Auto-correlation of stochastic processes
InDefinition 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 anNormalization
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 auto-correlation coefficient of a stochastic process is If the function is well defined, its value must lie in the range_Wiener–Khinchin_theorem
The_ Wiener–Khinchin_theorem_relates_the_autocorrelation_function__Auto-correlation_of_random_vectors
The_(potentially_time-dependent)_auto-correlation_matrix_(also_called_second_moment)_of_a_(potentially_time-dependent)_ random_vector__Properties_of_the_autocorrelation_matrix
*_The_autocorrelation_matrix_is_a_ Hermitian_matrix_for_complex_random_vectors_and_a_ symmetric_matrix_for_real_random_vectors.__Auto-correlation_of_deterministic_signals_
In___Auto-correlation_of_continuous-time_signal_
Given_a_ signal___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_ discrete_sequence,_it_is_frequently_necessary_to_compute_the_autocorrelation_with_high_ computational_efficiency._A_ brute_force_method_based_on_the_signal_processing_definition__Estimation
For_a_ discrete_process_with_known_mean_and_variance_for_which_we_observe__Regression_analysis
In__Applications
*_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___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
*_ Autocorrelation_matrix *__References
_Further_reading
*_ *_ *_Mojtaba_Soltanalian,_and_Petre_Stoica.Autocorrelation of white noise
The autocorrelation of a continuous-timeWiener–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 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 wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry,Multi-dimensional autocorrelation
Multi-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 based on the signal processing definitionEstimation
For a discrete process with known mean and variance for which we observeRegression analysis
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 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. ASee also
* Autocorrelation matrix *References
Further reading
* * * Mojtaba Soltanalian, and Petre Stoica.Autocorrelation of white noise
The autocorrelation of a continuous-timeWiener–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 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 wide-sense stationary processes. * A fundamental property of the autocorrelation is symmetry,Multi-dimensional autocorrelation
Multi-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 based on the signal processing definitionEstimation
For a discrete process with known mean and variance for which we observeRegression analysis
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 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. ASee also
* Autocorrelation matrix *References
Further reading
* * * Mojtaba Soltanalian, and Petre Stoica.