ARProcess
ARProcess[{a_{1},…,a_{p}},v]
represents a weakly stationary autoregressive process of order p with normal white noise variance v.
ARProcess[{a_{1},…,a_{p}},Σ]
represents a weakly stationary vector AR process with multinormal white noise covariance matrix Σ.
ARProcess[{a_{1},…,a_{p}},v,init]
represents an AR process with initial data init.
ARProcess[c,…]
represents an AR process with a constant c.
Details
 ARProcess is also known as AR or VAR (vector AR).
 ARProcess is a discretetime and continuousstate random process.
 The AR 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 singlepath TemporalData object with time stamps understood as {…,2,1}.
 A scalar AR process can have real coefficients a_{i} and c, a positive variance v, and a nonnegative integer order p.
 An dimensional vector AR process can have real coefficient matrices a_{i} of dimensions ×, real vector c of length , and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
 The AR process with zero constant has transfer function , where:

scalar process vector process; is the × identity matrix  ARProcess[tproc,p] for a time series process tproc gives an AR process of order p such that the series expansions about zero of the corresponding transfer functions agree up to degree p.
 Possible time series processes tproc include ARProcess, ARMAProcess, and SARIMAProcess.
 ARProcess[p] represents an autoregressive process of order p for use in EstimatedProcess and related functions.
 ARProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.
Examples
open allclose allBasic Examples (3)
Scope (37)
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:
Compare the serial dependence between consecutive values on scatter plots:
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 that of the estimated process:
Use TimeSeriesModel to automatically find orders:
Compare the sample covariance functions with the best time series model:
Find the 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 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)
For low order it is possible to find the closed form of the correlation function:
Partial correlation function is zero for lags larger than the process order:
Inverse of the covariance matrix of an ARProcess is symmetric multidiagonal:
Covariance function for a vectorvalued process:
Vector ARMAProcess:
Stationarity and Invertibility (4)
Estimation Methods (6)
The available methods for estimating an ARProcess:
Method of moments admits the following solvers:
Use a general solver for moments when fixing or repeating parameters:
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 specification of windows used for PowerSpectralDensity calculation:
Spectral estimator allows the following solvers:
This method allows for fixed parameters:
Process Slice Properties (5)
Univariate SliceDistribution:
Multivariate slice distributions:
Slice distribution of a vectorvalued time series:
Firstorder probability density function with zero initial conditions:
Compare with the density function of a normal distribution:
Compute the expectation of an expression:
Skewness and kurtosis functions are constant:
CentralMoment and its generating function:
FactorialMoment has no closed form for symbolic order:
Cumulant and its generating function:
Representations (5)
Approximate an MA process with an AR process of order 3:
Compare the covariance function for the original and the approximate processes:
Approximate an ARMA process with an AR process:
Approximate an ARMA with fixed initial values:
Approximate a SARIMA process with an AR process:
TransferFunctionModel representation:
StateSpaceModel representation:
Applications (6)
Use ARProcess to estimate an ARMAProcess:
Transform the estimated process to ARMA with given orders:
Compare loglikelihood values:
Consider the mean daily temperature for Champaign in August 2012:
Compare CorrelationFunction of the model and the data:
The hourly readings of temperature in June 2011 near your location:
Create TimeSeriesModel with estimated process:
Check goodness of fit by investigating residuals:
The daily exchange rates of the euro to the dollar from May 2012 through September 2012:
The scatter plot of consecutive values indicates strong serial correlation:
Fit an AR process to the exchange rates:
Forecast for 20 business days ahead:
Plot the forecast with original data:
Daily mean temperature readings in years 2000–2011 near your location:
Check stationarity assuming Automatic initial conditions:
Compare CorrelationFunction and PartialCorrelationFunction of the model and the sample:
The following data represents the return on DJIA and return on market capitalization for eight months during 1961. Fit a VAR model to this data:
Properties & Relations (7)
ARProcess is a special case of an ARMAProcess:
ARProcess is a special case of an ARIMAProcess:
ARProcess is a special case of a FARIMAProcess:
ARProcess is a special case of a SARMAProcess:
ARProcess is a special case of a SARIMAProcess:
Squared values of an ARCHProcess follow an AR process:
CorrelationFunction and PartialCorrelationFunction of squared values:
The corresponding autoregressive process:
CorrelationFunction and PartialCorrelationFunction of the AR process:
Cumulated AR process is equivalent to an ARMAProcess:
Possible Issues (5)
Some properties are defined only for widesense stationary processes:
Use FindInstance to find an example of a weakly stationary AR process:
A process without specified initial values must satisfy weak stationarity conditions:
Some properties will work after specifying initial value(s):
Levinson–Durbin estimation method is not always applicable:
The method of moments may not find a solution in estimation:
Maximum entropy estimation method does not allow fixed or repeated parameters:
Neat Examples (2)
Simulate a weakly stationary threedimensional ARProcess:
Nonweakly stationary process, starting at the origin:
Simulate paths from an AR process:
Take a slice at 50 and visualize its distribution:
Plot paths and histogram distribution of the slice distribution at 50: