↑ arvau

Wikipedia / Autocorrelation

Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at different points in its domain (commonly, time). The analysis of autocorrelation is a mathematical tool for identifying repeating patterns or hidden periodicities within a signal obscured by noise. Autocorrelation is widely used in signal processing, time domain and time 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 with autocovariance.

Various time series models incorporate autocorrelation, such as unit root processes, trend-stationary processes, autoregressive processes, and moving average processes.

Autocorrelation of stochastic processes

In statistics, the autocorrelation of a real or complex random process is the Pearson correlation may be an integer for a discrete-time process or a real number for a continuous-time process.) Then the definition of the autocorrelation function is

is the expected value operator and the bar represents complex conjugation. Note that the expectation may not be well defined.

Suppose that the process has mean and variance. Subtracting the mean before multiplication yields the auto-covariance function:

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).

Definition for wide-sense stationary stochastic process

is a wide-sense stationary process: the autocovariance depends only on the time-distance between the pair of values but not on their position in time. This further implies that the autocovariance and autocorrelation can be expressed as a function of the time-lag, and that this would be an even function. This gives the more familiar forms for the autocorrelation function

and the auto-covariance function:

In particular, note that

Normalization

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

The definition of the autocorrelation coefficient of a stochastic process is

, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

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

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

Properties

Symmetry property

is an even function can be stated as respectively for a WSS process:

Maximum at zero

For a WSS process: is always real.

Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality, inequality for stochastic processes:

Autocorrelation of white noise

The autocorrelation of a continuous-time white noise signal will have a strong peak (represented by a Dirac delta function.

Wiener–Khinchin theorem

The Wiener–Khinchin theorem to the power spectral density via the Fourier transform:

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

Autocorrelation of random vectorsMatrix

The (potentially time-dependent) autocorrelation matrix (also called second moment) of a (potentially time-dependent) random vector. The autocorrelation matrix is used in various digital signal processing algorithms.

For a random vector containing random elements whose expected value and variance exist, the autocorrelation matrix is defined by

denotes transposition of the vector.

:

.

is a complex random vector, the autocorrelation matrix is instead defined by

denotes Hermitian transpose.

Properties of the autocorrelation matrix

  • The autocorrelation matrix is a Hermitian matrix for complex random vectors and a symmetric matrix for real random vectors.

  • The autocorrelation matrix is a positive semidefinite matrix, in 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:Respectively for complex random vectors:

Autocorrelation of deterministic signals

In signal 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 the autocorrelation coefficient or autocovariance function.

Autocorrelation of continuous-time signal

Given a signal is most often defined as the continuous cross-correlation.

represents the complex conjugate 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

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

For processes that are not stationary.

For processes that are also ergodic, the expectation can be replaced by the limit of a time average. The autocorrelation of an ergodic process is sometimes defined as or equated to

These definitions have the advantage that they give sensible well-defined single-parameter results for periodic functions, even when those functions are not the output of stationary ergodic processes.

Alternatively, signals that last forever can be treated by a short-time autocorrelation function analysis, using finite time integrals. (See short-time Fourier transform for a related process.)

Definition for periodic signals

:

which is equivalent to

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.

, which is easy to prove from the definition. In the continuous case,

Multi-dimensional autocorrelation

Multi-dimensional autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable discrete signal would be

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 sequence, it is frequently necessary to compute the autocorrelation with high computational efficiency. A brute force method for all other values of

then we get a circular autocorrelation (similar to circular convolution The procedure can be regarded as an application of the convolution property of Z-transform of a discrete signal.

While the brute force algorithm is order , several efficient algorithms exist which can compute the autocorrelation in order . For example, the Wiener–Khinchin theorem allows computing the autocorrelation from the raw data with two fast Fourier transforms (FFT):

where IFFT denotes the inverse fast Fourier transform. The asterisk denotes complex conjugate.

Alternatively, a multiple correlation can be performed by using brute force calculation for low values, and then progressively binning the data with a logarithmic density to compute higher values, resulting in the same efficiency, but with lower memory requirements.

Estimation

For a discrete, an estimate of the autocorrelation coefficient may be obtained as

are known, this estimate is unbiased. If the true mean and variance of the process are not known there are several possibilities:

  • are replaced by the standard formulae for sample mean and sample variance, then this is a biased estimate.

  • A periodogram. This estimate is always biased; however, it usually has a smaller mean squared error.

  • separately and calculating separate sample means and/or sample variances for use in defining the estimate.

's, the variance calculated may turn out to be negative.

Hassani −1/2 theorem

In time series analysis, the Hassani −1/2 theorem is a finite-sample identity concerning the conventional estimator of the sample autocorrelation function, Hassani showed that the sum of the sample autocorrelations over all positive lags is constant:

The identity follows from the fact that the sample autocovariances are computed after subtracting the sample mean. Consequently, the centered observations sum to zero, which imposes an algebraic constraint on the full set of sample autocorrelations. The theorem is therefore a property of the estimator and the finite sample, rather than a property of the underlying stochastic process.

regardless of the underlying stationary time series.

The theorem has been discussed in relation to diagnostic checking and model selection in time series analysis. In particular, later work has examined its implications for tests based on sample autocorrelations, including the Ljung–Box statistic, and for the interpretation of empirical ACF patterns in short-memory and long-memory processes. The identity has also been used to emphasize the distinction between theoretical autocorrelations of a process and empirical autocorrelations estimated from a finite sample.

The theorem applies to the standard sample-mean-corrected ACF estimator. It should not be confused with statements about the sum of the theoretical autocorrelation function of the underlying process, which may vary depending on the model and its parameters.

Regression analysis

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. 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 \chi^2 with k degrees of freedom.

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

In the estimation of a moving average model (MA), the autocorrelation function is used to determine the appropriate number of lagged error terms to be included. This is based on the fact that for an MA process of order q.

Applications

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

  • Autocorrelation analysis is used heavily in fluorescence correlation spectroscopy to provide quantitative insight into molecular-level diffusion and chemical reactions.

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

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

  • Utilized in the GPS system to correct for the propagation delay, 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.

  • The small-angle X-ray scattering intensity of a nanostructured system is the Fourier transform of the spatial autocorrelation function of the electron density.

  • In surface science and scanning probe microscopy, 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 of an electromagnetic field.

  • In astronomy, autocorrelation can determine the frequency of pulsars.

  • In music, 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). When applied at time scales larger than a second, autocorrelation can identify the musical beat, for example to determine tempo.

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

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

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

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

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

  • In analysis of Markov chain Monte Carlo data, autocorrelation must be taken into account for correct error determination.

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

  • In medical ultrasound imaging, autocorrelation is used to visualize blood flow.

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

  • In numerical relays, autocorrelation has been used to accurately measure power system frequency.

Serial dependence

Serial dependence is closely linked to the notion of autocorrelation, but represents a distinct concept (see Correlation and dependence). In 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 in the series is statistically dependent. A series is serially independent if there is no dependence between any pair.

is stationary.

See also

References

Further reading