# SARIMAProcess

SARIMAProcess[{a_{1},…,a_{p}},d,{b_{1},…,b_{q}},{s,{α_{1},…,α_{m}},δ,{β_{1},…,β_{r}}},v]

represents a seasonal integrated autoregressive moving-average process with ARIMA coefficients a_{i}, d, and b_{j}; seasonal order s; seasonal ARIMA coefficients α_{i}, δ, and β_{j}; seasonal integration order δ; and normal white noise with variance v.

SARIMAProcess[{a_{1},…,a_{p}},d,{b_{1},…,b_{q}},{s,{α_{1},…,α_{m}},δ,{β_{1},…,β_{r}}},Σ]

represents a vector SARIMA process with coefficient matrices a_{i}, b_{j}, α_{i}, and β_{j} and covariance matrix Σ.

SARIMAProcess[{a_{1},…},{d_{1},…},{b_{1},…},{{s_{1},…},{α_{1},…},{δ_{1},…},{β_{1},…}},Σ]

represents a vector SARIMA process with multiple integration orders d_{i}, seasonal orders s_{j}, and seasonal integration orders δ_{k}.

SARIMAProcess[{a_{1},…,a_{p}},d,{b_{1},…,b_{q}},{s,{α_{1},…,α_{m}},δ,{β_{1},…,β_{r}}},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 a
_{i}, b_{j}, α_{i}, β_{j}, 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 a
_{i}, b_{j}, α_{i}, and β_{j}of dimensions ×; vector c of length ; positive integer seasonality orders s_{i}or s; non-negative integer integration orders d_{i}or d, as well as δ_{i}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 allclose all## Basic Examples (3)

## 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:

Use TimeSeriesModel to automatically find orders:

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:

### Stationarity and Invertibility (4)

### 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:

### 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:

CentralMoment and its generating function:

FactorialMoment and its generating function:

Cumulant and its generating function:

### Representations (5)

Approximate with an ARProcess:

Approximate with an MAProcess:

Represent as equivalent ARMAProcess:

TransferFunctionModel representation:

StateSpaceModel representation:

## Applications (4)

#### Weather Data (1)

#### Airline Passengers (2)

#### 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:

Find forecast for the next seven years:

Calculate 95% confidence bands for the forecast:

## 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:

## 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:

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: