FARIMAProcess

FARIMAProcess[{a1, ..., ap}, d, {b1, ..., bq}, v]
represents the fractional, autoregressive, integrated moving-average process such that its d^(th) difference is an ARMAProcess[{a1, ..., ap}, {b1, ..., bq, v].

FARIMAProcess[{a1, ..., ap}, d, {b1, ..., bq}, ]
represents the vector-fractional, autoregressive, integrated moving-average process such that its ^(th) difference is a vector ARMAProcess.

FARIMAProcess[{a1, ..., ap}, {d1, ..., dn}, {b1, ..., bq}, ]
represents the vector-fractional, autoregressive, integrated moving-average process such that its ^(th) difference is a vector ARMAProcess.

DetailsDetails

  • 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 and , real integrating parameter d such that , and a positive variance v.
  • An -dimensional vector FARIMA process should have real coefficient matrices and of dimensions ×, real integrating parameters such that or real integrating parameter d such that , and the covariance matrix should be symmetric positive definite of dimensions ×.
  • FARIMAProcess can be used with such functions as CovarianceFunction, PDF, Probability, and RandomFunction.

ExamplesExamplesopen allclose all

Basic Examples (3)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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