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Can someone help interpret the Mar's Law:

Everything is linear if plotted log-log with a fat magic marker

I know that in some social network analysis, Log-Log scale does make certain things look linear. But I have a hard time convincing myself that the above statement is true in general.
PS: I am looking for an explanation more from an application point of view (with real data) rather than theoretical functions as shown in Alexis' nice plots)

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    $\begingroup$ It's definitely not true in general (I'll write up some counterexamples and post an answer when I get a chance), but it may be nearly true - perhaps even often true - in some areas of work ... the same ones that make people want to model everything as a power law. I'd read the quote as somewhat ironic in intention (though I've never seen it before so I don't have the context to be sure). $\endgroup$ – Glen_b -Reinstate Monica Feb 12 '15 at 2:28
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    $\begingroup$ The basic principle of the differential Calculus is that "everything" (that is, every differentiable function) locally looks linear when plotted with with a fat magic marker. (This is a tautological expression, because it restates what "differentiable" means.) "Mar's Law" merely applies that to the case of functions of the form $\log f(e^x)$. $\endgroup$ – whuber Oct 6 '17 at 16:32
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Log-log plots of "everything" do not look linear:

$F$ PDFs for a few different values of $\nu_{n}$ and $\nu_{d}$:

Some F distributions

$y = \sin (x)^{2}$:

y = sin(x)^2

$y = \frac{1}{\sin (x)}$:

y = 1/sin(x)

$y=e^{x}$:

y=e^x

Eh... I'll pass on buying Mar's "Law".

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    $\begingroup$ The quote is about plotting data, not functions. Therefore it's necessarily about less information and usually about a more constrained x-axis. Select between 5 and 10 random points between 0 and pi/2 and see if those plot on straightish lines using a fat marker (which you also didn't do). This will both demonstrate the law and the point the law is making. $\endgroup$ – John Feb 12 '15 at 2:53
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    $\begingroup$ @John $y=\frac{1}{\sin(x)}$ between $10$ and $10^{2}$, The red $F$ distribution between 0.5 and 1.5. If you need to define Mar's "Law" along tautological lines like "all functions look linear on log-log scales when they look linear", then why bother with specifying log-log at all? Things look linear on a linear-linear scale if you zoom in close enough (unless they are fractal functions, in which case, nope, zooming in doesn't help). Also the OP said nothing about $n=5$ data; said "everything". $\endgroup$ – Alexis Feb 12 '15 at 2:58
  • $\begingroup$ I am giving a +1 to Alexis for his nice work. But as John pointed out, I am more of looking toward application in real data (sorry for not explicitly point it out in the original post). $\endgroup$ – kindadolf Feb 12 '15 at 3:05
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    $\begingroup$ @kindadolf "her" And thank you. I think the point still holds for data based on functions, for example those shown above. If Mar's Law holds only "when zoomed in," "with noisy data," and "when things are basically linear" then it is not really telling us anything about log-log scale plots. It certainly holds for certain kinds of functions or data based on them under certain conditions, such as polynomials, quite nicely. But that's a far cry from "everything." $\endgroup$ – Alexis Feb 12 '15 at 3:12
  • $\begingroup$ Ummm... yes @Alexis... your reaction is most appropriate but you seem to miss the point. Didn't this "law" make you want to prove it wrong? $\endgroup$ – John Feb 12 '15 at 11:50
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It's a bit of a joke, not a hard-and-fast truth. And it deals with data sets, not functions. It basically means you massage data (apply statistical transformations) and/or apply visualization tricks to make relationships appear where there really aren't any.

By plotting log-log, you reduce the spread of data. You lose a lot of details. a simple linear regression on non-linear becomes a lot more linear.

Then by plotting with a fat magic marker, you (visually) increase the (physical) spread of you regression line (on paper). Points that are off of the regression line are now swept under the fat marker.

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    $\begingroup$ I don't think it is a joke and it deals with fitting data to a model (function). $\endgroup$ – Michael R. Chernick Oct 6 '17 at 16:39
  • $\begingroup$ @Michael It sounds like you may be over-interpreting the introduction of this answer. Do you seriously think the "magic marker" reference is made without some levity or has any actual rigor to it? $\endgroup$ – whuber Oct 6 '17 at 17:49
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    $\begingroup$ I think I was being a little nitpicky about this answer. $\endgroup$ – Michael R. Chernick Oct 6 '17 at 18:01

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