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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 "po ...
case, is 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 with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a
periodic signal A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
obscured by
noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arise ...
, or identifying the
missing fundamental frequency A harmonic sound is said to have a missing fundamental, suppressed fundamental, or phantom fundamental when its overtones suggest a fundamental frequency but the sound lacks a component at the fundamental frequency itself. The brain perceives the ...
in a signal implied by its
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
frequencies. It is often 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, and scientific measurements. Signal processing techniq ...
for analyzing functions or series of values, such as time domain signals. 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 process ...
. Unit root 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. Conversely ...
es, autoregressive processes, and
moving average process In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series. The moving-average model specifies that the output variable is cross-correlated with a ...
es are specific forms of processes with autocorrelation.


Auto-correlation of stochastic processes

In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the autocorrelation of a real or complex random process is the
Pearson correlation In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...
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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
for a discrete-time 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 "po ...
process). Then X_t is the value (or realization) produced by a given
run Run(s) or RUN may refer to: Places * Run (island), one of the Banda Islands in Indonesia * Run (stream), a stream in the Dutch province of North Brabant People * Run (rapper), Joseph Simmons, now known as "Reverend Run", from the hip-hop group ...
of the process at time t. Suppose that the process has
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...
\mu_t and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
\sigma_t^2 at time t, for each t. Then the definition of the auto-correlation 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, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
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, then) the complex conjugate of a + bi is equal to 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 proportional relative change in the other quantity, inde ...
).


Definition for wide-sense stationary stochastic process

If \left\ is a
wide-sense stationary process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
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 auto-correlation can be expressed as a function of the time-lag, and that this would be an
even function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power seri ...
of the lag \tau=t_2-t_1. This gives the more familiar forms for the auto-correlation 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) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. 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 auto-correlation 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 In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
(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 Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...
, and because the normalization has an effect on the statistical properties of the estimated autocorrelations.


Properties


Symmetry property

The fact that the auto-correlation function \operatorname_ is an
even function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power seri ...
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 considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
, 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, ...
signal will have a strong peak (represented by a
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
) 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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
: \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 .


Auto-correlation of random vectors

The (potentially time-dependent) auto-correlation matrix (also called second moment) of a (potentially time-dependent) random vector \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 ...
algorithms. For a random vector \mathbf = (X_1,\ldots,X_n)^ containing
random element In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansio ...
s whose
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
exist, the auto-correlation matrix is defined byPapoulis, Athanasius, ''Probability, Random variables and Stochastic processes'', McGraw-Hill, 1991 where ^ denotes transposition and has dimensions n \times n. Written component-wise: \operatorname_ = \begin \operatorname _1 X_1& \operatorname
_1 X_2 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
& \cdots & \operatorname
_1 X_n 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit (measurement), unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment ...
\\ \\ \operatorname _2 X_1& \operatorname
_2 X_2 Two by Two, two by two, 2×2 or 2by2 may refer to: Arts, entertainment and media Film and television * 2×2 (TV channel), a TV channel in Russia * ''Two by Two'', or ''Ooops! Noah Is Gone...'', a 2015 animated film * "Two By Two", a List of Rugra ...
& \cdots & \operatorname _2 X_n\\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname
_n X_1 N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
& \operatorname
_n X_2 N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
& \cdots & \operatorname
_n X_n N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
\\ \\ \end If \mathbf is a complex random vector, the autocorrelation matrix is instead defined by \operatorname_ \triangleq\ \operatorname mathbf \mathbf^. Here ^ denotes Hermitian transposition. 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 I, or i, is the ninth Letter (alphabet), letter and the third vowel letter of the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in Engl ...
/math>.


Properties of the autocorrelation matrix

* The autocorrelation matrix is a Hermitian matrix for complex random vectors and a symmetric matrix 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 z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...
, 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]^


Auto-correlation 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, and scientific measurements. Signal processing techniq ...
, 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.


Auto-correlation of continuous-time signal

Given a signal 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 fo ...
integral of f(t) with itself, at lag \tau. where \overline represents the complex conjugate 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.


