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Autocorrelation

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Correlation of a signal with a time-shifted copy of itself, as a function of shift

Correlation and covariance
Part of a series onStatistics

For random vectors Autocorrelation matrix Cross-correlation matrix Auto-covariance matrix Cross-covariance matrix
For stochastic processes Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
For deterministic signals Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
v t e

Above: A plot of a series of 100 random numbers concealing asine function. Below: The sine function revealed in acorrelogram produced by autocorrelation.

Visual comparison of convolution,cross-correlation, andautocorrelation. For the operations involving functionf, and assuming the height off is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point. Also, the symmetry off is the reason $g*f$ and $f\star g$ are identical in this example.

{\displaystyle g*f} — Visual comparison of convolution,cross-correlation, andautocorrelation. For the operations involving functionf, and assuming the height off is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point. Also, the symmetry off is the reason $g*f$ and $f\star g$ are identical in this example.

Autocorrelation, sometimes known asserial correlation in thediscrete time case, measures thecorrelation of asignal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of arandom variable at different points in time. The analysis of autocorrelation is a mathematical tool for identifying repeating patterns or hiddenperiodicities within a signal obscured bynoise. Autocorrelation is widely used insignal processing,time domain andtime series analysis to understand the behavior of data over time.

Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably withautocovariance.

Various time series models incorporate autocorrelation, such asunit root processes,trend-stationary processes,autoregressive processes, andmoving average processes.

Autocorrelation of stochastic processes

Instatistics, the autocorrelation of a real or complexrandom process is thePearson correlation between values of the process at different times, as a function of the two times or of the time lag. Let $\left\{X_{t}\right\}$ be a random process, and $t {\displaystyle t}$ be any point in time ( $t {\displaystyle t}$ may be aninteger for adiscrete-time process or areal number for acontinuous-time process). Then $X_{t}$ is the value (orrealization) produced by a givenrun of the process at time $t {\displaystyle t}$ . Suppose that the process hasmean $\mu _{t}$ andvariance $\sigma _{t}^{2}$ at time $t {\displaystyle t}$ , for each $t {\displaystyle t}$ . Then the definition of theautocorrelation function between times $t_{1}$ and $t_{2}$ is^[1]^: 388^[2]^: 165

$\operatorname {R} _{XX}(t_{1},t_{2})=\operatorname {E} \left[X_{t_{1}}{\overline {X}}_{t_{2}}\right]$

where $\operatorname {E}$ is theexpected value operator and the bar representscomplex conjugation. Note that the expectation may not bewell defined.

Subtracting the mean before multiplication yields theauto-covariance function between times $t_{1}$ and $t_{2}$ :^[1]^: 392^[2]^: 168

${\begin{aligned}\operatorname {K} _{XX}(t_{1},t_{2})&=\operatorname {E} \left[(X_{t_{1}}-\mu _{t_{1}}){\overline {(X_{t_{2}}-\mu _{t_{2}})}}\right]\\&=\operatorname {E} \left[X_{t_{1}}{\overline {X}}_{t_{2}}\right]-\mu _{t_{1}}{\overline {\mu }}_{t_{2}}\\&=\operatorname {R} _{XX}(t_{1},t_{2})-\mu _{t_{1}}{\overline {\mu }}_{t_{2}}\end{aligned}}$

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 ofpower law).

Definition for wide-sense stationary stochastic process

If $\left\{X_{t}\right\}$ is awide-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 autocorrelation can be expressed as a function of the time-lag, and that this would be aneven function of the lag $\tau =t_{2}-t_{1}$ . This gives the more familiar forms for theautocorrelation function^[1]^: 395

$\operatorname {R} _{XX}(\tau )=\operatorname {E} \left[X_{t+\tau }{\overline {X}}_{t}\right]$

and theauto-covariance function:

${\begin{aligned}\operatorname {K} _{XX}(\tau )&=\operatorname {E} \left[(X_{t+\tau }-\mu ){\overline {(X_{t}-\mu )}}\right]\\&=\operatorname {E} \left[X_{t+\tau }{\overline {X}}_{t}\right]-\mu {\overline {\mu }}\\&=\operatorname {R} _{XX}(\tau )-\mu {\overline {\mu }}\end{aligned}}$

In particular, note that

$\operatorname {K} _{XX}(0)=\sigma ^{2}.$

Normalization

It is common practice in some disciplines (e.g. statistics andtime series analysis) to normalize the autocovariance function to get a time-dependentPearson correlation coefficient. However, in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably.

The definition of the autocorrelation coefficient of a stochastic process is^[2]^: 169

${\begin{aligned}\rho _{XX}(t_{1},t_{2})&={\frac {\operatorname {K} _{XX}(t_{1},t_{2})}{\sigma _{t_{1}}\sigma _{t_{2}}}}\\&={\frac {\operatorname {E} \left[\left(X_{t_{1}}-\mu _{t_{1}}\right){\overline {\left(X_{t_{2}}-\mu _{t_{2}}\right)}}\right]}{\sigma _{t_{1}}\sigma _{t_{2}}}}.\end{aligned}}$

If the function $\rho _{XX}$ is well defined, its value must lie in the range $[-1,1]$ , with 1 indicating perfect correlation and −1 indicating perfectanti-correlation.

For awide-sense stationary (WSS) process, the definition is

$\rho _{XX}(\tau )={\frac {\operatorname {K} _{XX}(\tau )}{\sigma ^{2}}}={\frac {\operatorname {E} \left[(X_{t+\tau }-\mu ){\overline {(X_{t}-\mu )}}\right]}{\sigma ^{2}}}.$

The normalization is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength ofstatistical dependence, and because the normalization has an effect on the statistical properties of the estimated autocorrelations.

Properties

Symmetry property

The fact that the autocorrelation function $\operatorname {R} _{XX}$ is aneven function can be stated as^[2]^: 171 $\operatorname {R} _{XX}(t_{1},t_{2})={\overline {\operatorname {R} _{XX}(t_{2},t_{1})}}$ respectively for a WSS process:^[2]^: 173 $\operatorname {R} _{XX}(\tau )={\overline {\operatorname {R} _{XX}(-\tau )}}.$

Maximum at zero

For a WSS process:^[2]^: 174 $\left|\operatorname {R} _{XX}(\tau )\right|\leq \operatorname {R} _{XX}(0)$ Notice that $\operatorname {R} _{XX}(0)$ is always real.

Cauchy–Schwarz inequality

TheCauchy–Schwarz inequality, inequality for stochastic processes:^[1]^: 392 $\left|\operatorname {R} _{XX}(t_{1},t_{2})\right|^{2}\leq \operatorname {E} \left[|X_{t_{1}}|^{2}\right]\operatorname {E} \left[|X_{t_{2}}|^{2}\right]$

Autocorrelation of white noise

The autocorrelation of a continuous-timewhite noise signal will have a strong peak (represented by aDirac delta function) at $\tau =0$ and will be exactly $0 {\displaystyle 0}$ for all other $\tau$ .

Wiener–Khinchin theorem

TheWiener–Khinchin theorem relates the autocorrelation function $\operatorname {R} _{XX}$ to thepower spectral density $S_{XX}$ via theFourier transform:

${\begin{aligned}\operatorname {R} _{XX}(\tau )&=\int _{-\infty }^{\infty }S_{XX}(\omega )e^{i\omega \tau }\,{\rm {d}}\omega \\[1ex]S_{XX}(\omega )&=\int _{-\infty }^{\infty }\operatorname {R} _{XX}(\tau )e^{-i\omega \tau }\,{\rm {d}}\tau .\end{aligned}}$

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

${\begin{aligned}\operatorname {R} _{XX}(\tau )&=\int _{-\infty }^{\infty }S_{XX}(\omega )\cos(\omega \tau )\,{\rm {d}}\omega \\[1ex]S_{XX}(\omega )&=\int _{-\infty }^{\infty }\operatorname {R} _{XX}(\tau )\cos(\omega \tau )\,{\rm {d}}\tau .\end{aligned}}$

Autocorrelation of random vectors

The (potentially time-dependent)autocorrelation matrix (also called second moment) of a (potentially time-dependent)random vector $\mathbf {X} =(X_{1},\ldots ,X_{n})^{\rm {T}}$ is an $n\times n$ matrix containing as elements the autocorrelations of all pairs of elements of the random vector $\mathbf {X}$ . The autocorrelation matrix is used in variousdigital signal processing algorithms.

