# ARMAProcess

ARMAProcess[{a1,,ap},{b1,,bq},v]

represents a weakly stationary autoregressive moving-average process with AR coefficients ai, MA coefficients bj, and normal white noise variance v.

ARMAProcess[{a1,,ap},{b1,,bq},Σ]

represents a weakly stationary vector ARMA process with coefficient matrices ai and bj and covariance matrix Σ.

ARMAProcess[{a1,,ap},{b1,,bq},v,init]

represents an ARMA process with initial data init.

ARMAProcess[c,]

represents an ARMA process with a constant c.

# Details  • ARMAProcess is also known as ARMA and VARMA (vector ARMA).
• ARMAProcess is a discrete-time and continuousstate random process.
• The ARMA process is described by the difference equation , 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 ARMA process should have real coefficients ai, bj, and c, and a positive variance v.
• An -dimensional vector ARMA process should have real coefficient matrices ai and bj of dimensions × , real vector c of length n, and the covariance matrix Σ should be symmetric positive definite of dimensions × .
• The ARMA process with zero constant has transfer function , where equals:
• scalar process vector process; is the × identity matrix
• ARMAProcess[tproc,{p,q}] for a time series process tproc gives an ARMA process of orders p and q, such that its transfer function agrees with PadeApproximant about zero with degrees {q,p} of the transfer function of tproc.
• ARMAProcess[tproc] attempts to return an ARMA process such that its transfer function is the same as the one of tproc.
• Possible time series processes tproc include ARProcess, SARMAProcess, and SARIMAProcess.
• ARMAProcess[p,q] represents an ARMA process of orders p and q for use in EstimatedProcess and related functions.
• ARMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

# Examples

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## Basic Examples(3)

Simulate an ARMA process:

Covariance function:

Correlation function:

Partial correlation function:

## Scope(38)

### Basic Uses(11)

Simulate an ensemble of paths:

Simulate with given precision:

Simulate a first-order scalar process:

Sample paths for positive and negative values of the parameter:

Simulate a weakly stationary process with given initial values:

For a process with a trend, initial values influence the behavior of the whole path:

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 parameters:

Compare the sample covariance functions with the one of the estimated process:

Use TimeSeriesModel to automatically find orders:

Compare the sample covariance functions with the best time series model:

Find maximum likelihood estimator:

Fix the constant and the variance and estimate the remaining parameters:

Plot the log-likelihood function together with the position of the estimated parameters:

Estimate a vector autoregressive moving average process:

Compare covariance functions for each component:

Forecast future values:

Find the forecast for the next 10 steps:

Show the 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:

### Covariance and Spectrum(6)

Closed-form correlation function for low order:

Partial correlation function:

Correlation matrix:

Covariance matrix:

Covariance function for a vector-valued process:

Power spectral density:

Vector ARProcess:

### Stationarity and Invertibility(5)

Check if a time series is weakly stationary:

For a vector process:

Find conditions for a process to be weakly stationary:

Find conditions for higher autoregressive order:

Check if a time series is invertible:

For a vector process:

Find an invertible representation:

The moments are being conserved:

Find invertibility conditions:

Find conditions for higher moving-average order:

### Estimation Methods(5)

The available methods for estimating an ARMAProcess:

Compare log-likelihoods:

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 users 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 probability density function:

Stationary mean and variance:

Compare with the density function of a normal distribution:

Compute the expectation of an expression:

Calculate a probability:

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(6)

Approximate a SARMA process with an ARMAProcess[4,3]:

Compare the covariance function for the original and the approximate processes:

Approximate a vector process:

Approximate an ARIMA process with fixed initial conditions by an ARMA process:

Compare sample paths:

Approximate a SARIMA process with fixed initial conditions with an ARMA process:

Compare sample paths:

Represent an AR process as an equivalent ARMA process:

MA:

ARIMA with known integration order:

SARMA with known seasonal order:

SARIMA with known integration and seasonal orders:

TransferFunctionModel representation:

For a vector-valued process:

StateSpaceModel representation:

For a vector-valued process:

## Applications(4)

Fit an ARMA model for the hourly measurements of temperature in August:

Find process parameters:

Check if the process is weakly stationary:

The low-order ARMA model does not capture the seasonal trend well:

Find model parameters:

Check if the process is weakly stationary:

Plot forecast with original data:

The daily exchange rates of the euro to the dollar from May 2012 through September 2012:

Fit an ARMA process to the exchange rates:

Plot forecast with original data:

Monthly water levels on Lake Mead:

Create TemporalData starting with observation from February 1935:

Find a time series model:

Check if the residuals exhibit any significant serial correlation:

Forecast the water level for the next 10 years:

## Properties & Relations(7)

ARMAProcess is a generalization of MAProcess:

ARMAProcess is a generalization of ARProcess:

ARMAProcess is a special case of ARIMAProcess:

ARMAProcess is a special case of FARIMAProcess:

ARMAProcess is a special case of SARMAProcess:

ARMAProcess is a special case of SARIMAProcess:

Squared values of a GARCHProcess follow an ARMA process:

CorrelationFunction and PartialCorrelationFunction of squared values:

The corresponding ARMA process:

CorrelationFunction and PartialCorrelationFunction of the ARMA process:

## Possible Issues(3)

Some properties are defined only for weakly stationary processes: Use FindInstance to find a weakly stationary process:

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: 