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Autocorrelation, sometimes known as serial correlation in the
discrete time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
case, measures the
correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
of a
signal A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology. In ...
with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
at different points in time. The analysis of autocorrelation is a mathematical tool for identifying repeating patterns or hidden periodicities within a signal obscured by
noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
. Autocorrelation is widely used in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
,
time domain In mathematics and signal processing, the time domain is a representation of how a signal, function, or data set varies with time. It is used for the analysis of mathematical functions, physical signals or time series of economic or environmental ...
and
time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
to understand the behavior of data over time. Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with
autocovariance In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the proces ...
. Various time series models incorporate autocorrelation, such as
unit root In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if ...
processes,
trend-stationary process In the statistical analysis of time series, a trend-stationary process is a stochastic process from which an underlying trend (function solely of time) can be removed, leaving a stationary process. The trend does not have to be linear. Converse ...
es,
autoregressive process In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregre ...
es, and moving average processes.


Autocorrelation of stochastic processes

In
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the autocorrelation of a real or complex
random process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
is the Pearson correlation between values of the process at different times, as a function of the two times or of the time lag. Let \left\ be a random process, and t be any point in time (t may be an
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
for a
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
process or a
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
for a
continuous-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
process). Then X_t is the value (or realization) produced by a given run of the process at time t. Suppose that the process has
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
\mu_t and
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
\sigma_t^2 at time t, for each t. Then the definition of the autocorrelation function between times t_1 and t_2 isKun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, where \operatorname is the
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
operator and the bar represents
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
. Note that the expectation may not be well defined. Subtracting the mean before multiplication yields the auto-covariance function between times t_1 and t_2: Note that this expression is not well defined for all-time series or processes, because the mean may not exist, or the variance may be zero (for a constant process) or infinite (for processes with distribution lacking well-behaved moments, such as certain types of
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...
).


Definition for wide-sense stationary stochastic process

If \left\ is a
wide-sense stationary process In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. M ...
then the mean \mu and the variance \sigma^2 are time-independent, and further the autocovariance function depends only on the lag between t_1 and t_2: the autocovariance depends only on the time-distance between the pair of values but not on their position in time. This further implies that the autocovariance and autocorrelation can be expressed as a function of the time-lag, and that this would be an even function of the lag \tau=t_2-t_1. This gives the more familiar forms for the autocorrelation function and the auto-covariance function: In particular, note that \operatorname_(0) = \sigma^2 .


Normalization

It is common practice in some disciplines (e.g. statistics and
time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
) to normalize the autocovariance function to get a time-dependent
Pearson correlation coefficient In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviatio ...
. However, in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably. The definition of the autocorrelation coefficient of a stochastic process is \rho_(t_1,t_2) = \frac = \frac . If the function \rho_ is well defined, its value must lie in the range 1,1/math>, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. For a wide-sense stationary (WSS) process, the definition is \rho_(\tau) = \frac = \frac. The normalization is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength of statistical dependence, and because the normalization has an effect on the statistical properties of the estimated autocorrelations.


Properties


Symmetry property

The fact that the autocorrelation function \operatorname_ is an even function can be stated as \operatorname_(t_1,t_2) = \overline respectively for a WSS process: \operatorname_(\tau) = \overline .


Maximum at zero

For a WSS process: \left, \operatorname_(\tau)\ \leq \operatorname_(0) Notice that \operatorname_(0) is always real.


Cauchy–Schwarz inequality

The
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...
, inequality for stochastic processes: \left, \operatorname_(t_1,t_2)\^2 \leq \operatorname\left X_, ^2\right\operatorname\left white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...
signal will have a strong peak (represented by a
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
) at \tau=0 and will be exactly 0 for all other \tau.


Wiener–Khinchin theorem

The Wiener–Khinchin theorem relates the autocorrelation function \operatorname_ to the power spectral density S_ via the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
: \operatorname_(\tau) = \int_^\infty S_(f) e^ \, f S_(f) = \int_^\infty \operatorname_(\tau) e^ \, \tau . 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: \operatorname_(\tau) = \int_^\infty S_(f) \cos(2 \pi f \tau) \, f S_(f) = \int_^\infty \operatorname_(\tau) \cos(2 \pi f \tau) \, \tau .


Autocorrelation of random vectors

The (potentially time-dependent) autocorrelation matrix (also called second moment) of a (potentially time-dependent)
random vector In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge ...
\mathbf = (X_1,\ldots,X_n)^ is an n \times n matrix containing as elements the autocorrelations of all pairs of elements of the random vector \mathbf. The autocorrelation matrix is used in various
digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a ...
algorithms. For a
random vector In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge ...
\mathbf = (X_1,\ldots,X_n)^ containing random elements whose
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
and
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
exist, the autocorrelation matrix is defined byPapoulis, Athanasius, ''Probability, Random variables and Stochastic processes'', McGraw-Hill, 1991 where ^ denotes the
transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...
d matrix of dimensions n \times n. Written component-wise: \operatorname_ = \begin \operatorname _1 X_1& \operatorname _1 X_2& \cdots & \operatorname _1 X_n\\ \\ \operatorname _2 X_1& \operatorname _2 X_2& \cdots & \operatorname _2 X_n\\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname _n X_1& \operatorname _n X_2& \cdots & \operatorname _n X_n\\ \\ \end If \mathbf is a
complex random vector In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If Z_1,\ldots,Z_n are compl ...
, the autocorrelation matrix is instead defined by \operatorname_ \triangleq\ \operatorname mathbf \mathbf^. Here ^ denotes Hermitian transpose. For example, if \mathbf = \left( X_1,X_2,X_3 \right)^ is a random vector, then \operatorname_ is a 3 \times 3 matrix whose (i,j)-th entry is \operatorname _i X_j/math>.


