# 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

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## Basic Examples(1)

Find an invertible representation:

Check if the new process is invertible:

## Scope(4)

Compare a non-invertible times series with its invertible representation:

Find the invertible representation:

Compare means:

Compare covariance functions:

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:

Generate random samples:

Compare samples:

The sample correlation functions are comparable:

## 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(1)

ToInvertibleTimeSeries may not exist: There are zeros of TransferFunctionModel lying on the unit circle: