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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 statisti ...
of a
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
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 p ...
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 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 aris ...
, or identifying the missing fundamental frequency 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'', ...
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 sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
for analyzing functions or series of values, such as
time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the ...
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 proce ...
.
Unit root In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if ...
processes,
trend-stationary process In the statistical analysis of time series, a trend-stationary process is a stochastic process from which an underlying trend (function solely of time) can be removed, leaving a stationary process. The trend does not have to be linear. Conversel ...
es, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation.


Auto-correlation of stochastic processes

In statistics, the autocorrelation of a real or complex
random process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
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 coefficien ...
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 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 or a
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 grou ...
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 '' ari ...
\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 number ...
\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 ...
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 In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A fun ...
. 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 functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one qua ...
).


Definition for wide-sense stationary stochastic process

If \left\ is a wide-sense stationary process 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 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 In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
) to normalize the autocovariance function to get a time-dependent
Pearson correlation coefficient In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficien ...
. 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 In statistics, there is a negative relationship or inverse relationship between two variables if higher values of one variable tend to be associated with lower values of the other. A negative relationship between two variables usually implies tha ...
. 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. ...
(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 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 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 f ...
, 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 In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
relates the autocorrelation function \operatorname_ to the
power spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
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 In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
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 In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
\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 ar ...
algorithms. For a
random vector In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
\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 ...
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 number ...
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& \cdots & \operatorname _1 X_n\\ \\ \operatorname _2 X_1& \operatorname _2 X_2& \cdots & \operatorname _2 X_n\\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname _n X_1& \operatorname _n X_2& \cdots & \operatorname _n X_n\\ \\ \end If \mathbf is a
complex random vector In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If Z_1,\ldots,Z_n are com ...
, 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/math>.


Properties of the autocorrelation matrix

* The autocorrelation matrix is a
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
for complex random vectors and a
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
for real random vectors. * The autocorrelation matrix is a
positive semidefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number 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 ...
, 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 sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, 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 In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
f(t), the continuous autocorrelation R_(\tau) is most often defined as the continuous
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
integral of f(t) with itself, at lag \tau. where \overline represents the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of f(t). Note that the parameter t in the integral is a dummy variable and is only necessary to calculate the integral. It has no specific meaning.


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 processes, the autocorrelations are defined as \begin R_(\tau) &= \operatorname\left (t)\overline\right\\ R_(\ell) &= \operatorname\left (n)\,\overline\right. \end For processes that are not stationary, these will also be functions of t, or n. For processes that are also
ergodic 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 t ...
, 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 The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divi ...
for a related process.)


Definition for periodic signals

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


Properties

In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases. These properties hold for wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry, R_(\tau) = R_(-\tau), which is easy to prove from the definition. In the continuous case, ** the autocorrelation is an even function R_(-\tau) = R_(\tau) when f is a real function, and ** the autocorrelation is a
Hermitian function In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: :f^*(x) = f(-x) (where the ^* indicates the complex conjugate) ...
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 algebrai ...
. * The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay \tau, , R_(\tau), \leq R_(0). This is a consequence of the rearrangement inequality. The same result holds in the discrete case. * The autocorrelation of a
periodic function A periodic function 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 d ...
is, itself, periodic with the same period. * The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all \tau) is the sum of the autocorrelations of each function separately. * Since autocorrelation is a specific type of
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
, it maintains all the properties of cross-correlation. * By using the symbol * to represent
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...
and g_ is a function which manipulates the function f and is defined as g_(f)(t)=f(-t), the definition for R_(\tau) may be written as:R_(\tau) = (f * g_(\overline))(\tau)


