Nitrogen rates for winter wheat are set before the season, under unknown prices and weather. The standard UK advice does not respond to prices, yet recent price swings moved the most profitable rate by tens of kilograms per hectare. Machine learning is often proposed as the fix. However, it is usually judged on prediction accuracy, and accurate prediction does not by itself make the recommended rate more profitable. Our insight is to score nitrogen advice directly by the profit it forgoes on measured yield response curves. We build a test bench on 892 such curves from two long running UK experiments, and sweep the nitrogen to grain price ratio to cover all price scenarios. On this bench, machine learning fails as a predictor. No model recovers the best rate within farm tolerance, and the benchmark noise shows none can. At normal prices, every model also loses to the standard advice on profit. The gain sits elsewhere. A simple correction step applied after the model cuts profit losses by a quarter, while better models and extra features give no gain. The same frozen correction cuts losses by 43% at the second site without any retraining. A hybrid of standard advice plus a damped correction removes bias and trims rare large losses. The same price sweep also prices emission cuts, at a cost comparable to current carbon prices. Machine learning therefore pays as a profit scored correction to standard advice, not as its replacement.
Figures & tables
Figure 1: Accurate prediction is not profitable advice. a , A measured yield curve from Broadbalk (black) and two synthetic predictions. Prediction A is off by 1.50 t ha -1 everywhere, but it matches the slope, so its advice lands exactly on the best rate. Prediction B matches the yields closely (error 0.24 t ha -1 ) but bends too gently, so its advice is 50 kg too high and loses 0.11 t ha -1 of profit. Prices at b=5 . b , The same decoupling in our six trained models: yield prediction error (leave one year out) against the median profit loss of the resulting advice at b=5 . Accuracy does not rank the advice (Spearman ρ=0.09 ).
Figure 2: Two field experiments; one moving target. a , The two data sources. Broadbalk gives 402 measured yield response curves over 53 years. Woburn gives 490 curves over 43 years and is used for testing only. b , The most profitable N rate, measured on the Broadbalk curves (blue, median). It falls as N gets dearer relative to grain. Gold rings mark the observed UK price ratios of 2020, 2022, and 2023. Those prices alone moved the best rate from 219 to 174 kg. The RB209 advice (grey) stays at 219 kg throughout.
Period
AN, £/t
Wheat, £/t
N price, £/kg
b
2020 (Jan)
234
165
0.68
4.1
2021 (RB209 ref.)
345
200
1.00
5.0
2021 (Sep)
395
200
1.14
5.7
2022 (Jul peak)
841
280
2.44
8.7
2023 (Jan)
700
190
2.03
10.7
2023 (May)
390
185
1.13
6.1
Table 1: Observed UK prices and the implied price ratio b . AN is ammonium nitrate, 34.5 % N. Sources: AHDB fertiliser price series; RB209 2022 revision reference prices.
Figure 3: Same models, same predictions, two verdicts. The blue band spans our six ML models: mean curve, field mean curve, ridge regression, Gaussian process, random forest, and TabPFN. The blue line is their median. Each point covers 317 to 402 held out curves. a , Scored by rate error, the whole band sits above the 20 kg pass zone. The gold squares show the benchmark’s own noise, about 23 kg. Even a perfect model would therefore fail. The test, not the models, is broken. b , Scored by profit loss, the verdict depends on prices. RB209 (black) beats the whole band at normal prices ( b≤6 ). At crisis prices ( b≥8 ) it becomes the worst option.
Design
Median loss, t ha -1
Worst 10%
Bias, kg N ha -1
Large errors
RB209 alone
0.035
0.562
+6
42
ML alone
0.070
0.872
+25
62
Hybrid: RB209 + damped ML
0.033
0.518
+1
37
Table 2: Three designs on held out Broadbalk curves. Numbers are medians over the b grid; n is 338 to 345 per design. Large errors are curves losing more than 0.3 t ha -1 . The hybrid’s median edge over RB209 (0.033 vs 0.035) is small and untested statistically; read it as “no worse”. The hybrid’s real gains are the bias and tail columns.
Figure 4: Where the gain comes from. a , We tried twelve improvements to one ML pipeline. Each was scored with every year held out in turn. Bars to the right mean lower profit loss than the plain model. Grey group: better models and extra features; none helped reliably. Blue group: keep the model and correct its advice afterwards; only these reached the green zone. The open ring improved here but not in the original run. b , The three way comparison of Table 2 . ML alone doubles the loss of the standard advice. Adding a damped ML correction to RB209 matches its median. It also removes bias and trims the worst losses.
Change tested
What the operation does
Effect
Change the model or its features
Pick safer predictions
Choose the rate with the smallest predicted downside, not the highest mean profit.
−10.9 %
Add protein features
Add grain protein measurements as extra model inputs.
+5.7 %
Add weather features
Add season rainfall, temperature, and radiation as extra inputs.
−13.2 %
Average several training runs
Train the model several times with different seeds; average the predictions.
−12.0 %
Use the median, not the mean
Combine those runs by median instead of mean.
−33.4 %
Table 3: The twelve attempted improvements, in the order of Fig. 4 a. Effect is the change in profit loss against the plain model. Positive means better.
Figure 5: Transfer with no retraining. The pipeline was frozen on Broadbalk and applied to 490 Woburn curves unchanged. Middle bar: the correction cuts the median profit loss by 43% ( p=0.034 ). Right bar, the control: scramble which field group each curve belongs to, keeping everything else. The whole gain disappears. So the correction works through real field groupings, not luck.
Figure 6: The price ratio is also a carbon dial. Fertiliser emissions at the best measured rate, across the price ratio b . The factor is 9.10kg CO 2 e per kg N: manufacture 3.42 ( Brentrup et al., 2018 ) plus field nitrous oxide 5.68 ( IPCC, 2019 ; IPCC, 2021 ) . Raising the target from b=5 to b=8 has the same effect as a £66 per tonne carbon price and saves about 180kg CO 2 e per hectare.
Quantity
t ha -1 grain
@160
@200
@240
@280
RB209 median loss, b=5
0.058
9
12
14
16
RB209 median loss, b=12
0.246
39
49
59
69
RB209 worst 10%, b=3
3.45
553
691
829
967
Hybrid vs ML alone
0.037
6
7
9
10
Correction step gain
0.019
3
4
5
5
Woburn transfer gain
0.032
5
6
8
9
Table 4: Headline results converted to £ per hectare. The conversion multiplies grain terms by the grain price.
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA · CTrees, Pasadena, CA 91105, USA