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

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

# CovarianceFunction

CovarianceFunction[data,hspec]
estimates the covariance function at lags hspec from data.

CovarianceFunction[proc,hspec]
represents the covariance function at lags hspec for the random process proc.

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

## DetailsDetails

• CovarianceFunction is also known as autocovariance function.
• 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,d τ} from to in steps of {{τ1,τ2,…}} use explicit
• CovarianceFunction at lag h for data with mean and data values is given by:
•  (xi+h- )(xi-) for scalar-valued data for vector-valued data
• When data is TemporalData containing an ensemble of paths, the output represents the average across all paths.
• CovarianceFunction for a process proc with mean function and value at time t is given by:
•  Expectation[(x[s]-μ[s])(x[t]-μ[t])] for a scalar-valued process Expectation[(x[s]-μ[s])⊗(x[t]-μ[t])] for a vector-valued process
• The symbol represents KroneckerProduct.
• CovarianceFunction[proc,h] is defined only if proc is a weakly stationary process and is equivalent to CovarianceFunction[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 covariance function at lag 2:

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

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Calculate the covariance function for a discrete-time process:

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Calculate the covariance function for a continuous-time process:

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