This Julia package simulates stochastic timeseries that follow the ARFIMA process, or any of its simplifications: ARIMA/ARMA/AR/MA/IMA.
The code base is also a proof-of-concept of using Julia's multiple dispatch.
julia> ] add https://github.com/Datseris/ARFIMA.jl julia> using ARFIMA
see the examples below for usage.
ARFIMA and its variants
The ARFIMA process is composed out of several components and each can be included or not included in the process, resulting in simplified versions like ARMA.
This package exports a single function
arfima. This function generates the timeseries
Xₜ using Julia's multiple dispatch.
Here is its documentation string:
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.
Julia's multiple dispatch system decides which will be the simulated variant
of the process, based on the types of
d, φ, θ.
The ARFIMA parameters are (p, d, q) with
p = length(φ) and
q = length(θ),
p, q describing the autoregressive or moving average "orders" while
d is the differencing "order".
φ, θ can be of two types:
SVector. If they are
the corresponding components of autoregressive (φ) and moving average (θ)
are not done. Otherwise, the static vectors simply contain their values.
Nothing, then the differencing (integrated)
part is not done and the process is in fact AR/MA/ARMA.
d is of type
Int, then the simulated process is in fact ARIMA,
AbstractFloat then the process is ARFIMA.
In the last case it must hold that
d ∈ (-0.5, 0.5).
d, φ, θ are
nothing, white noise is returned.
arma(N, σ, φ, θ = nothing) is provided for convienience.
N, σ = 10_000, 0.5 X = arfima(N, σ, 0.4) # ARFIMA(0,d,0) X = arfima(N, σ, 0.4, SVector(0.8, 1.2)) # ARFIMA(2,d,0) X = arfima(N, σ, 1, SVector(0.8)) # ARIMA(1,d,0) X = arfima(N, σ, 1, SVector(0.8), SVector(1.2)) # ARIMA(1,d,1) X = arfima(N, σ, 0.4, SVector(0.8), SVector(1.2)) # ARFIMA(1,d,1) X = arfima(N, σ, nothing, SVector(0.8)) # ARFIMA(1,0,0) ≡ AR(1)
Some benchmark code is included in
src/benchmarks.jl. These results come
from running the code on a laptop with Windows 10, Julia 1.2.0, Intel i5-6200U @2.30GHz CPU, 8192MB RAM. Results that need microseconds to run are not timed accurately.
Process: ARFIMA(0, d=0.4, 0) For N = 10000, 0.2449999 seconds For N = 50000, 1.467 seconds For N = 100000, 7.5779998 seconds Process: ARIMA(0, d=2, 0) For N = 10000, 0.0 seconds For N = 50000, 0.0009999 seconds For N = 100000, 0.0020001 seconds Process: ARFIMA(φ=0.8, d=0.4, 0) For N = 10000, 0.267 seconds For N = 50000, 1.388 seconds For N = 100000, 7.4589999 seconds Process: ARMA(φ=0.8, θ=2.0) For N = 10000, 0.0 seconds For N = 50000, 0.0009999 seconds For N = 100000, 0.003 seconds
Thanks to Katjia Polotzek for providing an initial code base for ARFIMA and to Philipp Meyer for validation of parts of the code