AbsoluteCorrelationFunction [data,hspec]
estimates the absolute correlation function at lags hspec from data.
AbsoluteCorrelationFunction [proc,hspec]
represents the absolute correlation function at lags hspec for the random process proc.
AbsoluteCorrelationFunction [proc,s,t]
represents the absolute correlation function at times s and t for the random process proc.
AbsoluteCorrelationFunction
AbsoluteCorrelationFunction [data,hspec]
estimates the absolute correlation function at lags hspec from data.
AbsoluteCorrelationFunction [proc,hspec]
represents the absolute correlation function at lags hspec for the random process proc.
AbsoluteCorrelationFunction [proc,s,t]
represents the absolute correlation function at times s and t for the random process proc.
Details
- AbsoluteCorrelationFunction is also known as the autocorrelation 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,…}
- AbsoluteCorrelationFunction [{x1,…,xn},h] is equivalent to .
- When data is TemporalData containing an ensemble of paths, the output represents the average across all paths.
- AbsoluteCorrelationFunction for a process proc with value x[t] at time t is given by:
-
- The symbol ⊗ represents KroneckerProduct .
- AbsoluteCorrelationFunction [proc,h] is defined only if proc is a weakly stationary process and is equivalent to AbsoluteCorrelationFunction [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 absolute correlation function at lag 2:
Sample the absolute correlation function for a random sample from an autoregressive time series:
The absolute correlation function for a discrete-time process:
The absolute correlation function for a continuous-time process:
Scope (13)
Empirical Estimates (7)
Estimate the absolute correlation function for some data at lag 5:
Obtain empirical estimates of the correlation function up to lag 9:
Compute the absolute correlation function for lags 1 to 9 in steps of 2:
Compute the absolute correlation function for a time series:
The absolute correlation function of a time series for multiple lags is given as a time series:
Estimate the absolute correlation function for an ensemble of paths:
Compare empirical and theoretical absolute correlation functions:
Plot the absolute cross-correlation for vector data:
Random Processes (6)
The absolute correlation function for a weakly stationary discrete-time process:
The absolute correlation function only depends on the antidiagonal :
The absolute correlation function for a weakly stationary continuous-time process:
The absolute correlation function only depends on the antidiagonal :
The absolute correlation function for a non-weakly stationary discrete-time process:
The absolute correlation function depends on both time arguments:
The absolute correlation function for a non-weakly stationary continuous-time process:
The absolute correlation function depends on both time arguments:
The correlation function for some time series processes:
Absolute cross-correlation plots for a vector ARProcess :
Applications (2)
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 absolute correlation function of the data decays slowly:
ARProcess is clearly a better candidate model than MAProcess :
Use the absolute correlation function to determine if a process is mean ergodic:
The process is weakly stationary:
Calculate the absolute correlation function:
Find the value of the strip integral:
Check if the limit of the integral is 0 to conclude mean ergodicity:
Properties & Relations (13)
Sample absolute correlation function is a biased estimator for the process absolute correlation function:
Calculate the sample absolute correlation function:
Absolute correlation function for the process:
Plot both functions:
Absolute correlation function for a list can be calculated using AbsoluteCorrelation :
Calculate absolute correlation function for the data:
Use absolute correlation:
AbsoluteCorrelationFunction is the off-diagonal entry in the absolute correlation matrix:
Sample absolute correlation function at lag 0 estimates the second Moment :
Sample absolute correlation function is related to CovarianceFunction :
Sample absolute correlation function is related to CorrelationFunction :
Scale by the first element:
Compare to the sample correlation function:
Use Expectation to calculate the absolute correlation function:
The absolute correlation function is related to the Moment function:
Verify equality , where is the ^(th) moment function:
The absolute correlation function is related to the CovarianceFunction :
Verify equality , where is the mean function:
The absolute correlation function equals CovarianceFunction when the mean of the process is zero:
The absolute correlation function is invariant for ToInvertibleTimeSeries :
The absolute correlation function is not invariant to centralizing:
The data has nonzero mean:
Centralize data:
Compare absolute correlation functions:
PowerSpectralDensity is a transform of the absolute correlation function for mean zero processes:
Use FourierSequenceTransform with appropriate parameters:
Compare to the power spectrum:
Possible Issues (1)
AbsoluteCorrelationFunction output may contain DifferenceRoot :
Use FunctionExpand to recover explicit powers:
Related Guides
History
Text
Wolfram Research (2012), AbsoluteCorrelationFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/AbsoluteCorrelationFunction.html.
CMS
Wolfram Language. 2012. "AbsoluteCorrelationFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AbsoluteCorrelationFunction.html.
APA
Wolfram Language. (2012). AbsoluteCorrelationFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AbsoluteCorrelationFunction.html
BibTeX
@misc{reference.wolfram_2025_absolutecorrelationfunction, author="Wolfram Research", title="{AbsoluteCorrelationFunction}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/AbsoluteCorrelationFunction.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_absolutecorrelationfunction, organization={Wolfram Research}, title={AbsoluteCorrelationFunction}, year={2012}, url={https://reference.wolfram.com/language/ref/AbsoluteCorrelationFunction.html}, note=[Accessed: 17-November-2025]}