BUILT-IN MATHEMATICA 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.
• CovarianceFunction for a process proc with mean function and value at time t is given by:
•  Expectation[(x[s]-[s])(x[t]-[t])] for a scalar-valued process Expectation[(x[s]-[s](x[t]-[t])] for a vector-valued process
• CovarianceFunction[proc, h] is defined only if proc is a weakly stationary process and is equivalent to CovarianceFunction[proc, 0, h].
• The process proc can be any random process such as ARMAProcess and WienerProcess.
• CovarianceFunction[{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.
• The following specifications can be given for hspec:
•  at time or lag {max} unit spaced from 0 to {min,max} unit spaced from to {min,max,d} from to in steps of d {{1,2,...}} use explicit

## 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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