For arandom vector $\mathbf {X} =(X_{1},\ldots ,X_{n})^{\rm {T}}$ containingrandom elements whoseexpected value andvariance exist, theautocorrelation matrix is defined by^[3]^: 190^[1]^: 334

$\operatorname {R} _{\mathbf {X} \mathbf {X} }\triangleq \ \operatorname {E} \left[\mathbf {X} \mathbf {X} ^{\rm {T}}\right]$

where ${}^{\rm {T}}$ denotes thetransposed matrix of dimensions $n\times n$ .

Written component-wise:

$\operatorname {R} _{\mathbf {X} \mathbf {X} }={\begin{bmatrix}\operatorname {E} [X_{1}X_{1}]&\operatorname {E} [X_{1}X_{2}]&\cdots &\operatorname {E} [X_{1}X_{n}]\\\\\operatorname {E} [X_{2}X_{1}]&\operatorname {E} [X_{2}X_{2}]&\cdots &\operatorname {E} [X_{2}X_{n}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} [X_{n}X_{1}]&\operatorname {E} [X_{n}X_{2}]&\cdots &\operatorname {E} [X_{n}X_{n}]\end{bmatrix}}$

If $\mathbf {Z}$ is acomplex random vector, the autocorrelation matrix is instead defined by

$\operatorname {R} _{\mathbf {Z} \mathbf {Z} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {Z} ^{\rm {H}}].$

Here ${}^{\rm {H}}$ denotesHermitian transpose.

For example, if $\mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T}}$ is a random vector, then $\operatorname {R} _{\mathbf {X} \mathbf {X} }$ is a $3\times 3$ matrix whose $(i,j)$ -th entry is $\operatorname {E} [X_{i}X_{j}]$ .

Properties of the autocorrelation matrix

The autocorrelation matrix is aHermitian matrix for complex random vectors and asymmetric matrix for real random vectors.^[3]^: 190
The autocorrelation matrix is apositive semidefinite matrix,^[3]^: 190 i.e. $\mathbf {a} ^{\mathrm {T} }\operatorname {R} _{\mathbf {X} \mathbf {X} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {R} ^{n}$ for a real random vector, and respectively $\mathbf {a} ^{\mathrm {H} }\operatorname {R} _{\mathbf {Z} \mathbf {Z} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {C} ^{n}$ in case of a complex random vector.
All eigenvalues of the autocorrelation matrix are real and non-negative.
Theauto-covariance matrix is related to the autocorrelation matrix as follows: ${\begin{aligned}\operatorname {K} _{\mathbf {X} \mathbf {X} }&=\operatorname {E} \left[(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\rm {T}}\right]\\&=\operatorname {R} _{\mathbf {X} \mathbf {X} }-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{\rm {T}}\end{aligned}}$ Respectively for complex random vectors: ${\begin{aligned}\operatorname {K} _{\mathbf {Z} \mathbf {Z} }&=\operatorname {E} \left[(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])^{\rm {H}}\right]\\&=\operatorname {R} _{\mathbf {Z} \mathbf {Z} }-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ]^{\rm {H}}\end{aligned}}$

Autocorrelation of deterministic signals

Insignal processing, the above definition is often used without the normalization, that is, without subtracting the mean and dividing by the variance. When the autocorrelation function is normalized by mean and variance, it is sometimes referred to as theautocorrelation coefficient^[4] or autocovariance function.

Autocorrelation of continuous-time signal

Given asignal $f(t)$ , the continuous autocorrelation $R_{ff}(\tau )$ is most often defined as the continuouscross-correlation integral of $f(t)$ with itself, at lag $\tau$ .^[1]^: 411

$R_{ff}(\tau )=\int _{-\infty }^{\infty }f(t+\tau ){\overline {f(t)}}\,{\rm {d}}t=\int _{-\infty }^{\infty }f(t){\overline {f(t-\tau )}}\,{\rm {d}}t$

where ${\overline {f(t)}}$ represents thecomplex conjugate of $f(t)$ . Note that the parameter $t {\displaystyle t}$ in the integral is a dummy variable and is only necessary to calculate the integral. It has no specific meaning.

