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ARIMAProcess[{a1,…,ap},d,{b1,…,bq},v]

represents an autoregressive integrated moving-average process such that its d^(th) difference is a weakly stationary ARMAProcess[{a1,…,ap},{b1,…,bq},v].

ARIMAProcess[{a1,…,ap},d,{b1,…,bq},Σ]

represents a vector ARIMA process (y1(t),… ,yn(t)) such that its (d,…,d)^(th) difference is a vector weakly stationary ARMAProcess.

ARIMAProcess[{a1,…,ap},{d1,…,dn},{b1,…,bq},Σ]

represents a vector ARIMA process (y1(t),… ,yn(t)) such that its (d1,…,dn)^(th) difference is a vector weakly stationary ARMAProcess.

ARIMAProcess[{a1,…,ap},d,{b1,…,bq},v,init]

represents an ARIMA process with initial data init.

ARIMAProcess[c,…]

represents an ARIMA process with a constant c.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Stationarity and Invertibility  
Estimation Methods  
Process Slice Properties  
Representations  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Related Guides
History
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ARIMAProcess

ARIMAProcess[{a1,…,ap},d,{b1,…,bq},v]

represents an autoregressive integrated moving-average process such that its d^(th) difference is a weakly stationary ARMAProcess[{a1,…,ap},{b1,…,bq},v].

ARIMAProcess[{a1,…,ap},d,{b1,…,bq},Σ]

represents a vector ARIMA process (y1(t),… ,yn(t)) such that its (d,…,d)^(th) difference is a vector weakly stationary ARMAProcess.

ARIMAProcess[{a1,…,ap},{d1,…,dn},{b1,…,bq},Σ]

represents a vector ARIMA process (y1(t),… ,yn(t)) such that its (d1,…,dn)^(th) difference is a vector weakly stationary ARMAProcess.

ARIMAProcess[{a1,…,ap},d,{b1,…,bq},v,init]

represents an ARIMA process with initial data init.

ARIMAProcess[c,…]

represents an ARIMA process with a constant c.

Details

  • ARIMAProcess is a discrete-time and continuous-state random process.
  • An ARIMAProcess[…,d,…,v] has a polynomial trend of degree d for d≥1.
  • The ARIMA process is described by the difference equation , where is the state output, is the white noise input, is the shift operator and the constant c is taken to be zero if not specified.
  • The initial data init can be given as a list {…,y[-2],y[-1]} or a single-path TemporalData object with time stamps understood as {…,-2,-1}.
  • A scalar ARIMA process should have real coefficients ai, bj, and c, non-negative integer integration order d, and a positive variance v.
  • An -dimensional vector ARIMA process should have real coefficient matrices ai and bj of dimensions ×, real vector c of length , integer non-negative integrating orders di or integer non-negative integrating order d, and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
  • The ARIMA process with zero constant has transfer function , where , , and where is an -dimensional unit.
  • ARIMAProcess[p,d,q] represents an ARIMA process with autoregressive and moving average orders p and q and integration order d for use in EstimatedProcess and related functions.
  • ARIMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

Examples

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Basic Examples  (2)

Simulate an ARIMA process with a linear trend:

Simulate an ARIMA process with a quadratic trend:

Scope  (25)

Basic Uses  (9)

Simulate an ensemble of paths:

Simulate with given precision:

Simulate a process with given initial values:

In the presence of a nonzero constant:

Simulate a two-dimensional process:

Create a 2D sample path function from the data:

The color of the path is the function of time:

Create a 3D sample path function with time:

Color of the path is the function of time:

Simulate a three-dimensional process:

Create a sample path function from the data:

The color of the path is the function of time:

Estimate process parameters:

Find model parameters:

Use TimeSeriesModel to automatically find orders:

Estimate a vector process:

Forecast future values:

Show forecast path:

Plot the data and the forecasted values:

Find a forecast for a vector-valued time series process:

Find the forecast for the next 10 steps:

Plot the data and the forecast for each component:

Stationarity and Invertibility  (2)

Find conditions for a process to be weakly stationary:

Find invertibility conditions:

Estimation Methods  (5)

The available methods for estimating an ARIMAProcess:

Method of moments admits the following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Maximum conditional likelihood method allows the following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Maximum likelihood method allows the following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Spectral estimator allows to specify windows used for PowerSpectralDensity calculation:

Spectral estimator allows following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Process Slice Properties  (5)

Single time SliceDistribution:

Multiple time slice distributions:

Slice distribution of a vector-valued time series:

First-order probability density function:

Compute the expectation of an expression:

Calculate the probability of an expression:

Skewness and kurtosis are constant:

Moment of order r:

Generating functions:

CentralMoment and its generating function:

FactorialMoment has no closed form for symbolic order:

Cumulant and its generating function:

Representations  (4)

Approximate with an MA process:

Approximate with an AR process:

Compare sample paths:

Approximate a vector process:

Represent as the equivalent ARMA process:

It is usually not weakly stationary:

TransferFunctionModel representation:

For a vector-valued process:

StateSpaceModel representation:

For a vector-valued process:

Applications  (3)

Forecast annual revenue of commercial airlines:

Data has a linear trend that can be confirmed using UnitRootTest:

Fit an ARIMA model to the time series:

Find the forecast for 10 years ahead:

Global yearly mean temperature compared to 1951–1980 baseline:

Find order of integration with UnitRootTest:

Estimate an ARIMA with integration order equal to 1:

Find the forecast for the next 20 years:

Forecast stock prices:

Check if regularly sampled:

Resample to obtain regularly sampled time series:

Plot the prices:

Fit an ARIMA process:

Forecast to the next half a year:

Properties & Relations  (4)

ARIMAProcess is a generalization of an ARMAProcess:

ARIMAProcess is a generalization of an ARProcess:

ARIMAProcess is a generalization of an MAProcess:

ARIMA process follows WienerProcess in discrete steps:

Single time slice properties:

Mixed moments:

Possible Issues  (5)

Multi-time-slice properties may not evaluate for symbolic time stamps:

Some properties are defined only for weakly stationary processes:

Use FindInstance to find a weakly stationary process:

Slice distribution properties with inexact parameters may be ill-conditioned for symbolic times:

The negative result is incorrect:

Use numeric times:

Or use exact values of parameters:

ToInvertibleTimeSeries does not always exist:

There are zeros of the TransferFunctionModel on the unit circle:

The method of moments may not find a solution in estimation:

Use the "FindRoot" method instead:

Neat Examples  (2)

Simulate a three-dimensional ARIMAProcess:

Simulate paths from an ARIMA process:

Take a slice at 50 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 50:

Wolfram Research (2012), ARIMAProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ARIMAProcess.html (updated 2014).

Text

Wolfram Research (2012), ARIMAProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ARIMAProcess.html (updated 2014).

CMS

Wolfram Language. 2012. "ARIMAProcess." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/ARIMAProcess.html.

APA

Wolfram Language. (2012). ARIMAProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ARIMAProcess.html

BibTeX

@misc{reference.wolfram_2025_arimaprocess, author="Wolfram Research", title="{ARIMAProcess}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ARIMAProcess.html}", note=[Accessed: 01-May-2026]}

BibLaTeX

@online{reference.wolfram_2025_arimaprocess, organization={Wolfram Research}, title={ARIMAProcess}, year={2014}, url={https://reference.wolfram.com/language/ref/ARIMAProcess.html}, note=[Accessed: 01-May-2026]}