above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that "last forever" are treated instead as random processes, in which case different definitions are needed, based on expected values. For wide-sense-stationary random processes, the autocorrelations are defined as In the mathematical sciences, a stationary process (or strict(ly) stationary process) is a stochastic process in which the probability density function of some random variable X does not change over time or position. ...

For processes that are not stationary, these will also be functions of t, or n. Stationary can mean: Look up stationary in Wiktionary, the free dictionary. ...

For processes that are also ergodic, the expectation can be replaced by the limit of a time average. The autocorrelation of an ergodic process is sometimes defined as or equated to^[3] The ergodic hypothesis is a postulate of thermodynamics. ...

These definitions have the advantage that they give sensible well-defined single-parameter results for periodic functions, even when those functions are not the output of stationary ergodic processes.

Alternatively, signals that last forever can be treated by a short-time autocorrelation function analysis, using finite time integrals. (See short-time Fourier transform for a related process.) The short-time Fourier transform (STFT), or short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of a signal as it changes over time. ...

Multi-dimensional autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable discrete signal would be 2-dimensional renderings (ie. ... The space we live in is three-dimensional space. ... A discrete signal is a signal that has been sampled from a continuous signal. ...

When mean values are subtracted from signals before computing an autocorrelation function, the resulting function is usually called an auto-covariance function.

Properties

In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases.

• A fundamental property of the autocorrelation is symmetry, R(i) = R(−i), which is easy to prove from the definition. In the continuous case, the autocorrelation is an even function

when f is a real function and the autocorrelation is a Hermitian function

when f is a complex function.

• The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay τ, . This is a consequence of the Cauchy–Schwarz inequality. The same result holds in the discrete case.

• The autocorrelation of a periodic function is, itself, periodic with the very same period.

• The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all τ) is the sum of the autocorrelations of each function separately.

• Since autocorrelation is a specific type of cross-correlation, it maintains all the properties of cross-correlation.

• The autocorrelation of a continuous-time white noise signal will have a strong peak (represented by a Dirac delta function) at τ = 0 and will be absolutely 0 for all other τ.

• The Wiener–Khinchin theorem relates the autocorrelation function to the power spectral density via the Fourier transform:

• For real-valued functions, the symmetric autocorrelation function has a real symmetric transform, so the Wiener–Khinchin theorem can be re-expressed in terms of real cosines only:

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ... A Hermitian function is complex conjugated when its variable changes sign: for all in the domain of definition of . ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Bunyakovski–Schwarz inequality, named after Augustin Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar... Calculated spectrum of a generated approximation of white noise White noise is a random signal (or process) with a flat power spectral density. ... The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... The Wiener–Khinchin theorem (also known as the Wiener–Khintchine theorem and sometimes as the Khinchin–Kolmogorov theorem) states that the power spectral density of a wide-sense-stationary random process is the Fourier Transform of the corresponding autocorrelation function. ... In applied mathematics and physics, the spectral density is a general concept applied to a signal which may have any physical dimensions or none at all. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... The Wiener–Khinchin theorem (also known as the Wiener–Khintchine theorem and sometimes as the Khinchin–Kolmogorov theorem) states that the power spectral density of a wide-sense-stationary random process is the Fourier Transform of the corresponding autocorrelation function. ...

Regression analysis

In regression analysis using time series data, autocorrelation of the residuals ("error terms", in econometrics) is a problem. In statistics, regression analysis examines the relation of a dependent variable (response variable) to specified independent variables (explanatory variables). ... In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ... Econometrics literally means economic measurement. It is a combination of mathematical economics and statistics. ...

Autocorrelation violates the OLS assumption that the error terms are uncorrelated. While it does not bias the OLS coefficient estimates, the standard errors tend to be underestimated (and the t-scores overestimated).

The traditional test for the presence of first-order autocorrelation is the Durbin-Watson statistic or, if the explanatory variables include a lagged dependent variable, Durbin's h statistic. A more flexible test, covering autocorrelation of higher orders and applicable whether or not the regressors include lags of the dependent variable, is the Breusch-Godfrey test. This involves an auxiliary regression, wherein the residuals obtained from estimating the model of interest are regressed on (a) the original regressors and (b) k lags of the residuals, where k is the order of the test. The simplest version of the test statistic from this auxiliary regression is TR², where T is the sample size and R² is the coefficient of determination. Under the null hypothesis of no autocorrelation, this statistic is asymptotically distributed as Χ² with k degrees of freedom. The Durbin-Watson statistic is a test statistic used to detect the presence of autocorrelation in the residuals from a regression analysis. ... The Durbin-Watson statistic is a test statistic used to detect the presence of autocorrelation in the residuals from a regression analysis. ... In statistics, the coefficient of determination R2 is the proportion of variability in a data set that is accounted for by a statistical model. ...

Responses to autocorrelation include generalized least squares or Newey-West standard errors. In statistics the linear model can be expressed by saying where Y is an n×1 column vector of random variables, X is an n×p matrix of known (i. ...

Applications

• One application of autocorrelation is the measurement of optical spectra and the measurement of very-short-duration light pulses produced by lasers, both using optical autocorrelators.

• In optics, normalized autocorrelations and cross-correlations give the degree of coherence of an electromagnetic field.

• In signal processing, autocorrelation can give information about repeating events like musical beats or pulsar frequencies, though it cannot tell the position in time of the beat.

The visible spectrum is the portion of the optical spectrum (light or electromagnetic spectrum) that is visible to the human eye. ... This article does not cite any references or sources. ... In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is on the order of the femtosecond ( second). ... Experiment with a laser (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: Light Amplification by Stimulated Emission of Radiation. ... In optics, various autocorrelation functions can be experimentally realized. ... In optics, correlation functions are used to characterize the statistical and coherence properties of an electromagnetic field. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... // Music is an art form consisting of sound and silence expressed through time. ... putang ina. ... It has been suggested that Radio pulsar be merged into this article or section. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...

See also

• Cross-correlation
• Fluorescence correlation spectroscopy
• Optical autocorrelation

In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar... Also known as FCS Categories: Science stubs | Spectroscopy | Biochemistry | Cell biology | Biotechnology | Microscopes | Biophysics ... In optics, various autocorrelation functions can be experimentally realized. ...

External links

• Eric W. Weisstein, Autocorrelation at MathWorld.
• Autocorrelation articles in Comp.DSP (DSP usenet group).

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

References

1. ^ Spectral analysis and time series, M.B. Priestley (London, New York : Academic Press, 1982)
2. ^ Percival, Donald B.; Andrew T. Walden (1993). Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge University Press, pp190--195. ISBN 0-521-43541-2. 
3. ^ ^{a b} Patrick F. Dunn, Measurement and Data Analysis for Engineering and Science, New York: McGraw–Hill, 2005 ISBN 0-07-282538-3