Autocorrelation of discrete-time signal

The discrete autocorrelation $R {\displaystyle R}$ at lag $\ell$ for a discrete-time signal $y(n)$ is

$R_{yy}(\ell )=\sum _{n\in Z}y(n)\,{\overline {y(n-\ell )}}$

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. Forwide-sense-stationary random processes, the autocorrelations are defined as

${\begin{aligned}R_{ff}(\tau )&=\operatorname {E} \left[f(t){\overline {f(t-\tau )}}\right]\\R_{yy}(\ell )&=\operatorname {E} \left[y(n)\,{\overline {y(n-\ell )}}\right].\end{aligned}}$

For processes that are notstationary, these will also be functions of $t {\displaystyle t}$ , or $n {\displaystyle n}$ .

For processes that are alsoergodic, 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^[4]

${\begin{aligned}R_{ff}(\tau )&=\lim _{T\rightarrow \infty }{\frac {1}{T}}\int _{0}^{T}f(t+\tau ){\overline {f(t)}}\,{\rm {d}}t\\R_{yy}(\ell )&=\lim _{N\rightarrow \infty }{\frac {1}{N}}\sum _{n=0}^{N-1}y(n)\,{\overline {y(n-\ell )}}.\end{aligned}}$

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 thatlast forever can be treated by a short-time autocorrelation function analysis, using finite time integrals. (Seeshort-time Fourier transform for a related process.)

Definition for periodic signals

If $f {\displaystyle f}$ is a continuous periodic function of period $T {\displaystyle T}$ , the integration from $-\infty$ to $\infty$ is replaced by integration over any interval $[t_{0},t_{0}+T]$ of length $T {\displaystyle T}$ :

$R_{ff}(\tau )\triangleq \int _{t_{0}}^{t_{0}+T}f(t+\tau ){\overline {f(t)}}\,dt$

which is equivalent to

$R_{ff}(\tau )\triangleq \int _{t_{0}}^{t_{0}+T}f(t){\overline {f(t-\tau )}}\,dt$

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 forwide-sense stationary processes.^[5]

A fundamental property of the autocorrelation is symmetry, $R_{ff}(\tau )=R_{ff}(-\tau )$ , which is easy to prove from the definition. In the continuous case,
- the autocorrelation is aneven function $R_{ff}(-\tau )=R_{ff}(\tau )$ when $f {\displaystyle f}$ is a real function, and
- the autocorrelation is aHermitian function $R_{ff}(-\tau )=R_{ff}^{*}(\tau )$ when $f {\displaystyle f}$ is acomplex function.
The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay $\tau$ , $|R_{ff}(\tau )|\leq R_{ff}(0)$ .^[1]^: 410 This is a consequence of therearrangement inequality. The same result holds in the discrete case.
The autocorrelation of aperiodic function 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 ofcross-correlation, it maintains all the properties of cross-correlation.
By using the symbol $*$ to representconvolution and $g_{-1}$ is a function which manipulates the function $f {\displaystyle f}$ and is defined as $g_{-1}(f)(t)=f(-t)$ , the definition for $R_{ff}(\tau )$ may be written as: $R_{ff}(\tau )=(f*g_{-1}({\overline {f}}))(\tau )$

Multi-dimensional autocorrelation

Multi-dimensional autocorrelation is defined similarly. For example, inthree dimensions the autocorrelation of a square-summablediscrete signal would be

$R(j,k,\ell )=\sum _{n,q,r}x_{n,q,r}\,{\overline {x}}_{n-j,q-k,r-\ell }.$

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 adiscrete sequence, it is frequently necessary to compute the autocorrelation with highcomputational efficiency. Abrute force method based on the signal processing definition ${\textstyle R_{xx}(j)=\sum _{n}x_{n}\,{\overline {x}}_{n-j}}$ 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 ofi) 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: ${\begin{array}{rrrrrr}&2&3&-1\\\times &2&3&-1\\\hline &-2&-3&1\\&&6&9&-3\\+&&&4&6&-2\\\hline &-2&3&14&3&-2\end{array}}$

Thus the required autocorrelation sequence is $R_{xx}=(-2,3,14,3,-2)$ , where $R_{xx}(0)=14,$ $R_{xx}(-1)=R_{xx}(1)=3,$ and $R_{xx}(-2)=R_{xx}(2)=-2,$ 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 tocircular convolution) where the left and right tails of the previous autocorrelation sequence will overlap and give $R_{xx}=(\ldots ,14,1,1,14,1,1,\ldots )$ which has the same period as the signal sequence $x . {\displaystyle x.}$ The procedure can be regarded as an application of the convolution property ofZ-transform of a discrete signal.

