ARMAProcess
ARMAProcess[{a_{1},…,a_{p}},{b_{1},…,b_{q}},v]
represents a weakly stationary autoregressive movingaverage process with AR coefficients a_{i}, MA coefficients b_{j}, and normal white noise variance v.
ARMAProcess[{a_{1},…,a_{p}},{b_{1},…,b_{q}},Σ]
represents a weakly stationary vector ARMA process with coefficient matrices a_{i} and b_{j} and covariance matrix Σ.
ARMAProcess[{a_{1},…,a_{p}},{b_{1},…,b_{q}},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 discretetime and continuous‐state 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 singlepath TemporalData object with time stamps understood as {…,2,1}.
 A scalar ARMA process should have real coefficients a_{i}, b_{j}, and c, and a positive variance v.
 An dimensional vector ARMA process should have real coefficient matrices a_{i} and b_{j} 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
open allclose allBasic Examples (3)
Scope (38)
Basic Uses (11)
Simulate an ensemble of paths:
Simulate with given precision:
Simulate a firstorder 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 twodimensional 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 threedimensional process:
Create a sample path function from the data:
The color of the path is the function of time:
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 loglikelihood function together with the position of the estimated parameters:
Estimate a vector autoregressive moving average process:
Compare covariance functions for each component:
Find the forecast for the next 10 steps:
Plot the data and the forecasted values:
Find a forecast for a vectorvalued time series process:
Covariance and Spectrum (6)
Closedform correlation function for low order:
Covariance function for a vectorvalued process:
Vector ARProcess:
Stationarity and Invertibility (5)
Estimation Methods (5)
The available methods for estimating an ARMAProcess:
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:
Process Slice Properties (5)
Single time SliceDistribution:
Multiple time slice distributions:
Slice distribution of a vectorvalued time series:
Firstorder probability density function:
Compare with the density function of a normal distribution:
Compute the expectation of an expression:
Skewness and kurtosis are constant:
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 an ARIMA process with fixed initial conditions by an ARMA process:
Approximate a SARIMA process with fixed initial conditions with an ARMA process:
Represent an AR process as an equivalent ARMA process:
ARIMA with known integration order:
SARMA with known seasonal order:
SARIMA with known integration and seasonal orders:
TransferFunctionModel representation:
StateSpaceModel representation:
Applications (4)
Fit an ARMA model for the hourly measurements of temperature in August:
Check if the process is weakly stationary:
The loworder ARMA model does not capture the seasonal trend well:
Daily mean temperature readings in September 2012 near your location:
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:
Forecast for 10 business days ahead:
Plot forecast with original data:
Monthly water levels on Lake Mead:
Create TemporalData starting with observation from February 1935:
Check if the residuals exhibit any significant serial correlation:
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:
Neat Examples (2)
Simulate a weakly stationary threedimensional ARMAProcess:
Nonweakly stationary process, starting at the origin:
Simulate paths from an ARMA process:
Take a slice at 50 and visualize its distribution:
Plot paths and histogram distribution of the slice distribution at 50: