Figure P (supplemental): Paris and Lyon

This example is described in the paper (near the end of Section 1.1), but there was no space for an illustration, or the details of how the explainers were selected.

This example is based on a data set that scores 140 cities around the world on 40 "liveability" criteria. The data set came from a contest on Buzzdata.com. To demonstrate explainers, I have tried to show that other properties of cities can be connected to these liveability criteria.

A family of "exemplar-explainers" was generated by computing functions that distinguished Paris from other cities in the world. That is, the positive class contained only Paris, and the negative class contained the other 139 cities.

For this example, we restrict ourselve to very simple functions (a limit of 3 variables, with coefficients that are small integers). We also restrict ourselves to functions that are "correct" in that Paris is scored as the most "Parisian" city in the world (or at least tied for it - ties are a common occurance in this data, because it is highly quantized).

There is a diverse set of explainers that are both simple and correctly score Paris as the most-Parisian city.

To distinguish what is distinctly "Paris," as opposed to simply being "French" we can look for expainers that help make this distinction. There is one other French city in the data: Lyon. We sort the list of (simple and correct) Paris explainers by the rank of Lyon, such that if Lyon scores low, that explainer is at the beginning of the list, and it it scores highly, it is at the end. The figure below shows the first and last 2 explainers from this sorted list.

(the embedded svg seems to have broken. you can link directly to the diagram - which still seems to work. fig3p.svg)

Warning: this diagram looks best in Chrome. If you load this page in Firefox or Internet Explorer, it seems to mess up the fonts. The SVGs themselves usually look OK in most browsers with good SVG support, but somethign weird is happening with SVG embedding. You'll also notice some weirdnesses with the interactions if you mouse-over things.

Note that for all 4 of these explainers, Paris (marked in red) is ranked as either the most Parisian city, or tied for the most Parians-city. The left two explainers score Lyon considerably lower, while the right two score Lyon so that it is tied for being the most-Parisian city in the world.