Multi-dimensional autocorrelation

Multi-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable
discrete signal In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
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 equi ...
based on the signal processing definition R_(j) = \sum_n x_n\,\overline_ can be used when the signal size is small. For example, to calculate the autocorrelation of the real signal sequence x = (2,3,-1) (i.e. x_0=2, x_1=3, x_2=-1, and x_i = 0 for all other values of ) by hand, we first recognize that the definition just given is the same as the "usual" multiplication, but with right shifts, where each vertical addition gives the autocorrelation for particular lag values: \begin & 2 & 3 & -1 \\ \times & 2 & 3 & -1 \\ \hline &-2 &-3 & 1 \\ & & 6 & 9 & -3 \\ + & & & 4 & 6 & -2 \\ \hline & -2 & 3 &14 & 3 & -2 \end Thus the required autocorrelation sequence is R_=(-2,3,14,3,-2), where R_(0)=14, R_(-1)= R_(1)=3, and R_(-2)= R_(2) = -2, the autocorrelation for other lag values being zero. In this calculation we do not perform the carry-over operation during addition as is usual in normal multiplication. Note that we can halve the number of operations required by exploiting the inherent symmetry of the autocorrelation. If the signal happens to be periodic, i.e. x=(\ldots,2,3,-1,2,3,-1,\ldots), then we get a circular autocorrelation (similar to
circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
) 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 In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-t ...
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 ...
, several efficient algorithms exist which can compute the autocorrelation in order . For example, the
Wiener–Khinchin theorem In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
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 t ...
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 t ...
. The asterisk denotes
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. Alternatively, a multiple correlation can be performed by using brute force calculation for low values, and then progressively binning the data with a
logarithm In mathematics, the logarithm 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 of ...
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. 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 number ...
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 c ...
-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 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 ...
(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) autoregre ...
(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 ...
(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 statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8� ...
.) Autocorrelation of the errors violates the ordinary least squares assumption that the error terms are uncorrelated, meaning that the Gauss Markov theorem does not apply, and that OLS estimators are no longer the Best Linear Unbiased Estimators (
BLUE Blue is one of the three primary colours in the RYB 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 ...
). 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 ...
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 tha ...
. 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 ...
. 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 In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordin ...
and the Newey–West HAC estimator (Heteroskedasticity and Autocorrelation Consistent). In the estimation of a
moving average model 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 ...
(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 Fluorescence correlation spectroscopy (FCS) is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis p ...
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 te ...
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 ...
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 firs ...
s, both using optical autocorrelators. * Autocorrelation is used to analyze
dynamic light scattering Dynamic light scattering (DLS) is a technique in physics that can be used to determine the size distribution profile of small particles in suspension or polymers in solution. In the scope of DLS, temporal fluctuations are usually analyzed usin ...
data, which notably enables determination of the
particle size distribution The particle-size distribution (PSD) of a powder, or granular material, or particles dispersed in fluid, is a list of values or a mathematical function that defines the relative amount, typically by mass, of particles present according to size. Si ...
s 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 col ...
s suspended in a fluid. A laser shining into the mixture produces a
speckle pattern Speckle, speckle pattern, or speckle noise is a granular noise texture degrading the quality as a consequence of interference among wavefronts in coherent imaging systems, such as radar, synthetic aperture radar (SAR), medical ultrasound and o ...
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 satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
system to correct for the
propagation delay Propagation delay is the time duration taken for a signal to reach its destination. It can relate to networking, electronics or physics. ''Hold time'' is the minimum interval required for the logic level to remain on the input after triggering ed ...
, or time shift, between the point of time at the transmission of the
carrier signal In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has ...
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 ...
in the incoming satellite signal, until the receiver replica signal and the satellite signal codes match up. * The
small-angle X-ray scattering Small-angle X-ray scattering (SAXS) is a small-angle scattering technique by which nanoscale density differences in a sample can be quantified. This means that it can determine nanoparticle size distributions, resolve the size and shape of (monodi ...
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 t ...
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 ba ...
, 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^. I ...
of an electromagnetic field. * In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, autocorrelation can give information about repeating events like
musical beat In music and music theory, the beat is the basic unit of time, the pulse (regularly repeating event), of the ''mensural level'' (or ''beat level''). The beat is often defined as the rhythm listeners would tap their toes to when listening to a p ...
s (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 (ofte ...
) 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 E ...
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 e ...
, 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 Sound recording and reproduction is the electrical, mechanical, electronic, or digital inscription and re-creation of sound waves, such as spoken voice, singing, instrumental music, or sound effects. The two main classes of sound recording t ...
, 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 octav ...
prior to vocal processing, as a
distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio s ...
effect or to eliminate undesired mistakes and inaccuracies. * Autocorrelation in space rather than time, via the
Patterson function The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT. The Patterson function i ...
, 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 A mass spectrum is a histogram plot of intensity vs. ''mass-to-charge ratio'' (''m/z'') in a chemical sample, usually acquired using an instrument called a ''mass spectrometer''. Not all mass spectra of a given substance are the same; for example ...
makes use of autocorrelation in conjunction with
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
to score the similarity of an observed spectrum to an idealized spectrum representing a
peptide Peptides (, ) are short chains of amino acids linked by peptide bonds. Long chains of amino acids are called proteins. Chains of fewer than twenty amino acids are called oligopeptides, and include dipeptides, tripeptides, and tetrapeptides. ...
. * 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 he ...
, 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 In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time. Time series and ...
, spatial autocorrelation refers to correlation of a variable with itself through space. * In analysis of
Markov chain Monte Carlo In statistics, Markov chain Monte Carlo (MCMC) 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 spher ...
(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, muscl ...
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 assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines ...
, the presence or absence of autocorrelation in an asset's
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons ...
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 relays.