Auto-correlation 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 process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
es, 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 In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point * Stationary process * Stationary state Meteorology * A stationary front is a weather front that is not moving ...
, these will also be functions of t, or n. For processes that are also
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
, 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 In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power seri ...
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 ...
. * 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 In mathematics, the rearrangement inequality states that x_n y_1 + \cdots + x_1 y_n \leq x_ y_1 + \cdots + x_ y_n \leq x_1 y_1 + \cdots + x_n y_n for every choice of real numbers x_1 \leq \cdots \leq x_n \quad \text \quad y_1 \leq \cdots \leq y_n ...
. The same result holds in the discrete case. * The autocorrelation of a
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
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 fo ...
, 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 ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
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 Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al autocorrelation is defined similarly. For example, in
three dimensions Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
the autocorrelation of a square-summable discrete signal 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, a ...
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 equiv ...
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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
, 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). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...
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). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...
. The asterisk denotes complex conjugate. 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 is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
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, a ...
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 closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
. If the true mean and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
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 In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most co ...
-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. * 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 In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
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 is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
(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 In statistics and econometrics, and in particular in time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model. Both of these models are fitted to time ser ...
(ARIMA). With multiple interrelated data series,
vector autoregression Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single-variable (univariate) autoregres ...
(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 model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...
(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 the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
.) 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 colour model (traditional colour theory), as well as in the RGB (additive) colour model. It lies between violet and cyan on the spectrum of visible light. The eye perceives blue when obs ...
). 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 In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals (prediction errors) from a regression analysis. It is named after James Durbin and Geoffrey Watson. The sm ...
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 In statistics, the Breusch–Godfrey test is used to assess the validity of some of the modelling assumptions inherent in applying regression-like models to observed data series. In particular, it tests for the presence of serial correlation th ...
. 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 ''TR''2, where ''T'' is the sample size and ''R''2 is the
coefficient of determination In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used i ...
. Under the null hypothesis of no autocorrelation, this statistic is asymptotically distributed as \chi^2 with ''k'' degrees of freedom. Responses to nonzero autocorrelation include generalized least squares 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 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 or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
pulses In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the nec ...
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" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
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
micelle A micelle () or micella () (plural micelles or micellae, respectively) is an aggregate (or supramolecular assembly) of surfactant amphipathic lipid molecules dispersed in a liquid, forming a colloidal suspension (also known as associated collo ...
s 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 The Global Positioning System (GPS), originally Navstar GPS, is a Radionavigation-satellite service, satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of t ...
system to correct for the propagation delay, or time shift, between the point of time at the transmission of the carrier signal 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 or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...
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. *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 fiel ...
and
scanning probe microscopy Scan may refer to: Acronyms * Schedules for Clinical Assessment in Neuropsychiatry (SCAN), a psychiatric diagnostic tool developed by WHO * Shared Check Authorization Network (SCAN), a database of bad check writers and collection agency for bad ...
, 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 properties of an electromagnetic field. The degree of coherence is the normalized correlation of electric fields; in its simplest form, termed g^. ...
of an electromagnetic field. * 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, and scientific measurements. Signal processing techniq ...
, autocorrelation can give information about repeating events like musical beats (for example, to determine
tempo In musical terminology, tempo (Italian, 'time'; plural ''tempos'', or ''tempi'' from the Italian plural) is the speed or pace of a given piece. In classical music, tempo is typically indicated with an instruction at the start of a piece (often ...
) or
pulsar A pulsar (from ''pulsating radio source'') is a highly magnetized rotating neutron star that emits beams of electromagnetic radiation out of its magnetic poles. This radiation can be observed only when a beam of emission is pointing toward Ea ...
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, though it cannot tell the position in time of the beat. It can also be used to estimate the pitch of a musical tone. * In music recording, autocorrelation is used as a
pitch detection algorithm Pitch may refer to: Acoustic frequency * Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch" ** Absolute pitch or "perfect pitch" ** Pitch class, a set of all pitches that are a whole number of octave ...
prior to vocal processing, as a distortion effect or to eliminate undesired mistakes and inaccuracies. * 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 fo ...
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. Long chains of amino acids are called proteins. Chains of fewer than twenty amino acids are called oligopeptides, and include dipeptides, tripeptides, and tetrapeptides. A ...
. * 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 said, Astrophysics "seeks to ascertain the nature of the h ...
, autocorrelation is used to study and characterize the spatial distribution of
galaxies A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
in the universe and in multi-wavelength observations of low mass
X-ray binaries X-ray binaries are a class of binary stars that are luminous in X-rays. The X-rays are produced by matter falling from one component, called the ''donor'' (usually a relatively normal star), to the other component, called the ''accretor'', which ...
. * In panel data, 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) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...
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 properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
) 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 diagnostic techniques (mainly imaging techniques) using ultrasound, as well as therapeutic applications of ultrasound. In diagnosis, it is used to create an image of internal body structures such as tendons, muscles ...
imaging, autocorrelation is used to visualize blood flow. * In
intertemporal portfolio choice Intertemporal portfolio choice is the process of allocating one's investable wealth to various asset In financial accountancy, financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything ...
, the presence or absence of autocorrelation in an asset's rate of return can affect the optimal portion of the portfolio to hold in that asset. * Autocorrelation has been used to accurately measure power system frequency in
numerical relay In utility and industrial electric power transmission and distribution systems, a numerical relay is a computer-based system with software-based protection algorithms for the detection of electrical faults. Such relays are also termed as microp ...
s.


