The Andover projectPoisson regression on 224 map squares
Count the postcodes in each square, measure its roads, fit three models and test them on squares held back.
The learner lays a grid of 500 m National Grid squares over Andover and the villages round it, 16 across and 14 up, 224 in all. Two open datasets fill it in. Code-Point Open from Ordnance Survey gives the position of 1,366 postcodes inside the grid, 959 of them in district SP10 and 407 in SP11. One Overpass request to OpenStreetMap gives the roads, 303.0 km of them inside the grid. The counts are lopsided, as counts usually are: 88 squares contain no postcode at all, and the fullest contains 76.
Three models predicting postcodes per square from road length (our Python fits; misses are mean absolute errors in postcodes)| Model | Prediction for a square with no road | Squares predicted below zero | Average miss, all squares | Average miss, unseen squares |
|---|
| Straight line | -1.36 | 55 | 3.35 | 3.48 |
| Poisson regression on road km | 1.37 | 0 | 3.94 | 4.49 |
| Poisson regression on log of road km | 0.03 | 0 | 2.99 | 3.16 |
The straight line says each kilometre of road adds 5.52 postcodes, starting from minus 1.36. For 55 of the 224 squares its answer is a negative number of postcodes. Poisson regression cannot do that. It predicts the logarithm of the count, and turning a logarithm back into a count always gives something above zero. But the first Poisson model is worse overall, and the reason is worth finding. Working in logarithms turns adding into multiplying, so it claims every extra kilometre of road nearly doubles the postcodes, by a factor of 1.98. For the busiest squares that runs away: its top prediction is 94.7 where the real maximum is 76.
The repair is to give the model the logarithm of road length as well, so that both sides speak the same language. The fitted rule becomes a power law: postcodes grow as road length to the power 1.53. That model never goes negative, predicts 0.03 for an empty square, and has the smallest average miss. The column that matters most is the last. Each model was fitted on 70% of the squares and scored on the other 30%, 200 times over with different random splits. The straight line beat the first Poisson model in 193 of the 200. The log version beat the straight line in 193 of the 200. One more check keeps everyone honest: the counts vary about three times as much as a pure Poisson model expects, so its error bars would be too narrow, and a statistician would reach next for a model that allows extra spread.
Ages 8 to 11
Count dots in the squares of a printed grid, draw a line through the tallies, and spot the square where the line dips under nought.
Ages 11 to 15
Build the grid in Python, fit a straight line, and list every square with a negative prediction.
Ages 15 and up
Code Poisson regression from its update rule in NumPy, try both inputs, and run the 200 held-out splits.
Sources and what this does not show
Postcode positions contain OS data © Crown copyright and database right 2026 and Royal Mail data © Royal Mail copyright and database right 2026. Roads are © OpenStreetMap contributors under the Open Database Licence. The grid, the fits and the scores are ours. A postcode is not a household, and a square of another size or place would give other numbers. The models describe this one grid and predict nothing about future building.
