Auto-correlation of stochastic processes
InDefinition for wide-sense stationary stochastic process
If is aNormalization
It is common practice in some disciplines (e.g. statistics and_Wiener–Khinchin_theorem
The__Auto-correlation_of_random_vectors
The_(potentially_time-dependent)_auto-correlation_matrix_(also_called_second_moment)_of_a_(potentially_time-dependent)__Properties_of_the_autocorrelation_matrix
*_The_autocorrelation_matrix_is_a___Auto-correlation_of_deterministic_signals_
In___Auto-correlation_of_continuous-time_signal_
Given_a___Auto-correlation_of_discrete-time_signal_
The_discrete_autocorrelation__Definition_for_periodic_signals
If__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,__Multi-dimensional_autocorrelation
Multi-_Efficient_computation
For_data_expressed_as_a__Estimation
For_a__Regression_analysis
In__Applications
*_Autocorrelation_analysis_is_used_heavily_in___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__See_also
*_Autocorrelation_matrix *_Autocorrelation_technique *_Autocorrelation_(words), Autocorrelation_of_a_formal_word *_Autocorrelator *_Correlation_function *_Correlogram *_Cross-correlation *_Galton's_problem *_Partial_autocorrelation_function *_Fluorescence_correlation_spectroscopy *_Optical_autocorrelation *_Pitch_detection_algorithm *_Triple_correlation *_CUSUM *_Cochrane–Orcutt_estimation_(transformation_for_autocorrelated_error_terms) *_Prais–Winsten_transformation *_Scaled_correlation *_Unbiased_estimation_of_standard_deviation#Effect_of_autocorrelation_(serial_correlation), Unbiased_estimation_of_standard_deviation_References
_Further_reading
*_ *_ *_Mojtaba_Soltanalian,_and_Petre_Stoica.Autocorrelation of white noise
The autocorrelation of a continuous-time white noise signal will have a strong peak (represented by a Dirac delta function) atWiener–Khinchin theorem
The Wiener–Khinchin theorem relates the autocorrelation functionAuto-correlation of random vectors
The (potentially time-dependent) auto-correlation matrix (also called second moment) of a (potentially time-dependent) random vectorProperties 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, i.e.Auto-correlation of deterministic signals
InAuto-correlation of continuous-time signal
Given a Signal (electronics), signalAuto-correlation of discrete-time signal
The discrete autocorrelationDefinition for periodic signals
IfProperties
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 Stationary process#Weak or wide-sense stationarity, wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry,Multi-dimensional autocorrelation
Multi-Efficient computation
For data expressed as a Discrete signal, discrete sequence, it is frequently necessary to compute the autocorrelation with high algorithmic efficiency, computational efficiency. A brute force method based on the signal processing definitionEstimation
For a Discrete signal, discrete process with known mean and variance for which we observeRegression analysis
In regression analysis using time series analysis, 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 errors and residuals in statistics, 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" inApplications
* 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 spectrum, optical spectra and the measurement of very-short-durationSerial 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 aSee also
* Autocorrelation matrix * Autocorrelation technique * Autocorrelation (words), Autocorrelation of a formal word * Autocorrelator * Correlation function * Correlogram * Cross-correlation * Galton's problem * Partial autocorrelation function * Fluorescence correlation spectroscopy * Optical autocorrelation * Pitch detection algorithm * Triple correlation * CUSUM * Cochrane–Orcutt estimation (transformation for autocorrelated error terms) * Prais–Winsten transformation * Scaled correlation * Unbiased estimation of standard deviation#Effect of autocorrelation (serial correlation), Unbiased estimation of standard deviationReferences
Further reading
* * * Mojtaba Soltanalian, and Petre Stoica.Autocorrelation of white noise
The autocorrelation of a continuous-time white noise signal will have a strong peak (represented by a Dirac delta function) atWiener–Khinchin theorem
The Wiener–Khinchin theorem relates the autocorrelation functionAuto-correlation of random vectors
The (potentially time-dependent) auto-correlation matrix (also called second moment) of a (potentially time-dependent) random vectorProperties 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, i.e.Auto-correlation of deterministic signals
InAuto-correlation of continuous-time signal
Given a Signal (electronics), signalAuto-correlation of discrete-time signal
The discrete autocorrelationDefinition for periodic signals
IfProperties
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 Stationary process#Weak or wide-sense stationarity, wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry,Multi-dimensional autocorrelation
Multi-Efficient computation
For data expressed as a Discrete signal, discrete sequence, it is frequently necessary to compute the autocorrelation with high algorithmic efficiency, computational efficiency. A brute force method based on the signal processing definitionEstimation
For a Discrete signal, discrete process with known mean and variance for which we observeRegression analysis
In regression analysis using time series analysis, 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 errors and residuals in statistics, 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" inApplications
* 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 spectrum, optical spectra and the measurement of very-short-durationSerial 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 aSee also
* Autocorrelation matrix * Autocorrelation technique * Autocorrelation (words), Autocorrelation of a formal word * Autocorrelator * Correlation function * Correlogram * Cross-correlation * Galton's problem * Partial autocorrelation function * Fluorescence correlation spectroscopy * Optical autocorrelation * Pitch detection algorithm * Triple correlation * CUSUM * Cochrane–Orcutt estimation (transformation for autocorrelated error terms) * Prais–Winsten transformation * Scaled correlation * Unbiased estimation of standard deviation#Effect of autocorrelation (serial correlation), Unbiased estimation of standard deviationReferences
Further reading
* * * Mojtaba Soltanalian, and Petre Stoica.