LRydin
commented
3 years ago Hey @FuadYoussef, thank you for the question. I'll try my best to give you an answer.
So, I think you code is absolutely correct and the figure you show looks like a typical plot from MFDFA. Now the issue if of course the interpretation of the Hurst coefficient, which is not too easy. You are using sentence lengths, which I am not familiar with, but nevertheless, the point here has to do with having to subtract -1 to get the Hurst coefficient H. This is correct to do so for stochastic processes, but not for all processes.
Let me illustrate this using Espen Ihlen's fantastic paper on MFDFA: Introduction to multifractal detrended fluctuation analysis in Matlab, Ihlen, E.A.F, Front. Physio. 3:141. This is Espen Ihlen's Fig, 8
Where you can see the fluctuation function for (B) a monofractal time series and (C) white noise. (B) will have H>1 (this is why one subtracts -1, to get the mathematically correct H value), but (C) has 0<H<1, and it is a perfectly reasonable time series. I think your results fall in the category of "noise", thus you shouldn't subtract 1 from your results.
Two notes:
- to be precise, the Hurst coefficient
His just the case forq=2. The other ones are called generalised Hurst coefficientsh(q), which help to understand whether the data is monofractal (like plot (B) and (C) where all lines are parallel) or if the data is multifractal (plot (A)). - I see that you are using a second-order polynomial fitting in your code (
order = 2). I suggest that when you want to fit the slopes of the curves you discard the first points of thedfaandlag. This is in your line L37 of themain.py. Try maybe:slopes = np.polyfit(np.log(lag[2:]), np.log(dfa[2:]), 1)[0]it is the case that the smallest lags are always overfitted by the polynomial fits, and this don't really represent the correct variance of the detrended processes. This is a simple workaround, to correct for that.
I hope this helps a little bit. From my view it seems to me your data is slightly multifractal, which seems reasonable from my understanding of sentences lengths.
Feel free to ask more things, if you have any doubts!
FuadYoussef
Hi, Thank you for all of the hard work you've done to make this library. I've tried following the instructions in the documentation as well as the instructions in this post https://github.com/LRydin/MFDFA/issues/20 in order to analyze some of my own data and have encountered an issue. For reference, the code I am using is in this repository in the main.py and file_reader.py files. https://github.com/FuadYoussef/ChineseSentenceLengths
When I run the MFDFA and attempt to plot the data, I am able to generate the following graph:
However the H values I get by subtracting 1 from each of the slopes is this: [0.24513458672164035, 0.2186072164297126, 0.1926072404229875, 0.17172103657125448, -0.34609042498381115, -0.3619098348631248, -0.375020070287282, -0.38637649591195067]
I obtain these H values using this code: slopes = np.polyfit(np.log(lag), np.log(dfa), 1)[0] slopes = slopes.tolist() hExponents = [] for slope in slopes: hExponents.append(slope - 1)
My confusion is about why I am getting negative H values from this. Any help or suggestions you could give would be greatly appreciated. Thank you.