While the brute force algorithm isordern², several efficient algorithms exist which can compute the autocorrelation in ordern log(n). For example, theWiener–Khinchin theorem allows computing the autocorrelation from the raw dataX(t) with twofast Fourier transforms (FFT):^[6]^{[page needed]}

${\begin{aligned}F_{R}(f)&=\operatorname {FFT} [X(t)]\\S(f)&=F_{R}(f)F_{R}^{*}(f)\\R(\tau )&=\operatorname {IFFT} [S(f)]\end{aligned}}$

where IFFT denotes the inversefast Fourier transform. The asterisk denotescomplex conjugate.

Alternatively, a multipleτ correlation can be performed by using brute force calculation for lowτ values, and then progressively binning theX(t) data with alogarithmic density to compute higher values, resulting in the samen log(n) efficiency, but with lower memory requirements.^[7]^[8]

Estimation

For adiscrete process with known mean and variance for which we observe $n {\displaystyle n}$ observations $\{X_{1},\,X_{2},\,\ldots ,\,X_{n}\}$ , an estimate of the autocorrelation coefficient may be obtained as

${\hat {R}}(k)={\frac {1}{(n-k)\sigma ^{2}}}\sum _{t=1}^{n-k}(X_{t}-\mu )(X_{t+k}-\mu )$

for any positive integer $k<n$ . When the true mean $\mu$ and variance $\sigma ^{2}$ are known, this estimate isunbiased. If the true mean andvariance 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 abiased estimate.
Aperiodogram-based estimate replaces $n-k$ in the above formula with $n {\displaystyle n}$ . This estimate is always biased; however, it usually has a smallermean squared error.^[9]^[10]
Other possibilities derive from treating the two portions of data $\{X_{1},\,X_{2},\,\ldots ,\,X_{n-k}\}$ and $\{X_{k+1},\,X_{k+2},\,\ldots ,\,X_{n}\}$ separately and calculating separate sample means and/or sample variances for use in defining the estimate.^{[citation needed]}

The advantage of estimates of the last type is that the set of estimated autocorrelations, as a function of $k {\displaystyle k}$ , 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 $X {\displaystyle X}$ 's, the variance calculated may turn out to be negative.^[11]

Hassani −1/2 theorem

In time series analysis, Hassani’s −1/2 theorem is a finite-sample identity for the conventional estimator of thesample autocorrelation function (ACF). For a series of length $T\geq 2$ , using the usual sample-mean–corrected estimator ${\hat {\rho }}(h)$ , Hassani showed that the sum of the sample autocorrelations over all positive lags is constant: $\sum _{h=1}^{T-1}{\hat {\rho }}(h)=-{\tfrac {1}{2}}.$ The result implies that sample autocorrelations across lags are not independent and that the sample ACF cannot be “positive overall” across all lags. It is often cited in discussions of finite-sample behavior of the ACF and in cautions against using the sum of estimated autocorrelations as a diagnostic measure of total dependence or long-memory behavior.

Regression analysis

Inregression analysis usingtime series data, autocorrelation in a variable of interest is typically modeled either with anautoregressive model (AR), amoving average model (MA), their combination as anautoregressive-moving-average model (ARMA), or an extension of the latter called anautoregressive integrated moving average model (ARIMA). With multiple interrelated data series,vector autoregression (VAR) or its extensions are used.

Inordinary least squares (OLS), the adequacy of a model specification can be checked in part by establishing whether there is autocorrelation of theregression 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" ineconometrics.) Autocorrelation of the errors violates the ordinary least squares assumption that the error terms are uncorrelated, meaning that theGauss Markov theorem does not apply, and that OLS estimators are no longer the Best Linear Unbiased Estimators (BLUE). While it does not bias the OLS coefficient estimates, thestandard errors tend to be underestimated (and thet-scores overestimated) when the autocorrelations of the errors at low lags are positive.