Properties of the autocorrelation matrix

* The autocorrelation matrix is a
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
for complex random vectors and a
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
for real random vectors. * The autocorrelation matrix is a
positive semidefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. M ...
, i.e. \mathbf^ \operatorname_ \mathbf \ge 0 \quad \text \mathbf \in \mathbb^n for a real random vector, and respectively \mathbf^ \operatorname_ \mathbf \ge 0 \quad \text \mathbf \in \mathbb^n in case of a complex random vector. * All eigenvalues of the autocorrelation matrix are real and non-negative. * The ''auto-covariance matrix'' is related to the autocorrelation matrix as follows:\operatorname_ = \operatorname \mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf">mathbf.html" ;"title="\mathbf - \operatorname[\mathbf">\mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf^] = \operatorname_ - \operatorname[\mathbf] \operatorname[\mathbf]^Respectively for complex random vectors:\operatorname_ = \operatorname \mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf">mathbf.html" ;"title="\mathbf - \operatorname[\mathbf">\mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf^] = \operatorname_ - \operatorname[\mathbf] \operatorname[\mathbf]^


Autocorrelation of deterministic signals

In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, 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 or autocovariance function.


Autocorrelation of continuous-time signal

Given a
signal A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology. In ...
f(t), the continuous autocorrelation R_(\tau) is most often defined as the continuous
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
integral of f(t) with itself, at lag \tau. where \overline represents the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
of f(t). Note that the parameter t in the integral is a dummy variable and is only necessary to calculate the integral. It has no specific meaning.


Autocorrelation of discrete-time signal

The discrete autocorrelation R at lag \ell for a discrete-time signal y(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 \begin R_(\tau) &= \operatorname\left (t)\overline\right\\ R_(\ell) &= \operatorname\left (n)\,\overline\right. \end For processes that are not stationary, these will also be functions of t, or n. 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 \begin R_(\tau) &= \lim_ \frac 1 T \int_0^T f(t+\tau)\overline\, t \\ R_(\ell) &= \lim_ \frac 1 N \sum_^ y(n)\,\overline . \end 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.)


Definition for periodic signals

If f is a continuous periodic function of period T, the integration from -\infty to \infty is replaced by integration over any interval _0,t_0+T/math> of length T: R_(\tau) \triangleq \int_^ f(t+\tau) \overline \,dt which is equivalent to R_(\tau) \triangleq \int_^ f(t) \overline \,dt


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. These properties hold for wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry, R_(\tau) = R_(-\tau), which is easy to prove from the definition. In the continuous case, ** the autocorrelation is an even function R_(-\tau) = R_(\tau) when f is a real function, and ** the autocorrelation is a Hermitian function R_(-\tau) = R_^*(\tau) when f is a
complex function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic g ...
. * The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay \tau, , R_(\tau), \leq R_(0). This is a consequence of the rearrangement inequality. The same result holds in the discrete case. * The autocorrelation of a
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
is, itself, periodic with the same period. * The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all \tau) is the sum of the autocorrelations of each function separately. * Since autocorrelation is a specific type of
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
, it maintains all the properties of cross-correlation. * By using the symbol * to represent
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
and g_ is a function which manipulates the function f and is defined as g_(f)(t)=f(-t), the definition for R_(\tau) may be written as:R_(\tau) = (f * g_(\overline))(\tau)


Multi-dimensional autocorrelation

Multi-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable
discrete signal In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
would be R(j,k,\ell) = \sum_ x_\,\overline_ . When mean values are subtracted from signals before computing an autocorrelation function, the resulting function is usually called an auto-covariance function.


