ToInvertibleTimeSeries[tproc]
returns an invertible version of a time series process tproc.


ToInvertibleTimeSeries
ToInvertibleTimeSeries[tproc]
returns an invertible version of a time series process tproc.
Details

- ToInvertibleTimeSeries produces an invertible time series process of the same type, with modified moving-average coefficients and noise covariance such that it preserves mean and covariance function.
- An invertible time series representation is constructed by reflecting zeros of the corresponding transfer function outside the unit circle with regard to the unit circle. An invertible representation does not exist if there are zeros on the unit circle.
- ToInvertibleTimeSeries requires numeric moving-average coefficients.
- TimeSeriesInvertibility gives invertibility conditions for symbolic coefficients.
- ToInvertibleTimeSeries can be used with time series processes such as MAProcess, ARMAProcess, ARIMAProcess, and FARIMAProcess.
Examples
open all close allScope (5)
Compare a non-invertible times series with its invertible representation:
Find the invertible representation:
Compare a non-invertible time series with its invertible representation:
Calculate TransferFunctionModel:
Compare random samples of both processes for a fixed random seed:
Consider a non-invertible process and its invertible representation:
Find the corresponding transfer functions:
Noise for each process simulation would differ, but dependence on past noise is now smaller:
Generate a random sample from a non-invertible process and its invertible representation:
Find the invertible representation:
The sample correlation functions are comparable:
Check if a given time series is invertible:
Plot zeros of transfer function:
Since there are no zeros on the unit circle, the time series has invertible representation:
Check if zeros of the transfer function are inside the unit circle:
Properties & Relations (2)
ToInvertibleTimeSeries of an invertible process returns the original process:
ToInvertibleTimeSeries scales only the moving average coefficients and the noise variance:
Possible Issues (2)
ToInvertibleTimeSeries requires numeric moving-average coefficients:

Use TimeSeriesInvertibility to find conditions for non-numeric coefficients:
ToInvertibleTimeSeries may not exist:

There are zeros of TransferFunctionModel lying on the unit circle:
Related Guides
History
Text
Wolfram Research (2012), ToInvertibleTimeSeries, Wolfram Language function, https://reference.wolfram.com/language/ref/ToInvertibleTimeSeries.html.
CMS
Wolfram Language. 2012. "ToInvertibleTimeSeries." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ToInvertibleTimeSeries.html.
APA
Wolfram Language. (2012). ToInvertibleTimeSeries. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ToInvertibleTimeSeries.html
BibTeX
@misc{reference.wolfram_2025_toinvertibletimeseries, author="Wolfram Research", title="{ToInvertibleTimeSeries}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ToInvertibleTimeSeries.html}", note=[Accessed: 15-August-2025]}
BibLaTeX
@online{reference.wolfram_2025_toinvertibletimeseries, organization={Wolfram Research}, title={ToInvertibleTimeSeries}, year={2012}, url={https://reference.wolfram.com/language/ref/ToInvertibleTimeSeries.html}, note=[Accessed: 15-August-2025]}