Autocorrelation

Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable 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, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.

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

Unit root processes, trend-stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation.

The autocorrelation matrix is a for complex random vectors and a symmetric matrix for real random vectors.^[3]^: p.190

Hermitian matrix

The autocorrelation matrix is a ,^[3]^: p.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.

positive semidefinite matrix

All eigenvalues of the autocorrelation matrix are real and non-negative.

The auto-covariance matrix is related to the autocorrelation matrix as follows:
$\operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\rm {T}}]=\operatorname {R} _{\mathbf {X} \mathbf {X} }-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {X} ]^{\rm {T}}$ Respectively for complex random vectors: $\operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])^{\rm {H}}]=\operatorname {R} _{\mathbf {Z} \mathbf {Z} }-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ]^{\rm {H}}$

even function

The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay $\tau$ , $|R_{ff}(\tau )|\leq R_{ff}(0)$ .^: p.410 This is a consequence of the rearrangement inequality. The same result holds in the discrete case.

[1]

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

periodic function

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 , it maintains all the properties of cross-correlation.

cross-correlation

By using the symbol $*$ to represent and $g_{-1}$ is a function which manipulates the function $f$ and is defined as $g_{-1}(f)(t)=f(-t)$ , the definition for $R_{ff}(\tau )$ may be written as:
$R_{ff}(\tau )=(f*g_{-1}({\overline {f}}))(\tau )$

convolution

If $\mu$ and $\sigma ^{2}$ are replaced by the standard formulae for sample mean and sample variance, then this is a .

biased estimate

A -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.^[9]^[10]

periodogram

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.

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

for any positive integer $k<n$ . When the true mean $\mu$ and variance $\sigma ^{2}$ are known, this estimate is unbiased. If the true mean and variance of the process are not known there are several possibilities:

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.^[11]

Regression analysis[edit]

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

In ordinary least squares (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.) 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). While it does not bias the OLS coefficient estimates, the standard errors tend to be underestimated (and the t-scores overestimated) when the autocorrelations of the errors at low lags are positive.

The traditional test for the presence of first-order autocorrelation is the Durbin–Watson statistic or, if the explanatory variables include a lagged dependent variable, Durbin's h statistic. The Durbin-Watson can be linearly mapped however to the Pearson correlation between values and their lags.^[12] A more flexible test, covering autocorrelation of higher orders and applicable whether or not the regressors include lags of the dependent variable, is the Breusch–Godfrey test. This involves an auxiliary regression, wherein the residuals obtained from estimating the model of interest are regressed on (a) the original regressors and (b) k lags of the residuals, where 'k' is the order of the test. The simplest version of the test statistic from this auxiliary regression is TR², where T is the sample size and R² is the coefficient of determination. Under the null hypothesis of no autocorrelation, this statistic is asymptotically distributed as $\chi ^{2}$ with k degrees of freedom.

Responses to nonzero autocorrelation include generalized least squares and the Newey–West HAC estimator (Heteroskedasticity and Autocorrelation Consistent).^[13]

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

Autocorrelation analysis is used heavily in ^[14] to provide quantitative insight into molecular-level diffusion and chemical reactions.^[15]

fluorescence correlation spectroscopy

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

optical spectra

Autocorrelation is used to analyze data, which notably enables determination of the particle size distributions of nanometer-sized particles or micelles suspended in a fluid. A laser shining into the mixture produces a speckle pattern that results from the motion of the particles. Autocorrelation of the signal can be analyzed in terms of the diffusion of the particles. From this, knowing the viscosity of the fluid, the sizes of the particles can be calculated.

dynamic light scattering

Utilized in the system to correct for the propagation delay, or time shift, between the point of time at the transmission of the carrier signal at the satellites, and the point of time at the receiver on the ground. This is done by the receiver generating a replica signal of the 1,023-bit C/A (Coarse/Acquisition) code, and generating lines of code chips [-1,1] 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 the doppler shift in the incoming satellite signal, until the receiver replica signal and the satellite signal codes match up.^[16]

GPS

The intensity of a nanostructured system is the Fourier transform of the spatial autocorrelation function of the electron density.

small-angle X-ray scattering

In and scanning probe microscopy, autocorrelation is used to establish a link between surface morphology and functional characteristics.^[17]

surface science

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

degree of coherence

In , autocorrelation can determine the frequency of pulsars.

astronomy

In , autocorrelation (when applied at time scales smaller than a second) is used as a pitch detection algorithm for both instrument tuners and "Auto Tune" (used as a distortion effect or to fix intonation).^[18] When applied at time scales larger than a second, autocorrelation can identify the musical beat, for example to determine tempo.

music

Autocorrelation in space rather than time, via the , is used by X-ray diffractionists to help recover the "Fourier phase information" on atom positions not available through diffraction alone.

Patterson function

In statistics, spatial autocorrelation between sample locations also helps one estimate when sampling a heterogeneous population.

mean value uncertainties

The algorithm for analyzing mass spectra makes use of autocorrelation in conjunction with cross-correlation to score the similarity of an observed spectrum to an idealized spectrum representing a peptide.

SEQUEST

In , autocorrelation is used to study and characterize the spatial distribution of galaxies in the universe and in multi-wavelength observations of low mass X-ray binaries.

astrophysics

In , spatial autocorrelation refers to correlation of a variable with itself through space.

panel data

In analysis of data, autocorrelation must be taken into account for correct error determination.

Markov chain Monte Carlo

In (specifically in geophysics) it can be used to compute an autocorrelation seismic attribute, out of a 3D seismic survey of the underground.

geosciences

In imaging, autocorrelation is used to visualize blood flow.

medical ultrasound

In , the presence or absence of autocorrelation in an asset's rate of return can affect the optimal portion of the portfolio to hold in that asset.

intertemporal portfolio choice

In , autocorrelation has been used to accurately measure power system frequency.^[19]

numerical relays

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

Serial dependence[edit]

Serial dependence is closely linked to the notion of autocorrelation, but represents a distinct concept (see Correlation 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.

A time series of a random variable 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 $\left\{X_{t}\right\}$ 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 $\tau =s-t$ .

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Marno Verbeek

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Computational design of sequences with good correlation properties

Solomon W. Golomb, and Guang Gong. . Cambridge University Press, 2005.

Signal design for good correlation: for wireless communication, cryptography, and radar

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"Autocorrelation". MathWorld.