# Autocorrelation
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## Microsims — p5.js
### Autocorrelation (p5.js) · `R_xx(τ)`
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<iframe src="https://editor.p5js.org/sciencenibber/full/GIqO0gnR1" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Autocorrelation — p5.js microsim"></iframe>
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*How similar a signal is to a time-shifted copy of itself — reveals periodicity and drives spectral estimation.*
**Open in the editor:** [▶ fork this sketch](https://editor.p5js.org/sciencenibber/sketches/GIqO0gnR1) · movement *IX · Foundations & the rest of the toolbox* · library `p5js`
### Related microsims
Live sims on neighbouring articles — 6 of them inside this article's own Wikipedia link tree:
- [[Convolution]] *(in tree)*
- [[Digital_signal_processing]] *(in tree)*
- [[Fast_Fourier_transform]] *(in tree)*
- [[Fourier_analysis]] *(in tree)*
- [[Frequency_domain]] *(in tree)*
- [[Least-squares_spectral_analysis]] *(in tree)*
*Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*
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## Links (Wikipedia order)
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`Accelerated_failure_time_model` · `Actuarial_science` · `Akaike_information_criterion` · [[Algorithmic_efficiency]] · `Analysis_of_covariance` · `Analysis_of_variance` · `Anderson–Darling_test` · `Anscombe_transform` · `Arithmetic_mean` · `Arithmetic–geometric_mean` · `Astronomy` · `Astrophysics` · `Asymptotic_theory_(statistics)` · `Autocorrelation_(words)` · `Autocorrelation_technique` · `Autocorrelator` · `Autocovariance` · `Autoregressive_conditional_heteroskedasticity` · `Autoregressive_model` · `Average_absolute_deviation` · [[Bar_chart]] · `Bayes_estimator` · `Bayes_factor` · `Bayesian_inference` · `Bayesian_information_criterion` · `Bayesian_linear_regression` · `Bayesian_probability` · `Beat_(music)` · `Bias_of_an_estimator` · `Big_O_notation` · `Binomial_regression` · [[Bioinformatics]] · `Biostatistics` · `Biplot` · `Blocking_(statistics)` · `Bootstrapping_(statistics)` · [[Box_plot]] · `Box–Jenkins_method` · `Breusch–Godfrey_test` · `CUSUM` · `Canonical_correlation` · `Cartography` · `Categorical_variable` · `Cauchy–Schwarz_inequality` · `Census` · `Central_limit_theorem` · `Chemometrics` · `Chi-squared_distribution` · `Chi-squared_test` · `Circular_convolution` · `Clinical_study_design` · `Clinical_trial` · `Cluster_analysis` · `Cluster_sampling` · `Cochrane–Orcutt_estimation` · `Cochran–Mantel–Haenszel_statistics` · `Coefficient_of_determination` · `Coefficient_of_variation` · `Cohen's_kappa` · `Cohort_study` · `Cointegration` · `Completeness_(statistics)` · `Complex_conjugate` · `Complex_random_vector` · `Computer_Physics_Communications` · `Confidence_interval` · `Confounding` · `Contingency_table` · `Contraharmonic_mean` · `Control_chart` · [[Convolution]] · `Correlation` · `Correlation_function` · `Correlogram` · `Count_data` · `Covariance_and_correlation` · `Covariance_matrix` · `Credible_interval` · `Crime_statistics` · `Cross-correlation` · `Cross-correlation_matrix` · `Cross-covariance` · `Cross-covariance_matrix` · `Cross-sectional_study` · `Cross-validation_(statistics)` · `Cubic_mean` · `Data` · `Data_collection` · `Data_preprocessing` · `Data_transformation_(statistics)` · `Decomposition_of_time_series` · `Degrees_of_freedom_(statistics)` · `Demographic_statistics` · `Density_estimation` · `Descriptive_statistics` · `Design_of_experiments` · `Dickey–Fuller_test` · [[Digital_signal_processing]] · `Dimension` · `Dimensionality_reduction` · `Dirac_delta_function` · `Distortion_(music)` · `Divergence_(statistics)` · `Domain_of_a_function` · `Durbin–Watson_statistic` · `Dynamic_light_scattering` · `Econometrics` · `Effect_size` · `Efficiency_(statistics)` · `Electron_density` · `Elliptical_distribution` · `Empirical_distribution_function` · `Engineering_statistics` · `Environmental_statistics` · `Epidemiology` · `Ergodic_process` · `Eric_W._Weisstein` · `Errors_and_residuals` · `Estimating_equations` · [[Expected_value]] · `Experiment` · `Exponential_family` · `Exponential_smoothing` · `F-test` · `Factor_analysis` · `Factorial_experiment` · `Failure_rate` · `Fan_chart_(statistics)` · [[Fast_Fourier_transform