SARIMAProcess [{a1,…,ap},d,{b1,…,bq},{s,{α1,…,αm},δ,{β1,…,βr}},v]
represents a seasonal integrated autoregressive moving-average process with ARIMA coefficients ai, d, and bj; seasonal order s; seasonal ARIMA coefficients αi, δ, and βj; seasonal integration order δ; and normal white noise with variance v.
SARIMAProcess [{a1,…,ap},d,{b1,…,bq},{s,{α1,…,αm},δ,{β1,…,βr}},Σ]
represents a vector SARIMA process with coefficient matrices ai, bj, αi, and βj and covariance matrix Σ.
SARIMAProcess [{a1,…},{d1,…},{b1,…},{{s1,…},{α1,…},{δ1,…},{β1,…}},Σ]
represents a vector SARIMA process with multiple integration orders di, seasonal orders sj, and seasonal integration orders δk.
SARIMAProcess [{a1,…,ap},d,{b1,…,bq},{s,{α1,…,αm},δ,{β1,…,βr}},v,init]
represents a SARIMA process with initial data init.
SARIMAProcess [c,…]
represents a SARIMA process with constant c.
SARIMAProcess
SARIMAProcess [{a1,…,ap},d,{b1,…,bq},{s,{α1,…,αm},δ,{β1,…,βr}},v]
represents a seasonal integrated autoregressive moving-average process with ARIMA coefficients ai, d, and bj; seasonal order s; seasonal ARIMA coefficients αi, δ, and βj; seasonal integration order δ; and normal white noise with variance v.
SARIMAProcess [{a1,…,ap},d,{b1,…,bq},{s,{α1,…,αm},δ,{β1,…,βr}},Σ]
represents a vector SARIMA process with coefficient matrices ai, bj, αi, and βj and covariance matrix Σ.
SARIMAProcess [{a1,…},{d1,…},{b1,…},{{s1,…},{α1,…},{δ1,…},{β1,…}},Σ]
represents a vector SARIMA process with multiple integration orders di, seasonal orders sj, and seasonal integration orders δk.
SARIMAProcess [{a1,…,ap},d,{b1,…,bq},{s,{α1,…,αm},δ,{β1,…,βr}},v,init]
represents a SARIMA process with initial data init.
SARIMAProcess [c,…]
represents a SARIMA process with constant c.
Details
- SARIMAProcess is a discrete-time and continuous-state random process.
- The SARIMA process is effectively the composition of an ARIMA process and a seasonal version of an ARIMA process.
- The SARIMA process is described by the difference equation , with , where is the state output, is 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 SARIMA process should have real coefficients ai, bj, αi, βj, and c, positive integer seasonality order s, non-negative integer integration orders d and δ, and a positive variance v.
- An -dimensional vector SARIMA process should have real coefficient matrices ai, bj, αi, and βj of dimensions ×; vector c of length ; positive integer seasonality orders si or s; non-negative integer integration orders di or d, as well as δi or δ; and symmetric positive definite covariance matrix Σ of dimension ×.
- The SARIMA process with zero constant has transfer function , where , , , , , and is an n-dimensional unit.
- SARIMAProcess [p,d,q,{s,sp,sd,sq}] represents a SARIMA process with autoregressive and moving-average orders p and q and integration order d, their seasonal counterparts sp, sq, and sd, and seasonality s for use in EstimatedProcess and related functions.
- SARIMAProcess can be used with such functions as CovarianceFunction , RandomFunction , and TimeSeriesForecast .
Examples
open all close allBasic Examples (3)
Simulate a SARIMA process:
Simulate SARIMA with seasonal trend:
Simulate SARIMA with linear trend:
Scope (28)
Basic Uses (9)
Simulate an ensemble of paths:
Simulate with given precision:
Simulate a scalar process with different seasonalities:
Sample paths for positive and negative values of the parameter:
Simulate a process with given initial values:
A process with both linear and seasonal trend:
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:
The 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:
Forecast future values:
Find the forecast for the next 20 steps:
Show the forecast path of the forecast:
Plot the data and the forecasted values:
Find a forecast for a vector-valued time series process:
Find the forecast for the next 15 steps:
Plot the data and the forecast for each component:
Stationarity and Invertibility (4)
Check if a time series is weakly stationary:
For a vector process:
Find conditions for a process to be weakly stationary:
Check if a time series is invertible:
Find its invertible representation:
For a vector process:
Find invertibility conditions:
Estimation Methods (5)
The available methods for estimating a SARIMAProcess :
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 you to specify windows used for PowerSpectralDensity calculation:
Spectral estimator allows the 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 stationary probability density function:
Compute the expectation of an expression:
Calculate a probability:
Skewness and kurtosis:
Moment of order r:
Generating functions:
CentralMoment and its generating function:
FactorialMoment and its generating function:
Cumulant and its generating function:
Representations (5)
Approximate with an ARProcess :
Compare random paths:
For a vector process:
Approximate with an MAProcess :
Compare random paths:
For a vector process:
Represent as equivalent ARMAProcess :
TransferFunctionModel representation:
For a vector-valued process:
StateSpaceModel representation:
For a vector-valued process:
Applications (4)
Weather Data (1)
Average temperature on the first day of a month in Chicago, IL:
Fit a SARIMA process:
Forecast the average temperatures on the first day of a month for the next three years:
Airline Passengers (2)
The following data contains the monthly total number of US international airline passengers (in thousands) from January, 1949 to December, 1960:
Find a time series model:
Forecast for the next five years:
Calculate prediction bands:
Plot a forecast within a 95% confidence interval:
Use a simulation to forecast the number of passengers:
The fitted model:
Simulate the next five years:
Find the mean function of the simulated paths:
Retail Sales (1)
Use SARIMAProcess to model seasonal data of monthly retail sales in the United States:
Create TimeSeries from the selection:
Plot the sales with grid lines at December peaks:
Fit a seasonal model:
The process parameters:
Find forecast for the next seven years:
Calculate 95% confidence bands for the forecast:
There is an upper and a lower band:
Plot the forecast within the 95% confidence region:
Properties & Relations (6)
SARIMAProcess is a generalization of an ARIMAProcess :
SARIMAProcess is a generalization of a SARMAProcess :
SARIMAProcess is a generalization of an ARMAProcess :
SARIMAProcess is a generalization of an ARProcess :
SARIMAProcess is a generalization of an MAProcess :
Compare integration orders:
Create random samples for each process:
Plot samples with various integrations:
Possible Issues (4)
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 TransferFunctionModel lying on the unit circle:
Neat Examples (2)
Simulate a three-dimensional SARIMAProcess :
Simulate paths from a SARIMA 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), SARIMAProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/SARIMAProcess.html (updated 2014).
CMS
Wolfram Language. 2012. "SARIMAProcess." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/SARIMAProcess.html.
APA
Wolfram Language. (2012). SARIMAProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SARIMAProcess.html
BibTeX
@misc{reference.wolfram_2025_sarimaprocess, author="Wolfram Research", title="{SARIMAProcess}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/SARIMAProcess.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_sarimaprocess, organization={Wolfram Research}, title={SARIMAProcess}, year={2014}, url={https://reference.wolfram.com/language/ref/SARIMAProcess.html}, note=[Accessed: 05-December-2025]}