# Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# CorrelationFunction

CorrelationFunction[data,hspec]
estimates the correlation function at lags hspec from data.

CorrelationFunction[proc,hspec]
represents the correlation function at lags hspec for the random process proc.

CorrelationFunction[proc,s,t]
represents the correlation function at times s and t for the random process proc.

## DetailsDetails

• CorrelationFunction is also known as autocorrelation or cross-correlation function (ACF or CCF).
• The following specifications can be given for hspec:
•  τ at time or lag τ {τmax} unit spaced from 0 to {τmin,τmax} unit spaced from to {τmin,τmax,dτ} from to in steps of dτ {{τ1,τ2,…}} use explicit
• CorrelationFunction[{x1,,xn},h] is equivalent to with =Mean[{x1,,xn}].
• When data is TemporalData containing an ensemble of paths, the output represents the average across all paths.
• CorrelationFunction of the process proc is the CovarianceFunction c normalized by the outer product of the standard deviation function σ at times s and t:
•  c[s,t]/(σ[s]σ[t]) for scalar-valued data or processes c[s,t]/(σ[s] ⊗ σ[t]) for vector-valued data or processes
• The symbol represents KroneckerProduct.
• CorrelationFunction[proc,h] is defined only if proc is a weakly stationary process and is equivalent to CorrelationFunction[proc,h,0].
• The process proc can be any random process, such as ARMAProcess and WienerProcess.

## ExamplesExamplesopen allclose all

### Basic Examples  (4)Basic Examples  (4)

Estimate the correlation function at lag 2:

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The sample correlation function for a random sample from an autoregressive time series:

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The correlation function for a discrete-time process:

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The correlation function for a continuous-time process:

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