Efficient computation

For data expressed as a
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...
sequence, it is frequently necessary to compute the autocorrelation with high computational efficiency. A
brute force method Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equi ...
based on the signal processing definition R_(j) = \sum_n x_n\,\overline_ can be used when the signal size is small. For example, to calculate the autocorrelation of the real signal sequence x = (2,3,-1) (i.e. x_0=2, x_1=3, x_2=-1, and x_i = 0 for all other values of ) by hand, we first recognize that the definition just given is the same as the "usual" multiplication, but with right shifts, where each vertical addition gives the autocorrelation for particular lag values: \begin & 2 & 3 & -1 \\ \times & 2 & 3 & -1 \\ \hline &-2 &-3 & 1 \\ & & 6 & 9 & -3 \\ + & & & 4 & 6 & -2 \\ \hline & -2 & 3 &14 & 3 & -2 \end Thus the required autocorrelation sequence is R_=(-2,3,14,3,-2), where R_(0)=14, R_(-1)= R_(1)=3, and R_(-2)= R_(2) = -2, the autocorrelation for other lag values being zero. In this calculation we do not perform the carry-over operation during addition as is usual in normal multiplication. Note that we can halve the number of operations required by exploiting the inherent symmetry of the autocorrelation. If the signal happens to be periodic, i.e. x=(\ldots,2,3,-1,2,3,-1,\ldots), then we get a circular autocorrelation (similar to circular convolution) where the left and right tails of the previous autocorrelation sequence will overlap and give R_=(\ldots,14,1,1,14,1,1,\ldots) which has the same period as the signal sequence x. The procedure can be regarded as an application of the convolution property of Z-transform of a discrete signal. While the brute force algorithm is order , several efficient algorithms exist which can compute the autocorrelation in order . For example, the Wiener–Khinchin theorem allows computing the autocorrelation from the raw data with two
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
s (FFT): \begin F_R(f) &= \operatorname (t)\\ S(f) &= F_R(f) F^*_R(f) \\ R(\tau) &= \operatorname (f)\end where IFFT denotes the inverse
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
. The asterisk denotes
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
. Alternatively, a multiple correlation can be performed by using brute force calculation for low values, and then progressively binning the data with a
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
ic density to compute higher values, resulting in the same efficiency, but with lower memory requirements.


Estimation

For a
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...
process with known mean and variance for which we observe n observations \, an estimate of the autocorrelation coefficient may be obtained as \hat(k)=\frac \sum_^ (X_t-\mu)(X_-\mu) for any positive integer k. When the true mean \mu and variance \sigma^2 are known, this estimate is
unbiased Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individ ...
. If the true mean and
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of the process are not known there are several possibilities: * If \mu and \sigma^2 are replaced by the standard formulae for sample mean and sample variance, then this is a biased estimate. * 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 squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
. * Other possibilities derive from treating the two portions of data \ and \ separately and calculating separate sample means and/or sample variances for use in defining the estimate. The advantage of estimates of the last type is that the set of estimated autocorrelations, as a function of k, then form a function which is a valid autocorrelation in the sense that it is possible to define a theoretical process having exactly that autocorrelation. Other estimates can suffer from the problem that, if they are used to calculate the variance of a linear combination of the X's, the variance calculated may turn out to be negative.


Regression analysis

In regression analysis using time series data, autocorrelation in a variable of interest is typically modeled either with an
autoregressive model In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregre ...
(AR), a moving average model (MA), their combination as an autoregressive-moving-average model (ARMA), or an extension of the latter called an autoregressive integrated moving average model (ARIMA). With multiple interrelated data series, vector autoregression (VAR) or its extensions are used. In
ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression In statistics, linear regression is a statistical model, model that estimates the relationship ...
(OLS), the adequacy of a model specification can be checked in part by establishing whether there is autocorrelation of the regression residuals. Problematic autocorrelation of the errors, which themselves are unobserved, can generally be detected because it produces autocorrelation in the observable residuals. (Errors are also known as "error terms" in
econometrics Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
.) Autocorrelation of the errors violates the ordinary least squares assumption that the error terms are uncorrelated, meaning that the Gauss Markov theorem does not apply, and that OLS estimators are no longer the Best Linear Unbiased Estimators (
BLUE Blue is one of the three primary colours in the RYB color model, RYB colour model (traditional colour theory), as well as in the RGB color model, RGB (additive) colour model. It lies between Violet (color), violet and cyan on the optical spe ...
). While it does not bias the OLS coefficient estimates, the standard errors tend to be underestimated (and the t-scores overestimated) when the autocorrelations of the errors at low lags are positive. 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. The Durbin-Watson can be linearly mapped however to the Pearson correlation between values and their lags. 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 Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
from this auxiliary regression is ''TR''2, where ''T'' is the sample size and ''R''2 is the coefficient of determination. Under the
null hypothesis The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
of no autocorrelation, this statistic is asymptotically distributed as \chi^2 with ''k'' degrees of freedom. Responses to nonzero autocorrelation include
generalized least squares In statistics, generalized least squares (GLS) is a method used to estimate the unknown parameters in a Linear regression, linear regression model. It is used when there is a non-zero amount of correlation between the Residual (statistics), resi ...
and the Newey–West HAC estimator (Heteroskedasticity and Autocorrelation Consistent). In the estimation of a moving average model (MA), the autocorrelation function is used to determine the appropriate number of lagged error terms to be included. This is based on the fact that for an MA process of order ''q'', we have R(\tau) \neq 0, for \tau = 0,1, \ldots , q, and R(\tau) = 0, for \tau >q.


Applications

Autocorrelation's ability to find repeating patterns in
data Data ( , ) are a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted for ...
yields many applications, including: * Autocorrelation analysis is used heavily in fluorescence correlation spectroscopy to provide quantitative insight into molecular-level diffusion and chemical reactions. * Another application of autocorrelation is the measurement of optical spectra and the measurement of very-short-duration
light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
pulses produced by
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
s, both using optical autocorrelators. * Autocorrelation is used to analyze dynamic light scattering data, which notably enables determination of the particle size distributions of nanometer-sized particles or micelles suspended in a fluid. A laser shining into the mixture produces a speckle pattern that results from the motion of the particles. Autocorrelation of the signal can be analyzed in terms of the diffusion of the particles. From this, knowing the viscosity of the fluid, the sizes of the particles can be calculated. * Utilized in the GPS system to correct for the
propagation delay Propagation delay is the time duration taken for a signal to reach its destination, for example in the electromagnetic field, a wire, speed of sound, gas, fluid or seismic wave, solid body. Physics * An electromagnetic wave travelling through ...
, or time shift, between the point of time at the transmission of the
carrier signal In telecommunications, a carrier wave, carrier signal, or just carrier, is a periodic waveform (usually sinusoidal) that conveys information through a process called ''modulation''. One or more of the wave's properties, such as amplitude or frequ ...
at the satellites, and the point of time at the receiver on the ground. This is done by the receiver generating a replica signal of the 1,023-bit C/A (Coarse/Acquisition) code, and generating lines of code chips 1,1in packets of ten at a time, or 10,230 chips (1,023 × 10), shifting slightly as it goes along in order to accommodate for the
doppler shift The Doppler effect (also Doppler shift) is the change in the frequency of a wave in relation to an observer who is moving relative to the source of the wave. The ''Doppler effect'' is named after the physicist Christian Doppler, who described t ...
in the incoming satellite signal, until the receiver replica signal and the satellite signal codes match up. * The small-angle X-ray scattering intensity of a nanostructured system is the Fourier transform of the spatial autocorrelation function of the
electron density Electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typical ...
. *In
surface science Surface science is the study of physical and chemical phenomena that occur at the interface of two phases, including solid–liquid interfaces, solid– gas interfaces, solid– vacuum interfaces, and liquid– gas interfaces. It includes the ...
and
scanning probe microscopy Scanning probe microscopy (SPM) is a branch of microscopy that forms images of surfaces using a physical probe that scans the specimen. SPM was founded in 1981, with the invention of the scanning tunneling microscope, an instrument for imaging ...
, autocorrelation is used to establish a link between surface morphology and functional characteristics. * In optics, normalized autocorrelations and cross-correlations give the
degree of coherence In quantum optics, correlation functions are used to characterize the statistical and Coherence (physics), coherence properties – the ability of waves to interfere – of electromagnetic radiation, like optical light. Higher order coherence or ...
of an electromagnetic field. * In
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, autocorrelation can determine the
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
of
pulsar A pulsar (''pulsating star, on the model of quasar'') is a highly magnetized rotating neutron star that emits beams of electromagnetic radiation out of its Poles of astronomical bodies#Magnetic poles, magnetic poles. This radiation can be obse ...
s. * In
music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
, autocorrelation (when applied at time scales smaller than a second) is used as a pitch detection algorithm for both instrument tuners and "Auto Tune" (used as a
distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal ...
effect or to fix intonation). When applied at time scales larger than a second, autocorrelation can identify the musical beat, for example to determine
tempo In musical terminology, tempo (Italian for 'time'; plural 'tempos', or from the Italian plural), measured in beats per minute, is the speed or pace of a given musical composition, composition, and is often also an indication of the composition ...
. * Autocorrelation in space rather than time, via the Patterson function, is used by X-ray diffractionists to help recover the "Fourier phase information" on atom positions not available through diffraction alone. * In statistics, spatial autocorrelation between sample locations also helps one estimate mean value uncertainties when sampling a heterogeneous population. * The SEQUEST algorithm for analyzing mass spectra makes use of autocorrelation in conjunction with
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
to score the similarity of an observed spectrum to an idealized spectrum representing a
peptide Peptides are short chains of amino acids linked by peptide bonds. A polypeptide is a longer, continuous, unbranched peptide chain. Polypeptides that have a molecular mass of 10,000 Da or more are called proteins. Chains of fewer than twenty am ...
. * In
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
, autocorrelation is used to study and characterize the spatial distribution of galaxies in the universe and in multi-wavelength observations of low mass X-ray binaries. * In
panel data In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time. Time series and ...
, spatial autocorrelation refers to correlation of a variable with itself through space. * In analysis of
Markov chain Monte Carlo In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that ...
data, autocorrelation must be taken into account for correct error determination. * In
geosciences Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres ...
(specifically in
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
) it can be used to compute an autocorrelation seismic attribute, out of a 3D seismic survey of the underground. * In
medical ultrasound Medical ultrasound includes Medical diagnosis, diagnostic techniques (mainly medical imaging, imaging) using ultrasound, as well as therapeutic ultrasound, therapeutic applications of ultrasound. In diagnosis, it is used to create an image of ...
imaging, autocorrelation is used to visualize blood flow. * In intertemporal portfolio choice, the presence or absence of autocorrelation in an asset's
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment over a specified time period, such as i ...
can affect the optimal portion of the portfolio to hold in that asset. * In numerical relays, autocorrelation has been used to accurately measure power system frequency.


Serial dependence

Serial dependence is closely linked to the notion of autocorrelation, but represents a distinct concept (see
Correlation and dependence In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
). In particular, it is possible to have serial dependence but no (linear) correlation. In some fields however, the two terms are used as synonyms. A
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
has serial dependence if the value at some time t in the series is statistically dependent on the value at another time s. A series is serially independent if there is no dependence between any pair. If a time series \left\ is stationary, then statistical dependence between the pair (X_t,X_s) would imply that there is statistical dependence between all pairs of values at the same lag \tau=s-t.


See also

* Autocorrelation matrix * Autocorrelation of a formal word * Autocorrelation technique * Autocorrelator * Cochrane–Orcutt estimation (transformation for autocorrelated error terms) * Correlation function * Correlogram *
Cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
*
CUSUM In statistical process control, statistical quality control, the CUSUM (or cumulative sum control chart) is a sequential analysis technique developed by E. S. Page of the University of Cambridge. It is typically used for monitoring change detecti ...
* Fluorescence correlation spectroscopy * Optical autocorrelation * Partial autocorrelation function * Phylogenetic autocorrelation (Galton's problem) * Pitch detection algorithm * Prais–Winsten transformation * Scaled correlation * Triple correlation * Unbiased estimation of standard deviation


References


Further reading

* * * * Solomon W. Golomb, and Guang Gong
Signal design for good correlation: for wireless communication, cryptography, and radar
Cambridge University Press, 2005. * Klapetek, Petr (2018).
Quantitative Data Processing in Scanning Probe Microscopy: SPM Applications for Nanometrology
' (Second ed.). Elsevier. pp. 108–112 . * {{Statistics, analysis Signal processing Time domain analysis>X_, ^2\right/math>


Autocorrelation of white noise

The autocorrelation of a continuous-time
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...
signal will have a strong peak (represented by a
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
) at \tau=0 and will be exactly 0 for all other \tau.


Wiener–Khinchin theorem

The Wiener–Khinchin theorem relates the autocorrelation function \operatorname_ to the power spectral density S_ via the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
: \operatorname_(\tau) = \int_^\infty S_(f) e^ \, f S_(f) = \int_^\infty \operatorname_(\tau) e^ \, \tau . 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: \operatorname_(\tau) = \int_^\infty S_(f) \cos(2 \pi f \tau) \, f S_(f) = \int_^\infty \operatorname_(\tau) \cos(2 \pi f \tau) \, \tau .


Autocorrelation of random vectors

The (potentially time-dependent) autocorrelation matrix (also called second moment) of a (potentially time-dependent)
random vector In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge ...
\mathbf = (X_1,\ldots,X_n)^ is an n \times n matrix containing as elements the autocorrelations of all pairs of elements of the random vector \mathbf. The autocorrelation matrix is used in various
digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a ...
algorithms. For a
random vector In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge ...
\mathbf = (X_1,\ldots,X_n)^ containing random elements whose
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
and
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
exist, the autocorrelation matrix is defined byPapoulis, Athanasius, ''Probability, Random variables and Stochastic processes'', McGraw-Hill, 1991 where ^ denotes the
transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...
d matrix of dimensions n \times n. Written component-wise: \operatorname_ = \begin \operatorname _1 X_1& \operatorname _1 X_2& \cdots & \operatorname _1 X_n\\ \\ \operatorname _2 X_1& \operatorname _2 X_2& \cdots & \operatorname _2 X_n\\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname _n X_1& \operatorname _n X_2& \cdots & \operatorname _n X_n\\ \\ \end If \mathbf is a
complex random vector In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If Z_1,\ldots,Z_n are compl ...
, the autocorrelation matrix is instead defined by \operatorname_ \triangleq\ \operatorname mathbf \mathbf^. Here ^ denotes Hermitian transpose. For example, if \mathbf = \left( X_1,X_2,X_3 \right)^ is a random vector, then \operatorname_ is a 3 \times 3 matrix whose (i,j)-th entry is \operatorname _i X_j/math>.


Properties of the autocorrelation matrix

* The autocorrelation matrix is a
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
for complex random vectors and a
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
for real random vectors. * The autocorrelation matrix is a
positive semidefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. M ...
, i.e. \mathbf^ \operatorname_ \mathbf \ge 0 \quad \text \mathbf \in \mathbb^n for a real random vector, and respectively \mathbf^ \operatorname_ \mathbf \ge 0 \quad \text \mathbf \in \mathbb^n in case of a complex random vector. * All eigenvalues of the autocorrelation matrix are real and non-negative. * The ''auto-covariance matrix'' is related to the autocorrelation matrix as follows:\operatorname_ = \operatorname \mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf">mathbf.html" ;"title="\mathbf - \operatorname[\mathbf">\mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf^] = \operatorname_ - \operatorname[\mathbf] \operatorname[\mathbf]^Respectively for complex random vectors:\operatorname_ = \operatorname \mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf">mathbf.html" ;"title="\mathbf - \operatorname[\mathbf">\mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf^] = \operatorname_ - \operatorname[\mathbf] \operatorname[\mathbf]^


Autocorrelation of deterministic signals

In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, 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 or autocovariance function.


