The Willenhall projectMultidimensional scaling: nine places, 36 distances, no map
Make two distance tables, hand each to the same program, and measure how far its map lands from the real one.
A single Overpass request fetches every road in a box around Willenhall, from motorway to residential street: 350.6 km of connected network. The learner pins the nine postcodes.io places to the road network, each within 5 m of a road point, and fills in two tables of 36 pairs. One holds straight-line distances. The other holds the shortest route along the roads, found with Dijkstra's algorithm. The straight lines average 1,818 m and the road routes 2,219 m.
Five of the 36 pairs, straight line against shortest road route, from our Python run on OpenStreetMap data| Pair of places | Straight line | By road |
|---|
| Spring Bank and Little London | 0.16 km | 0.24 km |
| Short Heath and Lane Head | 0.45 km | 0.59 km |
| Willenhall and Bentley | 2.22 km | 2.51 km |
| Bentley and Ashmore Lake | 2.39 km | 3.20 km |
| Willenhall and New Invention | 3.49 km | 3.87 km |
Classical multidimensional scaling squares the distances, centres the table, and takes the two strongest eigenvectors as x and y. The positions that come out have no north. They may be turned to any angle, or flipped like a reflection, because a distance table says nothing about direction. So a second step, Procrustes alignment, rotates or mirrors the result until it sits as closely as possible on the true positions. Before that step the straight-line map is 1,374 m out per place on average. After it the error is 0.0 m. Every position was already hidden in the 36 numbers.
How well each distance table rebuilds the map, after Procrustes alignment| Distance table | Held by top two eigenvalues | Negative eigenvalues | Mean error | Largest error |
|---|
| Straight line, 9 places | 100% | 0 | 0 m | 0 m |
| Road, 9 places, not rescaled | 93.7% | 3 | 285 m | 624 m |
| Road, 9 places, shrunk to 0.833 | 93.7% | 3 | 134 m | 316 m |
| Road, 60 random junctions, shrunk to 0.804 | 78.0% | 24 | 229 m | 891 m |
The road table gives a map that is too big, since every route is longer than the straight line it replaces. Shrinking it by the fitted factor of 0.833 halves the error, and what remains cannot be removed. Negative eigenvalues are the giveaway: they appear only when no arrangement on a flat sheet can honour all the distances at once. Bentley ends up furthest from its true spot, 316 m away, because its road routes to the other eight are stretched the most: 1.30 times the straight line on average, against 1.18 to 1.27 for the rest. With 60 junctions chosen at random the first picture came out as a mirror image, and the two axes held only 78.0% of the structure. More points, more bends, more that a flat drawing has to give up.
Ages 8 to 11
Three towns, three distances, a ruler and compasses: draw the triangle, then notice a friend drew it upside down.
Ages 11 to 15
Build the two tables in Python for a handful of places and compare the averages before any maths.
Ages 15 and up
Write the double-centring and eigenvector steps in NumPy, add Procrustes, and chase the negative eigenvalues.
Credit and limits
Road data is © OpenStreetMap contributors and used under the Open Database Licence. The tables, maps and errors are our own calculations. Routes ignore one-way rules, turn bans and speed, and they stop at the edge of our box, so they are not driving directions. A different choice of places would give different figures.
