({{, (RefLink[TimeSeriesInvertibility, paclet:ref/TimeSeriesInvertibility])[tproc] 
gives conditions for the time series process tproc to be invertible. }})

OpenersNotesSection(Spacer1)Details

A time series process is invertible if it can be written as an autoregressive time series, possibly of infinite order, such that the autoregressive coefficients are absolutely summable.

An invertible time series representation with preserved moments can be found using (RefLink[ToInvertibleTimeSeries, paclet:ref/ToInvertibleTimeSeries]).

(RefLink[TimeSeriesInvertibility, paclet:ref/TimeSeriesInvertibility]) can be used with time series processes such as (RefLink[MAProcess, paclet:ref/MAProcess]), (RefLink[ARMAProcess, paclet:ref/ARMAProcess]), (RefLink[ARIMAProcess, paclet:ref/ARIMAProcess]), and (RefLink[FARIMAProcess, paclet:ref/FARIMAProcess]).

OpenersPrimaryExamplesSection(Spacer1)Examples  (6)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Check if a process is invertible:

TimeSeriesInvertibility[ARMAProcess[.1,{.2,.3},{-.4},1]]

TimeSeriesInvertibility[ARMAProcess[.1,{.2,.3},{4},1]]

Generate conditions for a process to be invertible:

TimeSeriesInvertibility[MAProcess[{a,b},(σ^2)]]

OpenersExampleSection1(Spacer1)Scope  (2)

Visualize conditions for an (RefLink[MAProcess, paclet:ref/MAProcess]) to be invertible:

RegionPlot[TimeSeriesInvertibility[MAProcess[{a,b},v]],{a,-2,2},{b,-2,2}]

For three parameters:

RegionPlot3D[Evaluate@TimeSeriesInvertibility[MAProcess[{a,b,c},v]],{a,-2,2},{b,-2,2},{c,-2,2},ViewPoint->{0,0,-2},PlotPoints->50,AxesLabel->Automatic]

Find an invertible (RefLink[MAProcess, paclet:ref/MAProcess]):

proc=MAProcess[c,{a,b},v,{x}];

FindInstance[TimeSeriesInvertibility[proc]&&a≠0&&b≠0,{a,b}]

Check:

TimeSeriesInvertibility[proc/.%[[1]]]

OpenersExampleSection4(Spacer1)Properties & Relations  (2)

Every (RefLink[ARProcess, paclet:ref/ARProcess]) is invertible:

TimeSeriesInvertibility[ARProcess[c,{(Subscript[a, 1]),(Subscript[a, 2])},(σ^2)]]

The conditions for an (RefLink[ARMAProcess, paclet:ref/ARMAProcess]) to be invertible depend only on the moving-average parameters:

TimeSeriesInvertibility[ARMAProcess[c,{a,b},{k,m},v]]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[ToInvertibleTimeSeries, paclet:ref/ToInvertibleTimeSeries])  ▪  (RefLink[ARProcess, paclet:ref/ARProcess])  ▪  (RefLink[MAProcess, paclet:ref/MAProcess])  ▪  (RefLink[ARMAProcess, paclet:ref/ARMAProcess])  ▪  (RefLink[ARIMAProcess, paclet:ref/ARIMAProcess])  ▪  (RefLink[FARIMAProcess, paclet:ref/FARIMAProcess])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Time Series Processes, paclet:guide/TimeSeriesProcesses])}})}})