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. Exa ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
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 In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point * Stationary process * Stationary state Meteorology * A stationary front is a weather front that is not moving ...
, 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, sometimes known as serial correlation in the discrete time case, is the correlation of a Signal (information theory), signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observa ...
*
Autocorrelation technique The autocorrelation technique is a method for estimating the dominating frequency in a complex signal, as well as its variance. Specifically, it calculates the first two moments of the power spectrum, namely the mean and variance. It is also known ...
* Autocorrelation of a formal word *
Autocorrelator A real time interferometric autocorrelator is an electronic tool used to examine the autocorrelation of, among other things, optical beam intensity and spectral components through examination of variable beam path differences. ''See Optical autocorr ...
*
Correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...
*
Correlogram In the analysis of data, a correlogram is a chart of correlation statistics. For example, in time series analysis, a plot of the sample autocorrelations r_h\, versus h\, (the time lags) is an autocorrelogram. If cross-correlation is plo ...
*
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 fo ...
*
Galton's problem Galton's problem, named after Sir Francis Galton, is the problem of drawing inferences from cross-cultural data, due to the statistical phenomenon now called autocorrelation. The problem is now recognized as a general one that applies to all nonexp ...
*
Partial autocorrelation function In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values, regressed the values of the time series at all shorter lags. It contrasts with the autocorre ...
* Fluorescence correlation spectroscopy * Optical autocorrelation *
Pitch detection algorithm Pitch may refer to: Acoustic frequency * Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch" ** Absolute pitch or "perfect pitch" ** Pitch class, a set of all pitches that are a whole number of octave ...
*
Triple correlation The triple correlation of an ordinary function on the real line is the integral of the product of that function with two independently shifted copies of itself: : \int_^ f^(x) f(x+s_1) f(x+s_2) dx. The Fourier transform of triple correlation ...
*
CUSUM In 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 detection. CUSUM was announced in Bi ...
*
Cochrane–Orcutt estimation Cochrane–Orcutt estimation is a procedure in econometrics, which adjusts a linear model for serial correlation in the error term. Developed in the 1940s, it is named after statisticians Donald Cochrane and Guy Orcutt. Theory Consider the model ...
(transformation for autocorrelated error terms) * Prais–Winsten transformation * Scaled correlation * Unbiased estimation of standard deviation


References


Further reading

* * * Mojtaba Soltanalian, and Petre Stoica.
Computational design of sequences with good correlation properties
" IEEE Transactions on Signal Processing, 60.5 (2012): 2180–2193. * 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, ...
signal will have a strong peak (represented by a
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
) 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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
: \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 .


Auto-correlation of random vectors

The (potentially time-dependent) auto-correlation matrix (also called second moment) of a (potentially time-dependent) random vector \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 ...
algorithms. For a random vector \mathbf = (X_1,\ldots,X_n)^ containing
random element In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by who commented that the “development of probability theory and expansio ...
s whose
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
exist, the auto-correlation matrix is defined byPapoulis, Athanasius, ''Probability, Random variables and Stochastic processes'', McGraw-Hill, 1991 where ^ denotes transposition and has dimensions n \times n. Written component-wise: \operatorname_ = \begin \operatorname _1 X_1& \operatorname
_1 X_2 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
& \cdots & \operatorname
_1 X_n 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit (measurement), unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment ...
\\ \\ \operatorname _2 X_1& \operatorname
_2 X_2 Two by Two, two by two, 2×2 or 2by2 may refer to: Arts, entertainment and media Film and television * 2×2 (TV channel), a TV channel in Russia * ''Two by Two'', or ''Ooops! Noah Is Gone...'', a 2015 animated film * "Two By Two", a List of Rugra ...
& \cdots & \operatorname _2 X_n\\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname
_n X_1 N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
& \operatorname
_n X_2 N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
& \cdots & \operatorname
_n X_n N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
\\ \\ \end If \mathbf is a complex random vector, the autocorrelation matrix is instead defined by \operatorname_ \triangleq\ \operatorname mathbf \mathbf^. Here ^ denotes Hermitian transposition. 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 I, or i, is the ninth Letter (alphabet), letter and the third vowel letter of the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in Engl ...
/math>.


Properties of the autocorrelation matrix

* The autocorrelation matrix is a Hermitian matrix for complex random vectors and a symmetric matrix 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 z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a co ...
, 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]^


Auto-correlation 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, and scientific measurements. Signal processing techniq ...
, 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.


Auto-correlation of continuous-time signal

Given a signal 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 fo ...
integral of f(t) with itself, at lag \tau. where \overline represents the complex conjugate 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.


Auto-correlation 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 process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
es, 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 In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point * Stationary process * Stationary state Meteorology * A stationary front is a weather front that is not moving ...
, these will also be functions of t, or n. For processes that are also
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
, 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 In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power seri ...
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 ...
. * 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 In mathematics, the rearrangement inequality states that x_n y_1 + \cdots + x_1 y_n \leq x_ y_1 + \cdots + x_ y_n \leq x_1 y_1 + \cdots + x_n y_n for every choice of real numbers x_1 \leq \cdots \leq x_n \quad \text \quad y_1 \leq \cdots \leq y_n ...
. The same result holds in the discrete case. * The autocorrelation of a
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
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 fo ...
, 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 ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
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 Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al autocorrelation is defined similarly. For example, in
three dimensions Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
the autocorrelation of a square-summable discrete signal 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, a ...
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 equiv ...
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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
, 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). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...
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). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...
. The asterisk denotes complex conjugate. 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 is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
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, a ...
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 closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
. If the true mean and
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
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 In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most co ...
-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. * 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 In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
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 is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
(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 In statistics and econometrics, and in particular in time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model. Both of these models are fitted to time ser ...
(ARIMA). With multiple interrelated data series,
vector autoregression Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single-variable (univariate) autoregres ...
(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 model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...
(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 the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
.) 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 colour model (traditional colour theory), as well as in the RGB (additive) colour model. It lies between violet and cyan on the spectrum of visible light. The eye perceives blue when obs ...
). 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 In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals (prediction errors) from a regression analysis. It is named after James Durbin and Geoffrey Watson. The sm ...
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 In statistics, the Breusch–Godfrey test is used to assess the validity of some of the modelling assumptions inherent in applying regression-like models to observed data series. In particular, it tests for the presence of serial correlation th ...
. 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 ''TR''2, where ''T'' is the sample size and ''R''2 is the
coefficient of determination In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used i ...
. Under the null hypothesis of no autocorrelation, this statistic is asymptotically distributed as \chi^2 with ''k'' degrees of freedom. Responses to nonzero autocorrelation include generalized least squares 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 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 or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
pulses In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the nec ...
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" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
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
micelle A micelle () or micella () (plural micelles or micellae, respectively) is an aggregate (or supramolecular assembly) of surfactant amphipathic lipid molecules dispersed in a liquid, forming a colloidal suspension (also known as associated collo ...
s 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 The Global Positioning System (GPS), originally Navstar GPS, is a Radionavigation-satellite service, satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of t ...
system to correct for the propagation delay, or time shift, between the point of time at the transmission of the carrier signal 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 or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...
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. *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 fiel ...
and
scanning probe microscopy Scan may refer to: Acronyms * Schedules for Clinical Assessment in Neuropsychiatry (SCAN), a psychiatric diagnostic tool developed by WHO * Shared Check Authorization Network (SCAN), a database of bad check writers and collection agency for bad ...
, 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 properties of an electromagnetic field. The degree of coherence is the normalized correlation of electric fields; in its simplest form, termed g^. ...
of an electromagnetic field. * 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, and scientific measurements. Signal processing techniq ...
, autocorrelation can give information about repeating events like musical beats (for example, to determine
tempo In musical terminology, tempo (Italian, 'time'; plural ''tempos'', or ''tempi'' from the Italian plural) is the speed or pace of a given piece. In classical music, tempo is typically indicated with an instruction at the start of a piece (often ...
) or
pulsar A pulsar (from ''pulsating radio source'') is a highly magnetized rotating neutron star that emits beams of electromagnetic radiation out of its magnetic poles. This radiation can be observed only when a beam of emission is pointing toward Ea ...
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, though it cannot tell the position in time of the beat. It can also be used to estimate the pitch of a musical tone. * In music recording, autocorrelation is used as a
pitch detection algorithm Pitch may refer to: Acoustic frequency * Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch" ** Absolute pitch or "perfect pitch" ** Pitch class, a set of all pitches that are a whole number of octave ...
prior to vocal processing, as a distortion effect or to eliminate undesired mistakes and inaccuracies. * 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 fo ...
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. Long chains of amino acids are called proteins. Chains of fewer than twenty amino acids are called oligopeptides, and include dipeptides, tripeptides, and tetrapeptides. A ...
. * 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 said, Astrophysics "seeks to ascertain the nature of the h ...
, autocorrelation is used to study and characterize the spatial distribution of
galaxies A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
in the universe and in multi-wavelength observations of low mass
X-ray binaries X-ray binaries are a class of binary stars that are luminous in X-rays. The X-rays are produced by matter falling from one component, called the ''donor'' (usually a relatively normal star), to the other component, called the ''accretor'', which ...
. * In panel data, 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) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...
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 properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
) 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 diagnostic techniques (mainly imaging techniques) using ultrasound, as well as therapeutic applications of ultrasound. In diagnosis, it is used to create an image of internal body structures such as tendons, muscles ...
imaging, autocorrelation is used to visualize blood flow. * In
intertemporal portfolio choice Intertemporal portfolio choice is the process of allocating one's investable wealth to various asset In financial accountancy, financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything ...
, the presence or absence of autocorrelation in an asset's rate of return can affect the optimal portion of the portfolio to hold in that asset. * Autocorrelation has been used to accurately measure power system frequency in
numerical relay In utility and industrial electric power transmission and distribution systems, a numerical relay is a computer-based system with software-based protection algorithms for the detection of electrical faults. Such relays are also termed as microp ...
s.


