# FARIMAProcess

FARIMAProcess[{a1,,ap},d,{b1,,bq},v]

represents an autoregressive fractionally integrated moving-average process such that its d difference is an ARMAProcess[{a1,,ap},{b1,,bq,v].

FARIMAProcess[{a1,,ap},d,{b1,,bq},Σ]

represents a vector autoregressive fractionally integrated moving-average process (y1(t), ,yn(t)) such that its (d,,d) difference is a vector ARMAProcess.

FARIMAProcess[{a1,,ap},{d1,,dn},{b1,,bq},Σ]

represents a vector autoregressive fractionally integrated moving-average process (y1(t), ,yn(t)) such that its (d1,,dn) difference is a vector ARMAProcess.

# Details

• FARIMAProcess is also known as ARFIMA or long-memory time series.
• FARIMAProcess is a discrete-time and continuous-state random process.
• The FARIMA process is described by the difference equations , where is the state output, is the white noise input, and is the shift operator.
• The scalar FARIMA process has transfer function , where .
• The vector FARIMA process has transfer matrix , where , and where is the × identity matrix.
• A scalar FARIMA process should have real coefficients ai, bj, real integrating parameter d such that , and a positive variance v.
• An -dimensional vector FARIMA process should have real coefficient matrices ai and bj of dimensions ×, real integrating parameters di such that or real integrating parameter d such that , and the covariance matrix Σ should be symmetric positive definite of dimensions ×.
• FARIMAProcess[p,d,q] and FARIMAProcess[p,q] represent a FARIMA process of orders p and q with known or unknown integration order d for use in EstimatedProcess and related functions.
• FARIMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.

# Examples

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## Basic Examples(3)

Simulate a FARIMA process:

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Covariance function:

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Correlation function:

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Partial correlation function:

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# See Also

Introduced in 2012
(9.0)