juliadynamics / arfima.jl

Simulate stochastic timeseries that follow ARFIMA, ARMA, ARIMA, AR, etc. processes

JuliaDynamics

This repository serves the following purposes:

• Contains the source code for the JuliaDynamics website in the src and build folders.
• Hosts the website via GitHub-pages and Jekyll.
• Contains tutorials for all packages of JuliaDynamics in the tutorials folder.
• Contains video resources for all packages of JuliaDynamics in the videos folder.

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The website was modeled after the website of QuantumOptics.jl and most code that builds the site was re-used from the repository of QuantumOptics.jl (with permission).

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To build locally do follow the instructions from here: https://jekyllrb.com/docs/

(install Jekyll and then do bundle exec jekyll serve which serves by default to http://localhost:4000)

arfima.jl's Issues

Math image in README is hard to read in dark mode

Here's what it looks like:

Is it possible to somehow render this to be dark mode friendly, e.g. with LaTeX directly? I tried something like

$$\underbrace{\left(1 - \sum_{i=1}^p \varphi_i B^i\right)}_{\text{\textcolor{blue}{auto-regressive}}}\underbrace{\left(1 - B\right)^{\overbrace{d}^{\text{\textcolor{purple}{fractional if $d \in \mathbb R$}}}}}_{\text{\textcolor{orange}{integrated}}} \underbrace{X_t}_{\text{\textcolor{gray}{generated timeseries}}} = \underbrace{\left(1 - \sum_{i=1}^q \theta_i\theta^i\right)}_{\text{\textcolor{green}{moving average}}}\underbrace{\varepsilon_t}_{\text{\textcolor{red}{white noise}}},$$

but that's worse, obviously (since GitHub can't get the actual packages needed to create the same type of image).

Inconsistency with explicit ARMA simulation

I'm not sure what I am doing wrong in the source code for ARFIMA.jl, but take the following explicit simulation of an ARMA process and compare it with its ""equivalent" ARFIMA.jl syntax:

using Random
Random.seed!(77163)
η = randn(5000)
s = ones(5000)
for n in 4:5000
    s[n] = 1.625s[n-1] - 0.284s[n-2] - 0.355s[n-3] + η[n] - 0.96η[n-1]
end
s ./= std(s)

using ARFIMA
φ = SVector(1.625, -0.284, -0.355)
θ = SVector(0.96)
Random.seed!(77163)
s2 = arma(5000, 1.0, φ, θ)
s2 ./= std(s2)

using PyPlot
plot(s)
plot(s2)
xlim(0, 1000)

Both simulations have the characteristic oscillatory behavior of the specified ARMA process, but the one coming from ARFIMA.jl has much, much less noise.

Why...?

Stranbo integration

Leaving an issue here to consider whether we might integrate the functionalities I have in Stranbo and arfima.

Stranbo is designed for generating time series with "complicated" sarimax (integer $i$) processes. Complicated means many seasonal effects (think, daily + weekly + monthly + yearly, ..., each one with its own d, ar, and ma coefficients) and/or many auxiliary components (again, each one with its own d, ar, and ma coefficients).

It is slower than ARFIMA, a little bit due to the generality, and a tad due to the fact that I don't use any @fastmath tricks (I got scared because of this kind of things, but maybe here it's safe?).

The high-picture view of how Stranbo work is:

1. define all processes components as classes
2. detine a process as an vector of components
3. gather all coefficients, and expand the polynomials for x, z, and the auxiliary components using Polynomials
4. For each x[t]
   4.1 use a modified getindex to look in the past (modified so to handle negative indices without padding)
   4.2 dot product coefficients and backward shifted (and truncated) series
   4.3 sum up
5. (eventually add anomalies...)

I believe that 4 is in mostly shared between the two packages (but I can learn a bit to improve even more). I don't think the polynomial stuff is in ARFIMA (I believe you have hardcoded the loops, right?).

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SARIMAX

Hello,

would you be open on adding X into SARIMA model to get also SARIMAX? The only option as of now is statsmodels in python. I don't count using R as an option. https://www.statsmodels.org/stable/generated/statsmodels.tsa.statespace.sarimax.SARIMAX.html
Which is alright, but it is terribly slow, because the whole implementation is just in python. I was thinking about writing SARIMAX from scratch in rust language, but that would be indeed nontrivial amount of work. I found your package and it looks very well, except missing the input for exogenous variables.

Source: https://towardsdatascience.com/time-series-forecasting-with-arima-sarima-and-sarimax-ee61099e78f6

Would you be open to this feature?