The traditional test for the presence of first-order autocorrelation is theDurbin–Watson statistic or, if the explanatory variables include a lagged dependent variable,Durbin's h statistic. The Durbin-Watson can be linearly mapped however to the Pearson correlation between values and their lags.^[12] A more flexible test, covering autocorrelation of higher orders and applicable whether or not the regressors include lags of the dependent variable, is theBreusch–Godfrey test. This involves an auxiliary regression, wherein the residuals obtained from estimating the model of interest are regressed on (a) the original regressors and (b)k lags of the residuals, where 'k' is the order of the test. The simplest version of thetest statistic from this auxiliary regression isTR², whereT is the sample size andR² is thecoefficient of determination. Under thenull hypothesis of no autocorrelation, this statistic is asymptoticallydistributed as $\chi ^{2}$ withk degrees of freedom.

Responses to nonzero autocorrelation includegeneralized least squares and theNewey–West HAC estimator (Heteroskedasticity and Autocorrelation Consistent).^[13]

In the estimation of amoving average model (MA), the autocorrelation function is used to determine the appropriate number of lagged error terms to be included. This is based on the fact that for an MA process of orderq, we have $R(\tau )\neq 0$ , for $\tau =0,1,\ldots ,q$ , and $R(\tau )=0$ , for $\tau >q$ .

Applications

Autocorrelation's ability to find repeating patterns indata yields many applications, including:

Autocorrelation analysis is used heavily influorescence correlation spectroscopy^[14] to provide quantitative insight into molecular-level diffusion and chemical reactions.^[15]
Another application of autocorrelation is the measurement ofoptical spectra and the measurement of very-short-durationlight pulses produced bylasers, both usingoptical autocorrelators.
Autocorrelation is used to analyzedynamic light scattering data, which notably enables determination of theparticle size distributions of nanometer-sized particles ormicelles suspended in a fluid. A laser shining into the mixture produces aspeckle pattern that results from the motion of the particles. Autocorrelation of the signal can be analyzed in terms of the diffusion of the particles. From this, knowing the viscosity of the fluid, the sizes of the particles can be calculated.
Utilized in theGPS system to correct for thepropagation delay, or time shift, between the point of time at the transmission of thecarrier signal at the satellites, and the point of time at the receiver on the ground. This is done by the receiver generating a replica signal of the 1,023-bit C/A (Coarse/Acquisition) code, and generating lines of code chips [-1,1] in packets of ten at a time, or 10,230 chips (1,023 × 10), shifting slightly as it goes along in order to accommodate for thedoppler shift in the incoming satellite signal, until the receiver replica signal and the satellite signal codes match up.^[16]
Thesmall-angle X-ray scattering intensity of a nanostructured system is the Fourier transform of the spatial autocorrelation function of theelectron density.
Insurface science andscanning probe microscopy, autocorrelation is used to establish a link between surface morphology and functional characteristics.^[17]
In optics, normalized autocorrelations and cross-correlations give thedegree of coherence of an electromagnetic field.
Inastronomy, autocorrelation can determine thefrequency ofpulsars.
Inmusic, autocorrelation (when applied at time scales smaller than a second) is used as apitch detection algorithm for both instrument tuners and "Auto Tune" (used as adistortion effect or to fix intonation).^[18] When applied at time scales larger than a second, autocorrelation can identify themusical beat, for example to determinetempo.
Autocorrelation in space rather than time, via thePatterson function, is used by X-ray diffractionists to help recover the "Fourier phase information" on atom positions not available through diffraction alone.
In statistics, spatial autocorrelation between sample locations also helps one estimatemean value uncertainties when sampling a heterogeneous population.
TheSEQUEST algorithm for analyzingmass spectra makes use of autocorrelation in conjunction withcross-correlation to score the similarity of an observed spectrum to an idealized spectrum representing apeptide.
Inastrophysics, autocorrelation is used to study and characterize the spatial distribution ofgalaxies in the universe and in multi-wavelength observations of low massX-ray binaries.
Inpanel data, spatial autocorrelation refers to correlation of a variable with itself through space.
In analysis ofMarkov chain Monte Carlo data, autocorrelation must be taken into account for correct error determination.
Ingeosciences (specifically ingeophysics) it can be used to compute an autocorrelationseismic attribute, out of a 3D seismic survey of the underground.
Inmedical ultrasound imaging, autocorrelation is used to visualize blood flow.
Inintertemporal portfolio choice, the presence or absence of autocorrelation in an asset'srate of return can affect the optimal portion of the portfolio to hold in that asset.
Innumerical relays, autocorrelation has been used to accurately measure power system frequency.^[19]