]] · `Feature_scaling` · `First-hitting-time_model` · `Fisher_transformation` · `Fluorescence_correlation_spectroscopy` · `Forest_plot` · [[Fourier_analysis]] · `Fourier_transform` · `Frequency` · [[Frequency_domain]] · `Frequentist_inference` · `Friedman_test` · `G-test` · `Galaxy` · `Gauss–Markov_theorem` · `General_linear_model` · `Generalized_least_squares` · `Generalized_linear_model` · `Generalized_mean` · `Geographic_information_system` · `Geometric_mean` · `Geophysics` · `Geostatistics` · `Goodness_of_fit` · `Granger_causality` · `Graphical_model` · `Grouped_data` · `Guang_Gong` · `Harmonic_mean` · `Heinz_mean` · `Hermitian_function` · `Hermitian_matrix` · `Heronian_mean` · [[Histogram]] · `Hodges–Lehmann_estimator` · `Homoscedasticity_and_heteroscedasticity` · `Index_of_dispersion` · `Integer` · `Interaction_(statistics)` · `Interquartile_range` · `Intertemporal_portfolio_choice` · `Interval_estimation` · `Isotonic_regression` · `Jackknife_resampling` · `Jan_Kmenta` · `Jarque–Bera_test` · `Johansen_test` · `Jonckheere's_trend_test` · `Jurimetrics` · `Kaplan–Meier_estimator` · `Kendall_rank_correlation_coefficient` · `Kolmogorov–Smirnov_test` · `Kriging` · `Kruskal–Wallis_test` · `Kurtosis` · `L-moment` · `Laser` · [[Least-squares_spectral_analysis]] · `Lehmann–Scheffé_theorem` · `Lehmer_mean` · `Light` · `Likelihood-ratio_test` · `Likelihood_function` · `Lilliefors_test` · [[Line_chart]] · `Linear_discriminant_analysis` · `Linear_regression` · `List_of_fields_of_application_of_statistics` · `List_of_statistical_tests` · `List_of_statistics_articles` · `Ljung–Box_test` · `Location_parameter` · `Location–scale_family` · `Logarithm` · `Logistic_regression` · `Loss_function` · `Lp_space` · `M-estimator` · `Mann–Whitney_U_test` · `Markov_chain_Monte_Carlo` · `Marno_Verbeek` · `Mass_spectrum` · `MathWorld` · `Maximum_a_posteriori_estimation` · `McNemar's_test` · `Mean_squared_error` · `Median` · `Medical_statistics` · `Medical_ultrasound` · `Method_of_moments_(statistics)` · `Methods_engineering` · `Micelle` · `Minimum-variance_unbiased_estimator` · `Missing_data` · `Mixed_model` · `Mode_(statistics)` · `Model_selection` · `Moment_(mathematics)` · `Monotone_likelihood_ratio` · `Multivariate_analysis_of_variance` · `Multivariate_normal_distribution` · `Multivariate_statistics` · `Music` · `National_accounts` · `Natural_experiment` · `Nelson–Aalen_estimator` · `Noise_(signal_processing)` · `Nonlinear_regression` · `Nonparametric_regression` · `Nonparametric_statistics` · `Normalization_(statistics)` · `Null_hypothesis` · `Numerical_relay` · `Observational_study` · `Official_statistics` · `One-_and_two-tailed_tests` · `Opinion_poll` · `Optical_autocorrelation` · `Optimal_decision` · `Order_statistic` · `Ordinary_least_squares` · `Outlier` · `Outline_of_statistics` · `Panel_data` · `Parametric_statistics` · `Partial_autocorrelation_function` · `Partial_correlation` · `Partition_of_sums_of_squares` · `Patterson_function` · `Pearson_correlation_coefficient` · `Peptide` · `Percentile` · `Periodic_function` · `Periodogram` · `Permutation_test` · `Phylogenetic_autocorrelation` · `Pie_chart` · `Pitch_detection_algorithm` · `Pivotal_quantity` · `Point_estimation` · `Poisson_regression` · `Posterior_probability` · `Power_(statistics)` · `Power_law` · `Power_transform` · `Prediction_interval` · `Principal_component_analysis` · `Prior_probability` · `Probabilistic_design` · `Probability_distribution` · `Propagation_delay` · `Proportional_hazards_model` · `Psychometrics` · `Pulsar` · `Quality_control` · `Quasi-experiment` · `Questionnaire` · `Q–Q_plot` · `Radar_chart` · `Random_assignment` · `Random_element` · `Random_variable` · `Randomized_controlled_trial` · `Randomized_experiment` · `Range_(statistics)` · `Rank_correlation` · `Ranking_(statistics)` · `Rao–Blackwell_theorem` · `Rate_of_return` · `Real_number` · `Rearrangement_inequality` · `Regression_analysis` · `Regression_validation` · [[Reliability_engineering]] · `Replication_(statistics)` · `Resampling_(statistics)` · `Robust_regression` · `Robust_statistics` · `Run_chart` · `Sample_size_determination` · `Sampling_(statistics)` · `Sampling_distribution` · `Scale_parameter` · `Scaled_correlation` · `Scanning_probe_microscopy` · [[Scatter_plot]] · `Scientific_control` · `Score_test` · `Seasonal_adjustment` · `Seismic_attribute` · `Semiparametric_regression` · `Shape_parameter` · `Shapiro–Wilk_test` · `Short-time_Fourier_transform` · `Sign_test` · [[Signal_processing]] · `Simple_linear_regression` · `Simultaneous_equations_model` · `Skewness` · `Small-angle_X-ray_scattering` · `Social_statistics` · `Spatial_analysis` · `Spearman's_rank_correlation_coefficient` · `Spectral_density` · `Spectral_density_estimation` · `Standard_deviation` · `Standard_error` · `Standard_score` · `Stationary_process` · `Statistic` · `Statistical_classification` · `Statistical_dispersion` · `Statistical_distance` · `Statistical_graphics` · `Statistical_hypothesis_test` · `Statistical_inference` · `Statistical_model` · `Statistical_parameter` · `Statistical_population` · `Statistical_process_control` · `Statistical_theory` · `Statistics` · `Stem-and-leaf_display` · `Stochastic_approximation` · `Stratified_sampling` · `Structural_break` · `Structural_equation_modeling` · `Student's_t-test` · `Sufficient_statistic` · `Surface_science` · `Survey_methodology` · `Survival_analysis` · `Survival_function` · `Symmetric_matrix` · [[System_identification]] · `Tempo` · `Test_statistic` · `Three-dimensional_space` · `Time` · `Time_(magazine)` · [[Time_domain]] · [[Time_series]] · `Tolerance_interval` · `Transpose` · `Trend-stationary_process` · `Triple_correlation` · `Truncation_(statistics)` · `U-statistic` · `Ultrashort_pulse` · `Unbiased_estimation_of_standard_deviation` · `Uniformly_most_powerful_test` · `Unit_root` · `V-statistic` · `Van_der_Waerden_test` · `Variance` · `Variance-stabilizing_transformation` · `Vector_autoregression` · `Violin_plot` · `Wald_test` · `Wavelet` · `White_noise` · `Whittle_likelihood` · `Wiener–Khinchin_theorem` · `Wilcoxon_signed-rank_test` · `Winsorizing` · `X-ray_binary` · `Z-test` · [[Z-transform]]
> Signal Processing concept · part of the Signal Processing Portal · movement IX · !09 解析 kaiseki.svg
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*Rendered from the live microsim (▶ motion).*
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## See it next
[](Convolution)
*→ [[Convolution|Convolution]]*
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Back to Signal Processing Portal · the room · Semiotic gateway
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## What it is
Autocorrelation is the correlation of a signal with a time-delayed copy of itself, expressed as a function of the delay (lag); it measures how similar a signal is to itself across different time shifts.
## How it works / why it matters
For a signal $x$, the autocorrelation at lag $\tau$ is $R_x(\tau)=\mathbb{E}[x(t)\,x(t+\tau)]$ (or the corresponding sum for discrete signals), peaking at zero lag and revealing periodicities as secondary peaks. It is a cornerstone of spectral analysis because, by the Wiener–Khinchin theorem, the autocorrelation of a stationary random process and its power spectral density form a Fourier-transform pair — so autocorrelation is used to detect hidden periodicity, estimate spectra, and characterize noise.
## Signs & universals
Instantiates: signal · probability · frequency · noise — a statistical measure of a signal's self-similarity tied to its frequency content and noise.
## Related
It is a special case of Similarity (signal processing) and is computed as a [[Convolution]]-like operation; via Wiener–Khinchin it links to the Discrete Fourier transform and power spectra. It rests on [[Probability]] for random signals and underlies Autoregressive modeling within Signal Processing.
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## From the Real GENERATIVE library
> Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them. ([Wikipedia](https://en.wikipedia.org/wiki/Autocorrelation))
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*Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*
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## Wikipedia : Wikitube
**Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Autocorrelation) : [Wikitube](https://en.wikitube.io/wiki/Autocorrelation)
## Previous hub tags
Tree parents: [[Monte_Carlo_method]] · [[Reliability_engineering]].
Legacy hubs: none.
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*Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*