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MAProcess

MAProcess [{b₁,…,b_q},v]

represents a moving-average process of order q with normal white noise variance v.

MAProcess [{b₁,…,b_q},Σ]

represents a vector MA process with multinormal white noise covariance matrix Σ.

MAProcess [{b₁,…,b_q},v,init]

represents an MA process with initial data init.

MAProcess [c,…]

represents an MA process with a constant c.

Details

MAProcess is also known as a finite impulse response (FIR) filter.
MAProcess is a discrete-time and continuous-state random process.
The MA 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 MA process should have real coefficients b_i and c, and a positive variance v.
An -dimensional vector MA process should have real coefficient matrices b_i of dimensions ×, real vector c of length , and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
The MA process with zero constant has transfer function where:
scalar process

vector process; is the × identity matrix
MAProcess [tproc,q] for a time series process tproc gives an MA process of order q such that the series expansions about zero of the corresponding transfer functions agree up to degree q.
Possible time series processes tproc include ARProcess , ARMAProcess , and SARIMAProcess .
MAProcess [q] represents a moving-average process of order q for use in EstimatedProcess and related functions.
MAProcess can be used with such functions as CovarianceFunction , RandomFunction , and TimeSeriesForecast .

Examples

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Basic Examples (3)

Simulate an MA process:

Covariance function:

Correlation function:

Partial correlation function:

Scope (37)

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:

Initial values do not influence the process values:

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:

Estimation:

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 moving-average process:

Compare covariance functions for each component:

Forecast future values:

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)

Correlation function exists in closed form:

Closed form of the partial correlation function for the first order:

Covariance matrix:

Covariance matrix of an MAProcess is symmetric multidiagonal:

Correlation matrix:

Covariance function for a vector-valued process:

Power spectral density:

Vector MAProcess :

Stationarity and Invertibility (4)

MAProcess is weakly stationary for any choice of parameters:

For a vector process:

Check if a time series is invertible:

For a vector process:

Find invertible representation for a moving-average process:

The moments are being conserved:

Find invertibility conditions:

Find conditions for higher order: