Everything about Autocorrelation totally explained
Autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal which has been buried under noise, or identifying the
missing fundamental frequency in a signal implied by its
harmonic frequencies.
It is used frequently in
signal processing for analyzing functions or series of values, such as
time domain signals.
Informally, it's the similarity between observations as a function of the time separation between them.
More precisely, it's the
cross-correlation of a signal with itself.
Definitions
Different definitions of autocorrelation are in use depending on the field of study which is being considered and not all of them are equivalent. In some fields, the term is used interchangeably with
autocovariance.
Statistics
In
statistics, the autocorrelation function (ACF) of a random process describes the
correlation between the process at different points in time. Let
Xt be the value of the process at time
t (where
t may be an integer for a
discrete-time process or a real number for a continuous-time process).
If
Xt has
mean μ and
variance σ
2 then the definition of the ACF is
»
Regression analysis
In
regression analysis using
time series data, autocorrelation of the residuals ("error terms", in
econometrics) is a problem.
Autocorrelation violates the OLS assumption that the error terms are uncorrelated. While it doesn't 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. 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 Χ
2 with
k degrees of freedom.
Responses to nonzero autocorrelation include
generalized least squares and
Newey–West standard errors.
Applications
One 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.
In optics, normalized autocorrelations and cross-correlations give the degree of coherence of an electromagnetic field.
In signal processing, autocorrelation can give information about repeating events like musical beats or pulsar frequencies, though it can't tell the position in time of the beat. It can also be used to estimate the pitch of a musical tone.Further Information
Get more info on 'Autocorrelation'.
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