Serial dependence

Serial dependence is closely linked to the notion of autocorrelation, but represents a distinct concept (seeCorrelation and dependence). In particular, it is possible to have serial dependence but no (linear) correlation. In some fields however, the two terms are used as synonyms.

Atime series of arandom variable has serial dependence if the value at some time $t {\displaystyle t}$ in the series isstatistically dependent on the value at another time $s {\displaystyle s}$ . A series is serially independent if there is no dependence between any pair.

If a time series $\left\{X_{t}\right\}$ isstationary, then statistical dependence between the pair $(X_{t},X_{s})$ would imply that there is statistical dependence between all pairs of values at the same lag $\tau =s-t$ .

See also

Autocorrelation matrix
Autocorrelation of a formal word
Autocorrelation technique
Autocorrelator
Cochrane–Orcutt estimation (transformation for autocorrelated error terms)
Correlation function
Correlogram
Cross-correlation
CUSUM
Fluorescence correlation spectroscopy
Optical autocorrelation
Partial autocorrelation function
Phylogenetic autocorrelation (Galton's problem)
Pitch detection algorithm
Prais–Winsten transformation
Scaled correlation
Triple correlation
Unbiased estimation of standard deviation

References

^^a ^b ^c ^d ^e ^f ^gGubner, John A. (2006).Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press.ISBN 978-0-521-86470-1.
^^a ^b ^c ^d ^e ^fKun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018,ISBN 978-3-319-68074-3
^^a ^b ^cPapoulis, Athanasius,Probability, Random variables and Stochastic processes, McGraw-Hill, 1991
^^a ^bDunn, Patrick F. (2005).Measurement and Data Analysis for Engineering and Science. New York: McGraw–Hill.ISBN 978-0-07-282538-1.
^Proakis, John (August 31, 2001).Communication Systems Engineering (2nd Edition) (2 ed.). Pearson. p. 168.ISBN 978-0130617934.
^Box, G. E. P.; Jenkins, G. M.; Reinsel, G. C. (1994).Time Series Analysis: Forecasting and Control (3rd ed.). Upper Saddle River, NJ: Prentice–Hall.ISBN 978-0130607744.
^Frenkel, D.; Smit, B. (2002). "chap. 4.4.2".Understanding Molecular Simulation (2nd ed.). London: Academic Press.ISBN 978-0122673511.
^Colberg, P.; Höfling, F. (2011). "Highly accelerated simulations of glassy dynamics using GPUs: caveats on limited floating-point precision".Comput. Phys. Commun.182 (5):1120–1129.arXiv:0912.3824.Bibcode:2011CoPhC.182.1120C.doi:10.1016/j.cpc.2011.01.009.S2CID 7173093.
^Priestley, M. B. (1982).Spectral Analysis and Time Series. London, New York: Academic Press.ISBN 978-0125649018.
^Percival, Donald B.; Andrew T. Walden (1993).Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge University Press. pp. 190–195.ISBN 978-0-521-43541-3.
^Percival, Donald B. (1993). "Three Curious Properties of the Sample Variance and Autocovariance for Stationary Processes with Unknown Mean".The American Statistician.47 (4):274–276.doi:10.1080/00031305.1993.10475997.
^"Serial correlation techniques".Statistical Ideas. 26 May 2014.
^Baum, Christopher F. (2006).An Introduction to Modern Econometrics Using Stata. Stata Press.ISBN 978-1-59718-013-9.
^Elson, Elliot L. (December 2011)."Fluorescence Correlation Spectroscopy: Past, Present, Future".Biophysical Journal.101 (12):2855–2870.Bibcode:2011BpJ...101.2855E.doi:10.1016/j.bpj.2011.11.012.PMC 3244056.PMID 22208184.
^Hołyst, Robert; Poniewierski, Andrzej; Zhang, Xuzhu (2017)."Analytical form of the autocorrelation function for the fluorescence correlation spectroscopy".Soft Matter.13 (6):1267–1275.Bibcode:2017SMat...13.1267H.doi:10.1039/C6SM02643E.ISSN 1744-683X.PMID 28106203.
^Van Sickle, Jan (2008).GPS for Land Surveyors (Third ed.). CRC Press. pp. 18–19.ISBN 978-0-8493-9195-8.
^Kalvani, Payam Rajabi; Jahangiri, Ali Reza; Shapouri, Samaneh; Sari, Amirhossein; Jalili, Yousef Seyed (August 2019). "Multimode AFM analysis of aluminum-doped zinc oxide thin films sputtered under various substrate temperatures for optoelectronic applications".Superlattices and Microstructures.132 106173.doi:10.1016/j.spmi.2019.106173.S2CID 198468676.
^Tyrangiel, Josh (2009-02-05)."Auto-Tune: Why Pop Music Sounds Perfect".Time. Archived fromthe original on February 10, 2009.
^Kasztenny, Bogdan (March 2016)."A New Method for Fast Frequency Measurement for Protection Applications"(PDF). Schweitzer Engineering Laboratories.Archived(PDF) from the original on 2022-10-09. Retrieved28 May 2022.

Further reading

Kmenta, Jan (1986).Elements of Econometrics (Second ed.). New York: Macmillan. pp. 298–334.ISBN 978-0-02-365070-3.
Marno Verbeek (10 August 2017).A Guide to Modern Econometrics. Wiley.ISBN 978-1-119-40110-0.
Soltanalian, Mojtaba; Stoica, Petre (2012). "Computational Design of Sequences with Good Correlation Properties".IEEE Transactions on Signal Processing.60 (5): 2180.Bibcode:2012ITSP...60.2180S.doi:10.1109/TSP.2012.2186134.
Solomon W. Golomb, andGuang 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. 108–112ISBN 9780128133477.

Hassani, Hossein (2009). Sum of the sample autocorrelation function. Random Operators and Stochastic Equations. 17 (2): pp. 125–130.doi:10.1515/ROSE.2009.008.

Hassani, Hossein (2010). A note on the sum of the sample autocorrelation function]. Physica A: Statistical Mechanics and its Applications. 389 (8): pp. 1601–1606.doi:10.1016/j.physa.2009.12.050.

Weisstein, Eric W."Autocorrelation".MathWorld.

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Descriptive statistics

Continuous data

Center	Mean Arithmetic Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode
Dispersion	Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance
Shape	Central limit theorem Moments Kurtosis L-moments Skewness

Index of dispersion

Summary tables

Data collection

Study design	Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power
Survey methodology	Sampling Cluster Stratified Opinion poll Questionnaire Standard error
Controlled experiments	Blocking Factorial experiment Interaction Random assignment Randomized controlled trial Randomized experiment Scientific control
Adaptive designs	Adaptive clinical trial Stochastic approximation Up-and-down designs
Observational studies	Cohort study Cross-sectional study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in
Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife
Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons
Parametric tests	Likelihood-ratio Score/Lagrange multiplier Wald

Z-test(normal) Student'st-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality(Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC
Rank statistics	Sign Sample median Signed rank(Wilcoxon) Hodges–Lehmann estimator Rank sum(Mann–Whitney) Nonparametric anova 1-way(Kruskal–Wallis) 2-way(Friedman) Ordered alternative(Jonckheere–Terpstra) Van der Waerden test

Bayesian inference

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS) Template:Least squares and regression analysis
Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression
Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity
Generalized linear model	Exponential families Logistic(Bernoulli) / Binomial / Poisson regressions
Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / multivariate / time-series / survival analysis

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality
Specific tests	Dickey–Fuller Johansen Q-statistic(Ljung–Box) Durbin–Watson Breusch–Godfrey
Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model(Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) (Autoregressive model (AR))
Frequency domain	Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics
Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification
Social statistics	Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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