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:
  • τat 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
    1/(n)sum_(i=1)^(n-h)(x_(i+h)-mu^^ ) tensor (x_(i)-mu^^) 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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Introduced in 2012
(9.0)