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

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Basic 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:

Introduced in 2012
 (9.0)
 |
Updated in 2014
 (10.0)