PartialCorrelationFunction [data,hspec]
estimates the partial correlation function at lags hspec from data.
PartialCorrelationFunction [tproc,hspec]
represents the partial correlation function at lags hspec for the time series process tproc.
PartialCorrelationFunction
PartialCorrelationFunction [data,hspec]
estimates the partial correlation function at lags hspec from data.
PartialCorrelationFunction [tproc,hspec]
represents the partial correlation function at lags hspec for the time series process tproc.
Details
- PartialCorrelationFunction is also known as the partial autocorrelation function (PACF).
- PartialCorrelationFunction represents the correlation between x(t) and x(t+h), conditioned on x(u) for t<u<t+h, and x(t) representing tproc at time t.
- PartialCorrelationFunction [tproc,hspec] is defined only if tproc is a weakly stationary process.
- The process tproc can be any process such that WeakStationarity [tproc] gives True .
- 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,…}
Examples
open all close allBasic Examples (3)
Estimate the partial correlation function at lag 2:
Sample partial correlation function for a random sample from an autoregressive time series:
Partial correlation function for an ARProcess :
Scope (9)
Empirical Estimates (6)
Estimate the partial correlation function for some data at lag 9:
Obtain empirical estimates of the partial correlation function up to lag 9:
Compute the partial correlation function for lags 1 to 9 in steps of 2:
Compute the partial correlation function for a time series:
The partial correlation function of a time series for multiple lags is given as a time series:
Estimate the partial correlation function for an ensemble of paths:
Compare empirical and theoretical correlation functions:
Random Processes (3)
Partial correlation function for a MAProcess has infinite support:
Partial correlation function for an ARProcess has finite support:
Partial correlation function for an ARMAProcess has infinite support:
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 partial correlation function of the data decays slowly:
MAProcess is clearly a better candidate model than ARProcess :
Create a PACF plot with white-noise confidence bands:
Plot the partial correlation to lag 20 with 95% white-noise confidence bands:
Compare to uncorrelated white noise:
Properties & Relations (3)
Sample partial correlation function is a biased estimator for the process partial correlation function:
Calculate the sample partial correlation function:
Partial correlation function for the process:
Plot both functions:
Use CorrelationFunction to directly calculate PartialCorrelationFunction :
Define a ToeplitzMatrix using the first components of the correlation vector:
Replace the last column in the matrix with the last components:
Calculate ratio of determinants:
Compare to the value of PartialCorrelationFunction :
Partial correlation function and correlation function agree for lag of 1:
For an ARProcess :
Possible Issues (2)
Partial correlation function does not exist for non-weakly stationary processes:
Partial correlation function is not defined for zero time difference:
Related Guides
History
Text
Wolfram Research (2012), PartialCorrelationFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/PartialCorrelationFunction.html.
CMS
Wolfram Language. 2012. "PartialCorrelationFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PartialCorrelationFunction.html.
APA
Wolfram Language. (2012). PartialCorrelationFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PartialCorrelationFunction.html
BibTeX
@misc{reference.wolfram_2025_partialcorrelationfunction, author="Wolfram Research", title="{PartialCorrelationFunction}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/PartialCorrelationFunction.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_partialcorrelationfunction, organization={Wolfram Research}, title={PartialCorrelationFunction}, year={2012}, url={https://reference.wolfram.com/language/ref/PartialCorrelationFunction.html}, note=[Accessed: 17-November-2025]}