An Uncertainty-Aware Hybrid Mathematical-Machine-Learning Model for Smart Irrigation Decision Support

Andrea Scariolo

cs.LGSep 12, 2026

An Uncertainty-Aware Hybrid Mathematical-Machine-Learning Model for Smart Irrigation Decision Support

Authors: Andrea Scariolo

Organizations: Independent Researcher

Abstract

Agriculture accounts for roughly 70% of global freshwater withdrawals, yet irrigation is still commonly scheduled reactively, with no forecast of where soil moisture is heading and no statement of confidence in that forecast. Data-driven models are accurate but opaque and point-valued; water-balance models are transparent but carry large structural error. Neither alone supports a defensible irrigation decision under uncertainty. This study coupled the two and carried uncertainty through to the decision: a four-parameter water-balance core, calibrated on training data only, was corrected by a Random Forest that learned nothing but the physical residual, conformal prediction attached 90%-nominal intervals, and a risk-aware rule converted the interval lower bound into an irrigation trigger. It was evaluated on three years of hourly in-situ measurements from a rainfed Mediterranean cropland station under a strict chronological split, scored against persistence, from one hour to one week. At the 24 h horizon the hybrid reached RMSE 0.00925 m^3 m^-3 and +9.4% skill, roughly double the best of nine baselines, of which only the Random Forest beat persistence. Skill did not grow with lead time: it peaked at +27.4% at three hours and fell to +1.2% at one week. Conformalised quantile regression was better calibrated and 11% sharper than constant-width conformal prediction. The risk-aware rule raised management-threshold crossings detected in advance from 0.905 to 1.000, at a precision cost of 0.975 to 0.950 and 3.7% more notional water, and beyond 72 h the point forecast fell below the no-forecast rule while the interval-based rule did not. Uncertainty quantification therefore governs the lead time over which forecast-driven irrigation advice remains trustworthy, and here transparency in the physical layer cost no measurable accuracy.

Figures & tables

Figure 1: (a) The measured hourly streams the model consumes at the Torano station: rainfall, air temperature and volumetric soil moisture at 30, 60 and 90 cm, shown here over one month of the record. (b) The two-stage architecture. A transparent four-parameter water-balance core produces a physically grounded forecast, a tuned Random Forest corrects its residual, and conformal calibration attaches a distribution-free 90 %-nominal interval. (c) Skill relative to persistence at the 24 h horizon for the models actually run; the hybrid leads at +9.4 %. (d) The forecast and its interval turned into a decision for one real issue time in the test partition, 6 April 2010 at 16:00. Measured moisture sits just above the management trigger, and the 90 %-nominal interval of the 24 h forecast already reaches below it, so the risk-aware rule calls for irrigation while the point forecast alone would not.

Figure 2: Processing chain. Cleaned station records were transformed into past-only features. A transparent water-balance core produced a physically grounded forecast whose residual was corrected by a tuned Random Forest. The resulting hybrid forecast received a calibrated conformal interval, which drove the risk-aware irrigation decision.

Model	RMSE ( $\mathrm{m^{3}\,m^{-3}}$ )	Skill vs persistence (%)
Random Forest	0.00972	+4.8
Persistence	0.01021	0
Ridge regression	0.01046	$-$ 2.4
Gated recurrent unit	0.01071	$-$ 4.9
Long short-term memory	0.01073	$-$ 5.1
Gradient boosting (XGBoost)	0.01078	$-$ 5.6

Table 1: Nine-model comparison at the 24 h horizon on the Torano test partition

Model	RMSE ( $\mathrm{m^{3}\,m^{-3}}$ )	$R^{2}$	Skill (%)
Hybrid, perfect forcing (ablation)	0.00680	0.971	+33.4
Physical core, perfect forcing (ablation)	0.00778	0.962	+23.8
Hybrid, tuned corrector	0.00925	0.946	+9.4
Random Forest, tuned	0.00935	0.945	+8.4
Tri-training, three learners plus unlabelled data	0.00957	0.942	+6.3
Random Forest, untuned	0.00972	0.940	+4.8

