# MAProcess

MAProcess[{b1,,bq},v]

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

MAProcess[{b1,,bq},Σ]

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

MAProcess[{b1,,bq},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 bi and c, and a positive variance v.
• An -dimensional vector MA process should have real coefficient matrices bi 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:

### Estimation Methods(6)

The available methods for estimating an MAProcess:

Compare log likelihoods:

Method of moments allows following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Maximum conditional likelihood method allows following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Maximum likelihood method allows 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:

Minimum prediction method:

This method allows for fixed parameters:

### 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(5)

Approximate an AR process with an MA process of order 5:

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

Approximate a vector process:

Approximate an ARMA process with an MA process:

Compare sample paths:

Approximate a SARIMA process with an MA process:

Compare sample paths:

TransferFunctionModel representation:

For a vector-valued process:

StateSpaceModel representation:

For a vector-valued process:

## Applications(1)

Consider the following time series data and determine whether it is adequately modeled by an MAProcess:

The correlation function drops off after lag 3. This is evidence of an MAProcess:

The partial correlation alternates and dampens slowly, which also indicates an MAProcess:

Fit an MAProcess model to the data:

Find residuals between the data and the model:

Test if residuals are normal white noise:

## Properties & Relations(5)

MAProcess is a special case of an ARMAProcess:

MAProcess is a special case of an ARIMAProcess:

MAProcess is a special case of a FARIMAProcess:

MAProcess is a special case of a SARMAProcess:

MAProcess is a special case of a SARIMAProcess:

## Possible Issues(3)

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: Use a different solver:

Minimum prediction error estimation method does not allow repeated parameters: Use a different method:

## Neat Examples(2)

Simulate a three-dimensional MAProcess:

Simulate paths from an MA process:

Take a slice at 50 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 50: