5 ms·
Clauset Shalizi Newman 2007 has not-nice things to say about the classic physicist's idiot trick of fitting power law distributions by drawing a straight line o
by curuinor 10y ago
Clauset Shalizi Newman 2007 has not-nice things to say about the classic physicist's idiot trick of fitting power law distributions by drawing a straight line on a log-log graph: it's got huge bias. https://arxiv.org/abs/0706.1062 https://arxiv.org/abs/0706.1062
However, the other difficult thing about power law distributions is that the dataset size requirements for proper determination of the fact that it's a power law distribution are occasionally incredibly difficult. So their critique is very strong, given the comparative lack of data. It is often the case that computer systems, with the overflowing reams of data, are still not enough. Note that the paper I cited up there suggests MLE and then a Kolmogorov-Smirnoff test, so it'll say a lot of things aren't power laws that could well be.
Another way to look at it is from a more geometric point of view. The metric entropy of any generic system of variables is defined as the sum of the positive Lyapunov exponents: and as an "entropy" that quantity does have a lot of commonalities with the other entropies. But to have positive Lyapunov exponents is often to have a chaotic dynamics, so it could just be conjectured that the time series of commits and merge octopus sizes in kernel git history is chaotic, so the evolution of the time series will be fractal in nature.
But it's also really fucking hard to confirm or deny that one, because there are varied and strange definitions of chaos itself and the methods that have been suggested to measure Lyapunov exponent in real systems are arcane and difficult. You could try some synchronization methods, but they remain arcane and crap. Fractal measurement methods are also shitty and full of dark magic.
One neat little trick might be to discretize the series, symbolic dynamics-style (it's already discretized but discretize further, into like percentiles or something) and run it through one of the dynamical machine learning dealies to see if there's patterns. Not too much literature on that but it's a thing that some randoes in like 2004 or something did
- gary_bernhardt 10y agoI removed the "power law" language from the post, other than a quick "this is often called a power law, which probably isn't correct" note. I don't want to get bogged down in statistics when all I really care about here is "it's fat and one sided".
- xapata 10y agoYeah, it's really hard to distinguish power-law from log-normal from statistical measurements, even in simulations when you know the generating process.
- researcher11 10y agoThis is my favorite Bad TED talk - by a physicist using the power law straight line trick; https://www.ted.com/talks/sean_gourley_on_the_mathematics_of_war https://www.ted.com/talks/sean_gourley_on_the_mathematics_of... It was also published in Science. Sean also founded Quid. I thought I was taking crazy pills when I saw it. I wondered how could this guy not know better and how could no one else around him know either.
- curuinor 10y agoThis is Richardson 1948 shit, so Richardson had the good excuse of doing all the work while people did their computations by hand. This numbnuts is not only like 60 years late, but does not have that excuse. Lot of people who don't like reading in complex systems, I find. Also a lot of people who aren't sufficiently cynical.
- captaincrowbar 10y agoThere's an old saying among scientists: any set of data points will fit a power law if plotted on log-log paper with a fat enough magic marker.
- danbruc 10y agoAs somebody who has not much to do with statistics, what are the bad consequences of misidentifying a distribution? After skimming the paper it seems that there are several distributions that are similar in the sense that it is hard to tell which one a given dataset actually follows. So naively it seems that it should not really matter which one one picks if they are similar enough that it is hard to tell them apart to begin with. Is picking the wrong distribution bad because it leads to wrong conclusions when trying to explain the mechanism causing the distribution? Because extrapolations may yield significantly different results? Or is it more specifically about log–log charts and that small deviations on a logarithmic scale can correspond to significant differences on a linear scale so that a seemingly good fit is not actually a good fit?
- curuinor 10y agoMean of lognormal distribution: e ^ (parameter + (other parameter ^ 2) / 2) Mean of pareto (power law) distribution, where parameter < 1: infinity
- danbruc 10y agoYou mean - pun not really intended - that making choices based on the mean of the supposed distribution, say for risk assessment purposes, may lead to wildly wrong conclusions? Or more generally, that characteristics of different distributions can be very different even if they fit a given dataset similarly well?
- curuinor 10y agoBoth, actually. And of mechanistic statements. For example, if you have this sort of multiplicative central limit theorem with no lower bound, where you imagine independent variables being multiplied together - the attractor of that dynamical process, the result of that universal phenomenon, is lognormal. Add a lower bound, bam, power law (champernowne 1953 "A Model of Income Distribution"). So you're really making different claims about reality.