Table 2: Hybrid and boosting results at the 24 h horizon

Lead time (h)	RMSE persistence	RMSE hybrid	$R^{2}$	Skill (%)
1	0.00178	0.00154	0.998	+13.6
3	0.00356	0.00258	0.996	+27.4
6	0.00524	0.00398	0.990	+24.0
12	0.00746	0.00624	0.975	+16.4
24	0.01021	0.00925	0.946	+9.4
48	0.01347	0.01286	0.896	+4.5

Table 3: Forecast performance versus lead time

Figure 3: Error and skill against lead time. (a) Root-mean-square error of persistence and of the tuned hybrid; both grow monotonically. (b) Skill relative to persistence; the hybrid peaks at three hours and decays towards zero within a week, while the physical core alone remains at the persistence level throughout.

Figure 4: Attribution of the tuned direct-prediction Random Forest. (a) Global mean absolute Shapley values for the ten leading predictors. (b) Shapley dependence for the dominant predictor, coloured by the one-hour lag, showing a near-monotone relationship between current moisture and its contribution to the forecast.

Method	Coverage	Mean width ( $\mathrm{m^{3}\,m^{-3}}$ )	Winkler score
Split conformal	0.826	0.01939	0.05172
Conformalised quantile regression	0.875	0.01725	0.04050

Table 4: Prediction-interval quality at the 24 h horizon, 90 % nominal level

Slice	n	Split coverage	Split width	CQR coverage	CQR width
All	3 713	0.826	0.01939	0.875	0.01725
Wet	1 834	0.744	0.01939	0.864	0.02371
Dry	1 879	0.906	0.01939	0.886	0.01094
December–February	1 543	0.730	0.01939	0.832	0.02030
March–May	2 170	0.894	0.01939	0.905	0.01508

Table 5: Conditional coverage and width by regime and season at the 24 h horizon

Figure 5: Observed and forecast soil moisture with the 90 % conformalised interval over the last 1 600 h of the test partition. The spring drydown crosses the management trigger in early April; the band widens markedly around the rainfall event of mid-May, when the state is least predictable.

Figure 6: Interval quality at the 24 h horizon. (a) Empirical against nominal coverage; the dashed line marks perfect calibration. (b) Mean interval width by regime at issue time; the adaptive method narrows in dry conditions and widens in wet ones, whereas the constant-width method cannot.

Rule	Alarms	Precision	Recall	F1	Notional water (mm)	Missed crossings
Reactive (no forecast)	961	0.975	0.975	0.975	24 397	24
Forecast (point)	969	0.982	0.991	0.987	24 493	9
Risk-aware (interval lower bound)	1 012	0.950	1.000	0.974	25 309	0

Table 6: Decision-rule comparison at the 24 h horizon

Figure 7: Decision performance at the 24 h horizon. (a) Notional water proxy against missed management-threshold crossings for the three rules. (b) Share of crossings detected in advance within the decision-relevant regime, where irrigation timing is genuinely difficult.

Figure 8: Uncertainty and decisions against lead time. (a) Mean interval width; both methods widen as the forecast reaches further ahead, with the adaptive method consistently sharper beyond 24 h. (b) Share of management-threshold crossings detected by each rule; the point forecast falls below the no-forecast rule beyond 72 h whereas the interval-based rule does not.

Lead time (h)	Interval half-width ( $\mathrm{m^{3}\,m^{-3}}$ )	Fires on regime (%)	Precision	Alarm rate
1	0.0009	60.0	0.979	0.263
24	0.0091	71.0	0.950	0.273
72	0.0175	92.7	0.888	0.292
168	0.0357	100.0	0.704	0.371

Table 7: Selectivity of the risk-aware rule across lead times

Decision-Making under Uncertainty

Decision-making under uncertainty studies how to choose actions when outcomes, states, or models are imperfectly known.

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Abstract

Decision-Making under Uncertainty

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