ARFIMA.arfima — Method
arfima([rng,] N, σ, d, φ=nothing, θ=nothing) -> Xₜ

Create a stochastic timeseries of length N that follows the ARFIMA process, or any of its subclasses, like e.g. ARMA, AR, ARIMA, etc., see below. σ is the standard deviation of the white noise used to generate the process. The first optional argument is an AbstractRNG, a random number generator to establish reproducibility.

The generating equation for Xₜ is:

\[\left( 1 - \sum_{i=1}^p \phi_i B^i \right) \left( 1-B \right)^d X_t = \left( 1 + \sum_{i=1}^q \theta_i B^i \right) \varepsilon_t\]

with $B$ the backshift operator and $\varepsilon_t$ white noise.

This equation encapsulates all possible variants of ARFIMA and Julia's multiple dispatch system decides which will be the simulated variant, based on the types of d, φ, θ.

Variants

The ARFIMA parameters are (p, d, q) with p = length(φ) and q = length(θ), with p, q describing the autoregressive or moving average "orders" while d is the differencing "order". Both φ, θ can be of two types: Nothing or SVector. If they are Nothing the corresponding components of autoregressive (φ) and moving average (θ) are not done. Otherwise, the static vectors simply contain their values.

If d is Nothing, then the differencing (integrated) part is not done and the process is in fact AR/MA/ARMA. If d is of type Int, then the simulated process is in fact ARIMA, while if d is AbstractFloat then the process is ARFIMA. In the last case it must hold that d ∈ (-0.5, 0.5). If all d, φ, θ are nothing, white noise is returned.

The function arma(N, σ, φ, θ = nothing) is provided for convienience.

Examples

N, σ = 10_000, 0.5
arfima(N, σ, 0.4)                                   # ARFIMA(0,d,0)
arfima(N, σ, 0.4, SVector(0.8))                     # ARFIMA(1,d,0)
arfima(N, σ, 1, SVector(0.8))                       # ARIMA(1,d,0)
arfima(N, σ, 1, SVector(0.8), SVector(1.2))         # ARIMA(1,d,1)
arfima(N, σ, 0.4, SVector(0.8), SVector(1.2))       # ARFIMA(1,d,1)
arfima(N, σ, nothing, SVector(0.8))                 # AR(1)
arfima(N, σ, nothing, nothing, SVector(1.2))        # MA(1)
arfima(N, σ, nothing, SVector(0.8), SVector(1.2))   # ARMA(1,1)
ARFIMA.autoregressive — Method
autoregressive(N, Z, φ::SVector{P}) -> X

Generate an autoregressive process of length N based on input noise term Z. This is used in both ARFIMA and ARMA. The noise term must have at least N+P elements.

ARFIMA.bdp — Method
bdp(φ::SVector, X::AbstractVector, t)

Perform the backshift dot product between φ and X, with starting index t: $\sum_i \phi_i X_{t-i}$.