Autocorrelation of continuous-time signal

Given a
signal A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology. In ...
f(t), the continuous autocorrelation R_(\tau) is most often defined as the continuous
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
integral of f(t) with itself, at lag \tau. where \overline represents the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
of f(t). Note that the parameter t in the integral is a dummy variable and is only necessary to calculate the integral. It has no specific meaning.


Autocorrelation of discrete-time signal

The discrete autocorrelation R at lag \ell for a discrete-time signal y(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 \begin R_(\tau) &= \operatorname\left (t)\overline\right\\ R_(\ell) &= \operatorname\left (n)\,\overline\right. \end For processes that are not stationary, these will also be functions of t, or n. 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 \begin R_(\tau) &= \lim_ \frac 1 T \int_0^T f(t+\tau)\overline\, t \\ R_(\ell) &= \lim_ \frac 1 N \sum_^ y(n)\,\overline . \end 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.)


Definition for periodic signals

If f is a continuous periodic function of period T, the integration from -\infty to \infty is replaced by integration over any interval _0,t_0+T/math> of length T: R_(\tau) \triangleq \int_^ f(t+\tau) \overline \,dt which is equivalent to R_(\tau) \triangleq \int_^ f(t) \overline \,dt


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. These properties hold for wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry, R_(\tau) = R_(-\tau), which is easy to prove from the definition. In the continuous case, ** the autocorrelation is an even function R_(-\tau) = R_(\tau) when f is a real function, and ** the autocorrelation is a Hermitian function R_(-\tau) = R_^*(\tau) when f is a
complex function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic g ...
. * The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay \tau, , R_(\tau), \leq R_(0). This is a consequence of the rearrangement inequality. The same result holds in the discrete case. * The autocorrelation of a
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
is, itself, periodic with the same period. * The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all \tau) is the sum of the autocorrelations of each function separately. * Since autocorrelation is a specific type of
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
, it maintains all the properties of cross-correlation. * By using the symbol * to represent
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
and g_ is a function which manipulates the function f and is defined as g_(f)(t)=f(-t), the definition for R_(\tau) may be written as:R_(\tau) = (f * g_(\overline))(\tau)


Multi-dimensional autocorrelation

Multi-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable
discrete signal In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
would be R(j,k,\ell) = \sum_ x_\,\overline_ . When mean values are subtracted from signals before computing an autocorrelation function, the resulting function is usually called an auto-covariance function.


Efficient computation

For data expressed as a
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...
sequence, it is frequently necessary to compute the autocorrelation with high computational efficiency. A
brute force method Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equi ...
based on the signal processing definition R_(j) = \sum_n x_n\,\overline_ can be used when the signal size is small. For example, to calculate the autocorrelation of the real signal sequence x = (2,3,-1) (i.e. x_0=2, x_1=3, x_2=-1, and x_i = 0 for all other values of ) by hand, we first recognize that the definition just given is the same as the "usual" multiplication, but with right shifts, where each vertical addition gives the autocorrelation for particular lag values: \begin & 2 & 3 & -1 \\ \times & 2 & 3 & -1 \\ \hline &-2 &-3 & 1 \\ & & 6 & 9 & -3 \\ + & & & 4 & 6 & -2 \\ \hline & -2 & 3 &14 & 3 & -2 \end Thus the required autocorrelation sequence is R_=(-2,3,14,3,-2), where R_(0)=14, R_(-1)= R_(1)=3, and R_(-2)= R_(2) = -2, the autocorrelation for other lag values being zero. In this calculation we do not perform the carry-over operation during addition as is usual in normal multiplication. Note that we can halve the number of operations required by exploiting the inherent symmetry of the autocorrelation. If the signal happens to be periodic, i.e. x=(\ldots,2,3,-1,2,3,-1,\ldots), then we get a circular autocorrelation (similar to circular convolution) where the left and right tails of the previous autocorrelation sequence will overlap and give R_=(\ldots,14,1,1,14,1,1,\ldots) which has the same period as the signal sequence x. The procedure can be regarded as an application of the convolution property of Z-transform of a discrete signal. While the brute force algorithm is order , several efficient algorithms exist which can compute the autocorrelation in order . For example, the Wiener–Khinchin theorem allows computing the autocorrelation from the raw data with two
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
s (FFT): \begin F_R(f) &= \operatorname (t)\\ S(f) &= F_R(f) F^*_R(f) \\ R(\tau) &= \operatorname (f)\end where IFFT denotes the inverse
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
. The asterisk denotes
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
. Alternatively, a multiple correlation can be performed by using brute force calculation for low values, and then progressively binning the data with a
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
ic density to compute higher values, resulting in the same efficiency, but with lower memory requirements.


Estimation

For a
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...
process with known mean and variance for which we observe n observations \, an estimate of the autocorrelation coefficient may be obtained as \hat(k)=\frac \sum_^ (X_t-\mu)(X_-\mu) for any positive integer k. When the true mean \mu and variance \sigma^2 are known, this estimate is
unbiased Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individ ...
. If the true mean and
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of the process are not known there are several possibilities: * If \mu and \sigma^2 are replaced by the standard formulae for sample mean and sample variance, then this is a biased estimate. * 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 squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
. * Other possibilities derive from treating the two portions of data \ and \ separately and calculating separate sample means and/or sample variances for use in defining the estimate. The advantage of estimates of the last type is that the set of estimated autocorrelations, as a function of k, then form a function which is a valid autocorrelation in the sense that it is possible to define a theoretical process having exactly that autocorrelation. Other estimates can suffer from the problem that, if they are used to calculate the variance of a linear combination of the X's, the variance calculated may turn out to be negative.


Regression analysis

In regression analysis using time series data, autocorrelation in a variable of interest is typically modeled either with an
autoregressive model In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregre ...
(AR), a moving average model (MA), their combination as an autoregressive-moving-average model (ARMA), or an extension of the latter called an autoregressive integrated moving average model (ARIMA). With multiple interrelated data series, vector autoregression (VAR) or its extensions are used. In
ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression In statistics, linear regression is a statistical model, model that estimates the relationship ...
(OLS), the adequacy of a model specification can be checked in part by establishing whether there is autocorrelation of the regression residuals. Problematic autocorrelation of the errors, which themselves are unobserved, can generally be detected because it produces autocorrelation in the observable residuals. (Errors are also known as "error terms" in
econometrics Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
.) Autocorrelation of the errors violates the ordinary least squares assumption that the error terms are uncorrelated, meaning that the Gauss Markov theorem does not apply, and that OLS estimators are no longer the Best Linear Unbiased Estimators (
BLUE Blue is one of the three primary colours in the RYB color model, RYB colour model (traditional colour theory), as well as in the RGB color model, RGB (additive) colour model. It lies between Violet (color), violet and cyan on the optical spe ...
). While it does not bias the OLS coefficient estimates, the standard errors tend to be underestimated (and the t-scores overestimated) when the autocorrelations of the errors at low lags are positive. 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. The Durbin-Watson can be linearly mapped however to the Pearson correlation between values and their lags. 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 Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
from this auxiliary regression is ''TR''2, where ''T'' is the sample size and ''R''2 is the coefficient of determination. Under the
null hypothesis The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
of no autocorrelation, this statistic is asymptotically distributed as \chi^2 with ''k'' degrees of freedom. Responses to nonzero autocorrelation include
generalized least squares In statistics, generalized least squares (GLS) is a method used to estimate the unknown parameters in a Linear regression, linear regression model. It is used when there is a non-zero amount of correlation between the Residual (statistics), resi ...
and the Newey–West HAC estimator (Heteroskedasticity and Autocorrelation Consistent). In the estimation of a moving average model (MA), the autocorrelation function is used to determine the appropriate number of lagged error terms to be included. This is based on the fact that for an MA process of order ''q'', we have R(\tau) \neq 0, for \tau = 0,1, \ldots , q, and R(\tau) = 0, for \tau >q.


Applications

Autocorrelation's ability to find repeating patterns in
data Data ( , ) are a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted for ...
yields many applications, including: * Autocorrelation analysis is used heavily in fluorescence correlation spectroscopy to provide quantitative insight into molecular-level diffusion and chemical reactions. * Another application of autocorrelation is the measurement of optical spectra and the measurement of very-short-duration
light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
pulses produced by
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
s, both using optical autocorrelators. * Autocorrelation is used to analyze dynamic light scattering data, which notably enables determination of the particle size distributions of nanometer-sized particles or micelles suspended in a fluid. A laser shining into the mixture produces a speckle pattern that results from the motion of the particles. Autocorrelation of the signal can be analyzed in terms of the diffusion of the particles. From this, knowing the viscosity of the fluid, the sizes of the particles can be calculated. * Utilized in the GPS system to correct for the
propagation delay Propagation delay is the time duration taken for a signal to reach its destination, for example in the electromagnetic field, a wire, speed of sound, gas, fluid or seismic wave, solid body. Physics * An electromagnetic wave travelling through ...
, or time shift, between the point of time at the transmission of the
carrier signal In telecommunications, a carrier wave, carrier signal, or just carrier, is a periodic waveform (usually sinusoidal) that conveys information through a process called ''modulation''. One or more of the wave's properties, such as amplitude or frequ ...
at the satellites, and the point of time at the receiver on the ground. This is done by the receiver generating a replica signal of the 1,023-bit C/A (Coarse/Acquisition) code, and generating lines of code chips 1,1in packets of ten at a time, or 10,230 chips (1,023 × 10), shifting slightly as it goes along in order to accommodate for the
doppler shift The Doppler effect (also Doppler shift) is the change in the frequency of a wave in relation to an observer who is moving relative to the source of the wave. The ''Doppler effect'' is named after the physicist Christian Doppler, who described t ...
in the incoming satellite signal, until the receiver replica signal and the satellite signal codes match up. * The small-angle X-ray scattering intensity of a nanostructured system is the Fourier transform of the spatial autocorrelation function of the
electron density Electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typical ...
. *In
surface science Surface science is the study of physical and chemical phenomena that occur at the interface of two phases, including solid–liquid interfaces, solid– gas interfaces, solid– vacuum interfaces, and liquid– gas interfaces. It includes the ...
and
scanning probe microscopy Scanning probe microscopy (SPM) is a branch of microscopy that forms images of surfaces using a physical probe that scans the specimen. SPM was founded in 1981, with the invention of the scanning tunneling microscope, an instrument for imaging ...
, autocorrelation is used to establish a link between surface morphology and functional characteristics. * In optics, normalized autocorrelations and cross-correlations give the
degree of coherence In quantum optics, correlation functions are used to characterize the statistical and Coherence (physics), coherence properties – the ability of waves to interfere – of electromagnetic radiation, like optical light. Higher order coherence or ...
of an electromagnetic field. * In
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, autocorrelation can determine the
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
of
pulsar A pulsar (''pulsating star, on the model of quasar'') is a highly magnetized rotating neutron star that emits beams of electromagnetic radiation out of its Poles of astronomical bodies#Magnetic poles, magnetic poles. This radiation can be obse ...
s. * In
music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
, autocorrelation (when applied at time scales smaller than a second) is used as a pitch detection algorithm for both instrument tuners and "Auto Tune" (used as a
distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal ...
effect or to fix intonation). When applied at time scales larger than a second, autocorrelation can identify the musical beat, for example to determine
tempo In musical terminology, tempo (Italian for 'time'; plural 'tempos', or from the Italian plural), measured in beats per minute, is the speed or pace of a given musical composition, composition, and is often also an indication of the composition ...
. * Autocorrelation in space rather than time, via the Patterson function, is used by X-ray diffractionists to help recover the "Fourier phase information" on atom positions not available through diffraction alone. * In statistics, spatial autocorrelation between sample locations also helps one estimate mean value uncertainties when sampling a heterogeneous population. * The SEQUEST algorithm for analyzing mass spectra makes use of autocorrelation in conjunction with
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
to score the similarity of an observed spectrum to an idealized spectrum representing a
peptide Peptides are short chains of amino acids linked by peptide bonds. A polypeptide is a longer, continuous, unbranched peptide chain. Polypeptides that have a molecular mass of 10,000 Da or more are called proteins. Chains of fewer than twenty am ...
. * In
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
, autocorrelation is used to study and characterize the spatial distribution of galaxies in the universe and in multi-wavelength observations of low mass X-ray binaries. * In
panel data In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time. Time series and ...
, spatial autocorrelation refers to correlation of a variable with itself through space. * In analysis of
Markov chain Monte Carlo In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that ...
data, autocorrelation must be taken into account for correct error determination. * In
geosciences Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres ...
(specifically in
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
) it can be used to compute an autocorrelation seismic attribute, out of a 3D seismic survey of the underground. * In
medical ultrasound Medical ultrasound includes Medical diagnosis, diagnostic techniques (mainly medical imaging, imaging) using ultrasound, as well as therapeutic ultrasound, therapeutic applications of ultrasound. In diagnosis, it is used to create an image of ...
imaging, autocorrelation is used to visualize blood flow. * In intertemporal portfolio choice, the presence or absence of autocorrelation in an asset's
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment over a specified time period, such as i ...
can affect the optimal portion of the portfolio to hold in that asset. * In numerical relays, autocorrelation has been used to accurately measure power system frequency.


Serial dependence

Serial dependence is closely linked to the notion of autocorrelation, but represents a distinct concept (see
Correlation and dependence In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
). In particular, it is possible to have serial dependence but no (linear) correlation. In some fields however, the two terms are used as synonyms. A
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
has serial dependence if the value at some time t in the series is statistically dependent on the value at another time s. A series is serially independent if there is no dependence between any pair. If a time series \left\ is stationary, then statistical dependence between the pair (X_t,X_s) would imply that there is statistical dependence between all pairs of values at the same lag \tau=s-t.


See also

* Autocorrelation matrix * Autocorrelation of a formal word * Autocorrelation technique * Autocorrelator * Cochrane–Orcutt estimation (transformation for autocorrelated error terms) * Correlation function * Correlogram *
Cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
*
CUSUM In statistical process control, statistical quality control, the CUSUM (or cumulative sum control chart) is a sequential analysis technique developed by E. S. Page of the University of Cambridge. It is typically used for monitoring change detecti ...
* Fluorescence correlation spectroscopy * Optical autocorrelation * Partial autocorrelation function * Phylogenetic autocorrelation (Galton's problem) * Pitch detection algorithm * Prais–Winsten transformation * Scaled correlation * Triple correlation * Unbiased estimation of standard deviation


References


Further reading

* * * * Solomon W. Golomb, and Guang Gong
Signal design for good correlation: for wireless communication, cryptography, and radar
Cambridge University Press, 2005. * Klapetek, Petr (2018).
Quantitative Data Processing in Scanning Probe Microscopy: SPM Applications for Nanometrology
' (Second ed.). Elsevier. pp. 108–112 . * {{Statistics, analysis Signal processing Time domain analysis