Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Autocorrelation
Summary
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. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.
Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance.
Unit root processes, trend-stationary processes, autoregressive processes, and moving average processes are specific forms of processes with autocorrelation.
In statistics, the autocorrelation of a real or complex random process is the Pearson correlation between values of the process at different times, as a function of the two times or of the time lag. Let be a random process, and be any point in time ( may be an integer for a discrete-time process or a real number for a continuous-time process). Then is the value (or realization) produced by a given run of the process at time . Suppose that the process has mean and variance at time , for each . Then the definition of the auto-correlation function between times and is
where is the expected value operator and the bar represents complex conjugation. Note that the expectation may not be well defined.
Subtracting the mean before multiplication yields the auto-covariance function between times and :
Note that this expression is not well defined for all time series or processes, because the mean may not exist, or the variance may be zero (for a constant process) or infinite (for processes with distribution lacking well-behaved moments, such as certain types of power law).
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (32)
ENV-342: Geographic information system (GIS)
Acquisition de concepts et compétences de base liées à la représentation numérique des données géographiques et à leur insertion dans des SIG. Apprentissage de processus d'analyse spatiale pour les in
MATH-342: Time series
A first course in statistical time series analysis and applications.
FIN-417: Quantitative risk management
This course is an introduction to quantitative risk management that covers standard statistical methods, multivariate risk factor models, non-linear dependence structures (copula models), as well as p
Related lectures (153)
Stochastic Models for Communications: Continuous-Time Linear Systems
Covers continuous-time stochastic processes in linear systems, including signal analysis and filtering.
Renormalization Group: Ising Model
Explores the renormalization group applied to the Ising model and the derivation of the central limit theorem.
Stochastic Models for Communications: Continuous-Time Stochastic Processes
Covers continuous-time stochastic processes and linear systems.
Related publications (258)
Thermodynamic bounds on time-reversal asymmetry
Quantifying irreversibility of a system using finite information constitutes a major challenge in stochastic thermodynamics. We introduce an observable that measures the time-reversal asymmetry between two states after a given time lag. Our central result ...
College Pk2023The two-point correlation function covariance with fewer mocks
We present FITCOV an approach for accurate estimation of the covariance of two-point correlation functions that requires fewer mocks than the standard mock-based covariance. This can be achieved by dividing a set of mocks into jackknife regions and fitting ...
Oxford2023The DESI One-Percent Survey: Constructing Galaxy-Halo Connections for ELGs and LRGs Using Auto and Cross Correlations
In the current Dark Energy Spectroscopic Instrument (DESI) survey, emission line galaxies (ELGs) and luminous red galaxies (LRGs) are essential for mapping the dark matter distribution at < N(M)>. We measure the auto and cross correlation functions of ELGs ...
IOP Publishing Ltd2023Related people (37)
Related units (5)
Related concepts (27)
Time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart).
Spectral density
The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.
Cross-correlation
In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions.
Related MOOCs (6)
Neuronal Dynamics - Computational Neuroscience of Single Neurons
The activity of neurons in the brain and the code used by these neurons is described by mathematical neuron models at different levels of detail.
Neuronal Dynamics - Computational Neuroscience of Single Neurons
The activity of neurons in the brain and the code used by these neurons is described by mathematical neuron models at different levels of detail.
Path Integral Methods in Atomistic Modelling
The course provides an introduction to the use of path integral methods in atomistic simulations.
The path integral formalism allows to introduce quantum mechanical effects on the equilibrium and (ap