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. Exa ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
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 In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point * Stationary process * Stationary state Meteorology * A stationary front is a weather front that is not moving ...
, 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, sometimes known as serial correlation in the discrete time case, is the correlation of a Signal (information theory), signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observa ...
*
Autocorrelation technique The autocorrelation technique is a method for estimating the dominating frequency in a complex signal, as well as its variance. Specifically, it calculates the first two moments of the power spectrum, namely the mean and variance. It is also known ...
* Autocorrelation of a formal word *
Autocorrelator A real time interferometric autocorrelator is an electronic tool used to examine the autocorrelation of, among other things, optical beam intensity and spectral components through examination of variable beam path differences. ''See Optical autocorr ...
*
Correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...
*
Correlogram In the analysis of data, a correlogram is a chart of correlation statistics. For example, in time series analysis, a plot of the sample autocorrelations r_h\, versus h\, (the time lags) is an autocorrelogram. If cross-correlation is plo ...
*
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 fo ...
*
Galton's problem Galton's problem, named after Sir Francis Galton, is the problem of drawing inferences from cross-cultural data, due to the statistical phenomenon now called autocorrelation. The problem is now recognized as a general one that applies to all nonexp ...
*
Partial autocorrelation function In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values, regressed the values of the time series at all shorter lags. It contrasts with the autocorre ...
* Fluorescence correlation spectroscopy * Optical autocorrelation *
Pitch detection algorithm Pitch may refer to: Acoustic frequency * Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch" ** Absolute pitch or "perfect pitch" ** Pitch class, a set of all pitches that are a whole number of octave ...
*
Triple correlation The triple correlation of an ordinary function on the real line is the integral of the product of that function with two independently shifted copies of itself: : \int_^ f^(x) f(x+s_1) f(x+s_2) dx. The Fourier transform of triple correlation ...
*
CUSUM In 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 detection. CUSUM was announced in Bi ...
*
Cochrane–Orcutt estimation Cochrane–Orcutt estimation is a procedure in econometrics, which adjusts a linear model for serial correlation in the error term. Developed in the 1940s, it is named after statisticians Donald Cochrane and Guy Orcutt. Theory Consider the model ...
(transformation for autocorrelated error terms) * Prais–Winsten transformation * Scaled correlation * Unbiased estimation of standard deviation


References


Further reading

* * * Mojtaba Soltanalian, and Petre Stoica.
Computational design of sequences with good correlation properties
" IEEE Transactions on Signal Processing, 60.5 (2012): 2180–2193. * 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