Auto-correlation of stochastic processes
In statistics, the autocorrelation of a real or complexDefinition for wide-sense stationary stochastic process
If is a wide-sense stationary process then the mean and the variance are time-independent, and further the autocovariance function depends only on the lag between and : 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 auto-correlation can be expressed as a function of the time-lag, and that this would be an even function of the lag . This gives the more familiar forms for the auto-correlation function and the auto-covariance function: In particular, note thatNormalization
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___See_also
*__References
_Further_reading
*_ *_ *_Mojtaba_Soltanalian,_and_Petre_Stoica.Autocorrelation of white noise
The autocorrelation of a continuous-timeWiener–Khinchin theorem
TheAuto-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 aAuto-correlation of deterministic signals
InAuto-correlation of continuous-time signal
Given aAuto-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 wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry,Multi-dimensional autocorrelation
Multi-Efficient computation
For data expressed as aEstimation
For aRegression analysis
InApplications
* Autocorrelation analysis is used heavily inSerial 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. ASee also
*References
Further reading
* * * Mojtaba Soltanalian, and Petre Stoica.Autocorrelation of white noise
The autocorrelation of a continuous-timeWiener–Khinchin theorem
TheAuto-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 aAuto-correlation of deterministic signals
InAuto-correlation of continuous-time signal
Given aAuto-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 wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry,Multi-dimensional autocorrelation
Multi-Efficient computation
For data expressed as aEstimation
For aRegression analysis
InApplications
* Autocorrelation analysis is used heavily inSerial 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. ASee also
*References
Further reading
* * * Mojtaba Soltanalian, and Petre Stoica.