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.
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.
Details
- 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 τmax{τmin,τmax} unit spaced from τmin to τmax{τmin,τmax,dτ} from τmin to τmax in steps of dτ{{τ1,τ2,…}} use explicit {τ1,τ2,…}
- CovarianceFunction at lag h for data with mean and data values xi is given by:
-
(xi+h- )(xi-) for scalar-valued data1/(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 μ[t] and value x[t] at time t is given by:
-
- 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 .
Examples
open all close allBasic Examples (4)
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:
Scope (13)
Empirical Estimates (7)
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:
Random Processes (6)
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 :
Applications (1)
Properties & Relations (14)
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:
Possible Issues (1)
CovarianceFunction output may contain DifferenceRoot :
Use FunctionExpand to recover explicit powers:
History
Text
Wolfram Research (2012), CovarianceFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CovarianceFunction.html.
CMS
Wolfram Language. 2012. "CovarianceFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CovarianceFunction.html.
APA
Wolfram Language. (2012). CovarianceFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CovarianceFunction.html
BibTeX
@misc{reference.wolfram_2025_covariancefunction, author="Wolfram Research", title="{CovarianceFunction}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/CovarianceFunction.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_covariancefunction, organization={Wolfram Research}, title={CovarianceFunction}, year={2012}, url={https://reference.wolfram.com/language/ref/CovarianceFunction.html}, note=[Accessed: 17-November-2025]}