# ARIMAProcess

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

represents an autoregressive integrated moving-average process such that its  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) 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) 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 d1.
• 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 19511980 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: 