Home > Graphics > Problem banks, part 3

Problem banks, part 3

In the two previous graphics posts I found a few problems with a graph and remade it, and then chose a background for it (#8).  The only thing left to do on this graph is choose the aspect ratio.  There’s not necessarily anything wrong with the aspect ratio I had been using, but it’s worth checking out other aspect ratios to see if another one works better.

How will I choose aspect ratios to try out?  I’ll be using William Cleveland’s idea of “banking to 45”.  Cleveland has shown that people tend to be able to differentiate slopes or orientations of lines easiest when they are centered around a value intermediate between horizontal and vertical (i.e., oriented to 45 or -45 degrees, or, equivalently, a slope of plus or minus 1).  Hence, Cleveland recommends using an aspect ratio such that slopes or orientations are centered at the intermediate value.  But there are many options for how exactly to do this.  Should we focus on the slopes of the lines (change in y over change in x) or their orientations (the angle of the line from the x-axis, the arctangent of the slope)?  Centers slopes will tend to have a different result than centering orientation, if only a little different (as below).  And how exactly should the centering be done?  Should the median be centered, or the average, or the avarage weighted by length of line?  A paper by Jeffrey Heer and Maneesh Agrawala considers these options, along with culling horizontal and vertical lines — neglecting them when calculating the centering — since they’ll be horizontal or vertical regardless of aspect ratio.

In every procedure here, we will either constrain orientations to lie between 0 and 90 degrees (by using the absolute value of the arctangent), or we will use the absolute values of slopes.  This allows us to center all lines at once, rather than somehow centering lines with negative and positive slopes separately.

Nine versions are shown below, for nine ways of choosing aspect ratio.  A few sentences of discussion follow. To understand the aspect ratio, the given number represents the scale of y relative to x. So if the aspect ratio is 0.5, then the distance on the page between 1 and 2 in the y direction will be half as much as the distance between 1 and 2 in the x direction. To understand the aspect ratio in this particular graph you need to know the x values I am using, which can’t be judged from the graphs. The x values are at 1,2, etc., up to 33.

Original: eyeball estimate of good aspect ratio
Aspect ratio: 0.035

Option 1: center the average absolute slope
Apect ratio: 0.011

Option 2: center the median absolute slope
Aspect ratio: 0.004

Option 3: center the average absolute slope, culled
Apect ratio: 0.013

Option 4: center the median absolute slope, culled
Aspect ratio: 0.005

Option 5: center the weighted average absolute orientation
Aspect ratio: 0.038

Option 6: center the weighted average absolute orientation, culled
Aspect ratio: 0.034

Option 7: center the average absolute orientation
Aspect ratio: 0.107

Option 8: center the average absolute orientation, culled
Aspect ratio: 0.084

The original aspect ratio, 0.035, was chosen with the idea of banking to 45 degrees in mind, but no calculations were made.  It seems to fare pretty well.  It’s very close to the aspect ratios from Options 5 and 6., and also very close to the average of Options 1-8, which is approximately 0.037.  (Maybe this similarity is luck or a coincidence, but I did spend a fair amount of time and concentration varying the aspect ratio to pull as much detail as possible out of the graph.)  I won’t necessarily advocate 0.035 as the best ratio here.  Neither will I advocate the average aspect ratio or Options 5 and 6 over the others.  I will, however, stay with my original aspect ratio because it is so close to two of the options and the average, and because most of the other aspect ratios have some drawbacks (which is not to say that my chosen aspect ratio is free from any drawbacks of its own).

The graphs with aspect ratio 0.005, Options 2 and 4, seem to have the early variations squeezed out of them.  I’d prefer more detail in the left half.  On the other hand Options 7 and 8 draw a lot of detail out of the left side of the graph, but otherwise might be thought to produce an overly dramatic graph.  Of course, maybe they’re just appropriately dramatic.

The aspect ratios of 0.012 and 0.013 are ok; they seem to me to emphasize the shape of the trend over the details.

I’ve changed my mind.  I will end this post by advocating my original aspect ratio.  Not because it’s close to two options and the average, but on a subjective measure: my original graph and the weighted-average graphs (Options 5 and 6) are the most jagged-looking throughout the graph, which indicates that the lines are the most-distinguishable.

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