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CovarianceFunction—Wolfram Language Documentation

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• See Also
  • CorrelationFunction
  • AbsoluteCorrelationFunction
  • PartialCorrelationFunction
  • Covariance
  • Correlation
  • WeakStationarity
  • PowerSpectralDensity
• Related Guides
  • Random Processes
  • Fourier Analysis
  • Stochastic Differential Equation Processes
  • Time Series Processes
  • Probability & Statistics
  • Queueing Processes

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.

• CovarianceFunction is also known as autocovariance function.
• The following specifications can be given for hspec:
• τ at time or lag τ {τ_max} unit spaced from 0 to τ_max {τ_min,τ_max} unit spaced from τ_min to τ_max {τ_min,τ_max,d τ} from τ_min to τ_max in steps of d τ {{τ₁,τ₂,…}} use explicit {τ₁,τ₂,…}
• CovarianceFunction at lag h for data with mean and data values x_i is given by:
• 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 μ[t] and value x[t] 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.

Estimate the covariance function at lag 2:

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

Calculate the covariance function for a discrete-time process:

Calculate the covariance function for a continuous-time process:

Estimate the covariance function for some data at lag 9:

Obtain empirical estimates of the covariance function up to lag 9:

Compute the covariance function for lags 1 to 9 in steps of 2:

Compute the covariance function for a time series:

The covariance function of a time series for multiple lags is given as a time series:

Estimate the covariance function for an ensemble of paths:

Compare empirical and theoretical covariance functions:

Plot the cross-covariance for vector data:

The covariance function for a weakly stationary discrete-time process:

The covariance function only depends on the antidiagonal :

The covariance function for a weakly stationary continuous-time process:

The covariance function only depends on the antidiagonal :

The covariance function for a non-weakly stationary discrete-time process:

The covariance function depends on both time arguments:

The covariance function for a non-weakly stationary continuous-time process:

The covariance function depends on both time arguments:

The covariance function for some time-series processes:

Cross-covariance plots for a vector ARProcess:

Determine whether the following data is best modeled with an MAProcess or an ARProcess:

It is difficult to determine the underlying process from sample paths:

The covariance function of the data decays slowly:

ARProcess is clearly a better candidate model than MAProcess:

Sample covariance function is a biased estimator for the process covariance function:

Calculate the sample covariance function:

Covariance function for the process:

Plot both functions:

Covariance function for a process is the off-diagonal entry in the Covariance matrix:

Sample covariance function at lag 0 is a variance estimator:

Compare to the estimate using Variance:

The scaling factors are different:

Sample covariance function is related to CorrelationFunction:

Scaled sample correlation function:

Sample covariance function is related to AbsoluteCorrelationFunction:

Use Expectation to calculate the covariance function:

Covariance function for equal times reduces to Variance:

The covariance function is related to the AbsoluteCorrelationFunction :

For , the mean function is :

The covariance function is related to the Covariance:

It is the off-diagonal entry in the covariance matrix:

The covariance function is related to the CorrelationFunction :

For , the standard deviation function is :

Covariance function is invariant for ToInvertibleTimeSeries:

Covariance function is invariant to centralizing:

The data has nonzero mean:

Centralize data:

Compare covariance functions:

PowerSpectralDensity of a time series is a transform of the covariance function:

Use FourierSequenceTransform:

Compare to the power spectrum:

PowerSpectralDensity of data is a transform of the sample covariance function:

Apply ListFourierSequenceTransform:

Compare to SamplePowerSpectralDensity:

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