Phase 3 · January initialization · FMAM and its months

Rainfall-distribution models (Pathway B)

Each model predicts a full rainfall distribution from the raw ensemble (pooled over cells in units relative to each cell's climatology); terciles at the training CHIRPS boundaries, quantiles and intervals come from it. References, in the same folds: the published probabilities (tercile RPS) and the MVS-corrected ensemble (CRPS, quantiles). ✓ = better than the reference with Holm-adjusted p < 0.05 across the models and targets, and not worse over 2017–2025. Confirmed better — tercile probabilities: none; rainfall amounts (CRPS): none.

Tercile probabilities

Tercile RPSS, outer leave-one-year-out 1993–2016
Method	Feb	Mar	Apr	May	FMAM
Published probabilities (incumbent)	+0.033	+0.002	+0.001	+0.028	+0.032
Bayesian joint probability	+0.026	-0.088	-0.005	+0.019	+0.030
Bayesian model averaging for precipitation	+0.014	-0.037	-0.010	+0.030	+0.044
Quantile regression forest	+0.028	-0.107	-0.087	-0.027	-0.025

Rainfall amounts

Amount CRPSS against the CHIRPS climatology, 1993–2016
Method	Feb	Mar	Apr	May	FMAM
MVS-corrected ensemble	+0.074	-0.044	-0.016	+0.007	+0.025
Bayesian joint probability	+0.085	-0.048	-0.018	-0.016	+0.019
Bayesian model averaging for precipitation	+0.084	-0.014	-0.011	+0.008	+0.044
Quantile regression forest	+0.018	-0.100	-0.071	-0.033	-0.022

Intervals and quantiles

Share of observations inside the 10–90% interval, 1993–2016 (80% is calibrated)
Method	Feb	Mar	Apr	May	FMAM
MVS-corrected ensemble	56%	54%	68%	69%	72%
Bayesian joint probability	83%	82%	81%	81%	82%
Bayesian model averaging for precipitation	82%	81%	82%	84%	80%
Quantile regression forest	74%	71%	72%	73%	72%

Pinball loss of the median (mm), 1993–2016 (lower is better)
Method	Feb	Mar	Apr	May	FMAM
MVS-corrected ensemble	5.09	14.09	20.47	20.44	33.73
Bayesian joint probability	5.09	14.77	20.24	20.56	33.97
Bayesian model averaging for precipitation	5.14	14.12	20.53	20.59	33.60
Quantile regression forest	5.50	15.36	21.74	21.36	35.85

Test years

Fit 1993–2016, scored 2017–2025 (exploratory)
Method	Feb	Mar	Apr	May	FMAM
Published probabilities / MVS ensemble	+0.020 / +0.002	+0.008 / +0.005	+0.002 / -0.006	+0.004 / -0.007	+0.017 / +0.016
Bayesian joint probability (RPSS / CRPSS)	-0.029 / +0.023	-0.037 / -0.034	-0.006 / -0.008	-0.006 / -0.056	+0.013 / +0.015
Bayesian model averaging for precipitation (RPSS / CRPSS)	-0.025 / +0.029	+0.013 / +0.023	-0.041 / -0.017	-0.051 / -0.042	+0.012 / +0.023
Quantile regression forest (RPSS / CRPSS)	-0.034 / -0.016	-0.007 / +0.004	-0.052 / -0.031	-0.086 / -0.049	-0.006 / +0.017

Fits and model definitions

Bayesian joint probability: did not converge in every fold.
Bayesian model averaging for precipitation: converged in every fold.
Quantile regression forest: converged in every fold.

Models
Model	Estimates	Predictors	Fitting	Outputs
Bayesian joint probability	rainfall distribution (log-sinh normal, censored at zero, Student-t predictive)	ensemble mean m' (relative)	censored maximum likelihood of the transforms and the joint normal; parameter uncertainty integrated with the number of training years as the effective sample	CDF, quantiles, tercile probabilities
Bayesian model averaging for precipitation	mixture over members: probability of zero and a gamma for the cube root of the amount	each member's amount f' (relative), equal weights for exchangeable members	maximum likelihood of the mixture, pooled	CDF, quantiles, tercile probabilities
Quantile regression forest	conditional distribution as leaf-weighted training amounts; quantiles at 39 levels (2.5–97.5%)	m', s', md', ensemble 10% and 90% quantiles (relative), log climatological mean, CHIRPS coefficient of variation	200 trees, at least 40 samples per leaf, half the predictors per split, pooled	CDF, quantiles, tercile probabilities

Tercile RPSS and amount CRPSS of the Phase 3 distribution models against the published pipeline.