I want to test the distribution that best fit a specific metric (that I call SD) extracted from the source code of systems. I have a guess that they follow a power-law behavior.
- My sample: 20 systems
- For each one of this 20 systems I want to test if the internal distribution of the occurrence of each SD value follows a power-law (or, at least have a good fit).
- The metric extracted from these systems are not a random sample, but all the occurrences inside a single system.
- The range of the value of this metric is not determined.
- The values are discrete.
I will test if in all systems this metric SD follows a PowerLaw (or not).
I'm using the methodology by Aaron Clauset
And the R package created by Colin S. Gillespie
In summary, my steps for each distribution (each system) are:
1.Estimate the parameters xmin and α (in the plots they are k) of the power-law model using MLE.
m_pl = displ$new(data) est = estimate_xmin(m_pl) m_pl$setXmin(est) plot(m_pl)
2.Calculate the goodness-of-fit between the data and the power law. If the resulting p-value is greater than 0.1 the power law is a plausible hypothesis for the data.
bs = bootstrap_p(fittedPowerLaw, no_of_sims=numberOfBootstrapSims, threads=8) bs$p
General Question 1: Is the methodology right? I see a lot of examples where people test distributions with sample data. In this case where I have all occurrences from each system being tested, can I use the same steps?
A- Good result: Looking at the picture it seems to follow the PowerLaw model. The p-value: 0.2368, so the Power-Law is a plausible hypothesis.
B - Bad result:
Looking at the picture it seems to follow the PowerLaw model. But with the p-value of 0.0292, the Power-Law is ruled out.
Question 2: Are these results (A & B) right?
C - Odd result: Clauset says that small samples can bias the results (e.g. samples where n<100). I have one example where the dataset is large (n > 3.000), but most of the data have the same value (in this case, 1). The plot looks like we have small data, because a small variation only appears on the right. The p-value: 0.5976, so the Power-Law is a plausible hypothesis.
Question 3: Is the result valid in this case?