This is a great graph: Plotting y(t) vs y'(t), tracing over time with a dot for each year

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2022-01-15 16:00:06

Gwynn points us to a new book, “Slow Down: The end of the Great Acceleration – and Why It’s Good for the Planet, the Economy, and Our Lives,” by Danny Dorling. The author is a geographer, so I assume he hasn’t claimed to have “discovered a new continent,” but I expect he’ll appreciate the above world map from xkcd.

I haven’t seen Dorling’s book, but what Gwynn really wanted to point out were the visualizations on its webpage. These are a series of plots, each tracing a time series over a series of years with y on the y-axis and y'(t) (that is, dy/dt) on the x-axis. Here’s an example:

We could keep going forever. The general theme is that if you plot y vs. y’, showing the direct passage of time using a dot for each year, you can visually convey second derivatives too. When teaching these ideas, I will typically show time series graphs of y(t), y'(t), y”(t), but that’s just not so intuitive. Showing y(t) and y'(t) on the two axes has just the right amount of redundancy to really make these patterns clear.

These graphs come up in physics—Dorling illustrates with the classic phase portrait of the pendulum—but there must also be a statistical literature on this—I was thinking I could ask Lee Wilkinson but then I remembered, sadly, that he’s no longer alive—so feel free to inform me of this in the comments.