BUILT-IN MATHEMATICA SYMBOL

FARIMAProcess

FARIMAProcess[{a₁, ..., a_p}, d, {b₁, ..., b_q}, v]
represents the fractional, autoregressive, integrated moving-average process such that its d difference is an ARMAProcess[{a₁, ..., a_p}, {b₁, ..., b_q, v].

FARIMAProcess[{a₁, ..., a_p}, d, {b₁, ..., b_q}, ]
represents the vector-fractional, autoregressive, integrated moving-average process such that its difference is a vector ARMAProcess.

FARIMAProcess[{a₁, ..., a_p}, {d₁, ..., d_n}, {b₁, ..., b_q}, ]
represents the vector-fractional, autoregressive, integrated moving-average process such that its difference is a vector ARMAProcess.

Details 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 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.