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 statistic ...
). 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. E ...
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 p ...
has serial dependence if the value at some time t in the series is statistically dependent on the value at another time s. A series is serially independent if there is no dependence between any pair. If a time series \left\ is stationary, then statistical dependence between the pair (X_t,X_s) would imply that there is statistical dependence between all pairs of values at the same lag \tau=s-t.


See also

* Autocorrelation matrix * Autocorrelation technique * Autocorrelation of a formal word *
Autocorrelator A real time interferometry, 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 ...
*
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 re ...
*
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 plotte ...
*
Cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
*
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 autocorr ...
*
Fluorescence correlation spectroscopy Fluorescence correlation spectroscopy (FCS) is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis p ...
*
Optical autocorrelation In optics, various autocorrelation functions can be experimentally realized. The field autocorrelation may be used to calculate the spectrum of a source of light, while the intensity autocorrelation and the interferometric autocorrelation are com ...
*
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 octav ...
* Triple correlation * CUSUM * Cochrane–Orcutt estimation (transformation for autocorrelated error terms) * Prais–Winsten transformation *
Scaled correlation In statistics, scaled correlation is a form of a coefficient of correlation applicable to data that have a temporal component such as time series. It is the average short-term correlation. If the signals have multiple components (slow and fast), sc ...
* 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 In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
relates the autocorrelation function \operatorname_ to the
power spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
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 In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
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 In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
\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 ar ...
algorithms. For a
random vector In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its valu ...
\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 ...
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 number ...
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& \cdots & \operatorname _1 X_n\\ \\ \operatorname _2 X_1& \operatorname _2 X_2& \cdots & \operatorname _2 X_n\\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname _n X_1& \operatorname _n X_2& \cdots & \operatorname _n X_n\\ \\ \end If \mathbf is a
complex random vector In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If Z_1,\ldots,Z_n are com ...
, 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/math>.


Properties of the autocorrelation matrix

* The autocorrelation matrix is a
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
for complex random vectors and a
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
for real random vectors. * The autocorrelation matrix is a
positive semidefinite matrix In mathematics, a symmetric matrix M with real entries is positive-definite if the real number 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 ...
, 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 sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, 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 In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
f(t), the continuous autocorrelation R_(\tau) is most often defined as the continuous
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
integral of f(t) with itself, at lag \tau. where \overline represents the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of f(t). Note that the parameter t in the integral is a dummy variable and is only necessary to calculate the integral. It has no specific meaning.


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 processes, the autocorrelations are defined as \begin R_(\tau) &= \operatorname\left (t)\overline\right\\ R_(\ell) &= \operatorname\left (n)\,\overline\right. \end For processes that are not stationary, these will also be functions of t, or n. For processes that are also
ergodic 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 t ...
, 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 The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divi ...
for a related process.)


Definition for periodic signals

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


Properties

In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases. These properties hold for wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry, R_(\tau) = R_(-\tau), which is easy to prove from the definition. In the continuous case, ** the autocorrelation is an even function R_(-\tau) = R_(\tau) when f is a real function, and ** the autocorrelation is a
Hermitian function In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: :f^*(x) = f(-x) (where the ^* indicates the complex conjugate) ...
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 algebrai ...
. * The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay \tau, , R_(\tau), \leq R_(0). This is a consequence of the rearrangement inequality. The same result holds in the discrete case. * The autocorrelation of a
periodic function A periodic function 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 d ...
is, itself, periodic with the same period. * The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all \tau) is the sum of the autocorrelations of each function separately. * Since autocorrelation is a specific type of
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
, it maintains all the properties of cross-correlation. * By using the symbol * to represent
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...
and g_ is a function which manipulates the function f and is defined as g_(f)(t)=f(-t), the definition for R_(\tau) may be written as:R_(\tau) = (f * g_(\overline))(\tau)


Multi-dimensional autocorrelation

Multi-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable
discrete signal In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
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 equi ...
based on the signal processing definition R_(j) = \sum_n x_n\,\overline_ can be used when the signal size is small. For example, to calculate the autocorrelation of the real signal sequence x = (2,3,-1) (i.e. x_0=2, x_1=3, x_2=-1, and x_i = 0 for all other values of ) by hand, we first recognize that the definition just given is the same as the "usual" multiplication, but with right shifts, where each vertical addition gives the autocorrelation for particular lag values: \begin & 2 & 3 & -1 \\ \times & 2 & 3 & -1 \\ \hline &-2 &-3 & 1 \\ & & 6 & 9 & -3 \\ + & & & 4 & 6 & -2 \\ \hline & -2 & 3 &14 & 3 & -2 \end Thus the required autocorrelation sequence is R_=(-2,3,14,3,-2), where R_(0)=14, R_(-1)= R_(1)=3, and R_(-2)= R_(2) = -2, the autocorrelation for other lag values being zero. In this calculation we do not perform the carry-over operation during addition as is usual in normal multiplication. Note that we can halve the number of operations required by exploiting the inherent symmetry of the autocorrelation. If the signal happens to be periodic, i.e. x=(\ldots,2,3,-1,2,3,-1,\ldots), then we get a circular autocorrelation (similar to
circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
) 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 In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-t ...
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 ...
, several efficient algorithms exist which can compute the autocorrelation in order . For example, the
Wiener–Khinchin theorem In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary ...
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 t ...
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 t ...
. The asterisk denotes
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. Alternatively, a multiple correlation can be performed by using brute force calculation for low values, and then progressively binning the data with a
logarithm In mathematics, the logarithm 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 of ...
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. 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 number ...
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 c ...
-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 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 ...
(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) autoregre ...
(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 ...
(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 statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8� ...
.) Autocorrelation of the errors violates the ordinary least squares assumption that the error terms are uncorrelated, meaning that the Gauss Markov theorem does not apply, and that OLS estimators are no longer the Best Linear Unbiased Estimators (
BLUE Blue is one of the three primary colours in the RYB 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 ...
). 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 ...
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 tha ...
. 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 ...
. 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 In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordin ...
and the Newey–West HAC estimator (Heteroskedasticity and Autocorrelation Consistent). In the estimation of a
moving average model 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 ...
(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 Fluorescence correlation spectroscopy (FCS) is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis p ...
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 te ...
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 ...
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 firs ...
s, both using optical autocorrelators. * Autocorrelation is used to analyze
dynamic light scattering Dynamic light scattering (DLS) is a technique in physics that can be used to determine the size distribution profile of small particles in suspension or polymers in solution. In the scope of DLS, temporal fluctuations are usually analyzed usin ...
data, which notably enables determination of the
particle size distribution The particle-size distribution (PSD) of a powder, or granular material, or particles dispersed in fluid, is a list of values or a mathematical function that defines the relative amount, typically by mass, of particles present according to size. Si ...
s 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 col ...
s suspended in a fluid. A laser shining into the mixture produces a
speckle pattern Speckle, speckle pattern, or speckle noise is a granular noise texture degrading the quality as a consequence of interference among wavefronts in coherent imaging systems, such as radar, synthetic aperture radar (SAR), medical ultrasound and o ...
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 satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
system to correct for the
propagation delay Propagation delay is the time duration taken for a signal to reach its destination. It can relate to networking, electronics or physics. ''Hold time'' is the minimum interval required for the logic level to remain on the input after triggering ed ...
, or time shift, between the point of time at the transmission of the
carrier signal In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has ...
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 ...
in the incoming satellite signal, until the receiver replica signal and the satellite signal codes match up. * The
small-angle X-ray scattering Small-angle X-ray scattering (SAXS) is a small-angle scattering technique by which nanoscale density differences in a sample can be quantified. This means that it can determine nanoparticle size distributions, resolve the size and shape of (monodi ...
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 t ...
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 ba ...
, 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^. I ...
of an electromagnetic field. * In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, autocorrelation can give information about repeating events like
musical beat In music and music theory, the beat is the basic unit of time, the pulse (regularly repeating event), of the ''mensural level'' (or ''beat level''). The beat is often defined as the rhythm listeners would tap their toes to when listening to a p ...
s (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 (ofte ...
) 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 E ...
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 e ...
, 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 Sound recording and reproduction is the electrical, mechanical, electronic, or digital inscription and re-creation of sound waves, such as spoken voice, singing, instrumental music, or sound effects. The two main classes of sound recording t ...
, 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 octav ...
prior to vocal processing, as a
distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio s ...
effect or to eliminate undesired mistakes and inaccuracies. * Autocorrelation in space rather than time, via the
Patterson function The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT. The Patterson function i ...
, 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 A mass spectrum is a histogram plot of intensity vs. ''mass-to-charge ratio'' (''m/z'') in a chemical sample, usually acquired using an instrument called a ''mass spectrometer''. Not all mass spectra of a given substance are the same; for example ...
makes use of autocorrelation in conjunction with
cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
to score the similarity of an observed spectrum to an idealized spectrum representing a
peptide Peptides (, ) are short chains of amino acids linked by peptide bonds. Long chains of amino acids are called proteins. Chains of fewer than twenty amino acids are called oligopeptides, and include dipeptides, tripeptides, and tetrapeptides. ...
. * 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 he ...
, 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 In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time. Time series and ...
, spatial autocorrelation refers to correlation of a variable with itself through space. * In analysis of
Markov chain Monte Carlo In statistics, Markov chain Monte Carlo (MCMC) 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 spher ...
(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, muscl ...
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 assets, especially financial assets, repeatedly over time, in such a way as to optimize some criterion. The set of asset proportions at any time defines ...
, the presence or absence of autocorrelation in an asset's
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons ...
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 relays.


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 statistic ...
). 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. E ...
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 p ...
has serial dependence if the value at some time t in the series is statistically dependent on the value at another time s. A series is serially independent if there is no dependence between any pair. If a time series \left\ is stationary, then statistical dependence between the pair (X_t,X_s) would imply that there is statistical dependence between all pairs of values at the same lag \tau=s-t.


See also

* Autocorrelation matrix * Autocorrelation technique * Autocorrelation of a formal word *
Autocorrelator A real time interferometry, 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 ...
*
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 re ...
*
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 plotte ...
*
Cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...
*
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 autocorr ...
*
Fluorescence correlation spectroscopy Fluorescence correlation spectroscopy (FCS) is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis p ...
*
Optical autocorrelation In optics, various autocorrelation functions can be experimentally realized. The field autocorrelation may be used to calculate the spectrum of a source of light, while the intensity autocorrelation and the interferometric autocorrelation are com ...
*
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 octav ...
* Triple correlation * CUSUM * Cochrane–Orcutt estimation (transformation for autocorrelated error terms) * Prais–Winsten transformation *
Scaled correlation In statistics, scaled correlation is a form of a coefficient of correlation applicable to data that have a temporal component such as time series. It is the average short-term correlation. If the signals have multiple components (slow and fast), sc ...
* 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