Slowly but surely we have finally reached ARFIMA model! Taking such small step (adding letter F to the acronym) took a lot of effort, but it was worth it. Why? Well, this exercise has allowed me to get a glimpse at fractional calculus and develop some intuition with this tool.

Last time we have looked into fractional derivatives and even managed to derive fractional derivative for \( f(x) = x \). For more complicated functions this is much more problematic. Here, in this post, we will show you a quick numerical method to calculate fractional derivative of any arbitrary series.

This video from Tom Rocks Maths Youtube channel talks about L-system. One can use L-system to artificially generate realistic botanical drawings. Fascinating isn't it?

Last time we have seen that ARMA modelis can be integrated (and differentiated) to deal with non-stationarity present in the empirical data. ARIMA model should be sufficient in most cases, but if the empirical data is know to exhibit "true" long-range memory, then ordinary calculus will not work. In those cases one would have to use fractional calculus. Here we will take a brief look at fractional derivatives (and integrals).

Last time we have combined AR(p) and MA(q) models to make ARMA(p, q) model. We are getting quite close to understanding what ARFIMA processes are, but to do that we still have things to look at. This time lets take into account letter I (which stands for "integrated") - lets look at ARIMA models.