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.
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
open all close allBasic 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:
Related Guides
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: 18-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_arimaprocess, organization={Wolfram Research}, title={ARIMAProcess}, year={2014}, url={https://reference.wolfram.com/language/ref/ARIMAProcess.html}, note=[Accessed: 18-November-2025]}