# What scheme is used for the axes in this plot?

What scheme is being used for the axes in this plot? Does it have a name?

It looks like the x axis is linear with a discrete change in spacing at 1 micron, but I can't figure out what's going on in the y axis.

• One thing that does have a name is "Lie Factor" (Tufte). The Lie Factor on the x-axis is 10:1, owing to the nearly hidden, nonsmooth jump in scale at 1 micrometer.
– whuber
Apr 24, 2013 at 21:43

One way to reconstruct axis scales is with--statistical analysis!

First capture the image in the finest detail possible and digitize reference marks as accurately as you reasonably can. Here's an excerpt of my effort, which looks good to a small fraction of a pixel:

Then create a table matching the digitized coordinates to the labels. Here's a piece of that table:

Because the x-coordinates are essentially constant, I'm confident there's not enough slanting in the image to mess things up. Now we only need find a numerical relationship between the two. For this, Eureqa formulize does a wonderful job. (It's a free download and takes almost no time to learn if you know anything about curve fitting.) Here's half of the information shown while it's searching:

On the left the best solutions found so far are listed. (By this time a third of a million generations in a genetic algorithm had been created, taking about ten minutes total.) "Size" is a measure of their complexity and "fit" measures the goodness of fit (in pixels). Given that I have digitized to a small fraction of a pixel, but certainly not as good as a tenth (the image isn't that good), I would be content with the solutions of size 9 or 10 shown here. The fit of solution 10 is shown at the right. (Solution 9's fit veers away from the points at the lower right: such a systematic departure is an indication that the solution is not correct.)

As you can see, there isn't a real nice relationship here: nothing like a linear transformation of an exponential or logarithm, for instance. As a double-check I looked for a nice inverse relationship relating the labeled value to the pixel height. Nothing any better is emerging (the software is running in the background--using all eight cores--as I write this). I tentatively conclude that the vertical scale is almost as imaginative as the horizontal one, created by the stroke of an artist's brush and not by the pen of any scientist. Regardless, I now have formulas that can be used to recreate (or even--at huge risk) extend the y-axis.

Here are the digitized points for anyone who would like to investigate further. Values for X and Y are in pixels within the image (transposed into separate lines in order to save screen space). The precision of the digitizer itself was approximately 0.06 pixels, and I made an effort to digitize the y-values accurately and consistently (paying little attention to the x-values), but I wouldn't trust individual points to more than about 0.5 pixels, given that the linewidth in the image is typically two pixels. Values of Z are inferred from the labels on the Y axis based on the supposition that the ticks subdivide each decade into equal intervals.

X   84.127  84.269  84.127  84.269  84.269  84.198  84.340  84.198  84.340  84.269  84.411  84.411  84.269  84.411  84.340  84.482  84.482  84.198  84.056  84.624  84.127
Y   407.016 383.463 359.484 335.647 311.243 287.264 270.522 253.637 236.894 220.010 203.126 190.072 176.167 162.830 149.776 136.297 125.088 114.305 102.954 91.887  80.678
Z   100 98  96  94  92  90  88  86  84  82  80  78  76  74  72  70  68  66  64  62  60
• Heh, I had a similar idea before I realized you'd already been down that route. To a first approximation, it's close to quadratic. Given the y axis is a proportion/percentage, some things to check are "is it a logit?" (nope, indeed because 100 is on-scale, clearly not, though the other points fit pretty well); "is it arcsin-sqrt?" (nope; and I realize it doesn't make sense to use it here, but someone might be confused enough to do that). Apr 25, 2013 at 1:05
• A quick google of spectral reflectance literature doesn't turn up any obvious suggestions that I could see, either. However, I did turn up a source of the image, which may give some further clues. Apr 25, 2013 at 1:12
• @Glen_b Yes, I had considered taking some of those polynomial fits and comparing them to Taylor series of common functions. But given the travesty already made of the x-axis, it just didn't seem worth the effort. And now you have found out it's just marketing literature, anyway. The interest in this question for me was in how it brought up an opportunity to describe a general procedure to reverse-engineer axes (and other elements found in graphical images), rather than solving this particular problem.
– whuber
Apr 25, 2013 at 1:51
• I agree, I think the main value in this is in your exploration of means for answering such questions. Apr 25, 2013 at 1:55
• Very cool! I'm starting to think it's just linear along the y axis but changes width every 10%. I doubt that Eureqa will look for formulas containing floor(y/10) or such. Is there any way I can have a look at the full table, or would you mind seeing if the differences along the y axis jump every 10%? Apr 25, 2013 at 2:02

The x-axis looks like a cheat to me. To construct it, just make two linear scales and weld them together in a manner that hides the seam.

Unless it is a standard type for that type of data (in which case it is a very odd standard), it could be replaced with a logarithmic scale.

• I'm trying to datathief the plot, so I need to understand exactly how that y axis is constructed. Apr 24, 2013 at 21:40