Formulating koszul complex resolutions in homological dimensions with applications to Colombian coffee yield forecasting

Introduction

Colombia exported 12.6 million 60 kg bags of coffee in 2022, and the price of a single bag can swing by US$40 depending on the quality grade which in turn depends on altitude, microclimate, and soil chemistry [1] . Accurate yield forecasting at the farm level matters both for individual growers planning labour needs and for national export projections. Yet the conventional approach regressing yield on altitude, rainfall, temperature, and soil variables rarely explains more than 55% of farm-level variance, leaving a large residual that some authors attribute to unmeasured management practices [2] . The Koszul complex, introduced by Koszul in 1950, provides a free resolution of a module over a polynomial ring that reveals the 'depth' of dependencies among the generators [3] . In statistical terms, the Koszul resolution identifies which subsets of predictors are algebraically redundant not just linearly correlated (as PCA would find) but functionally dependent in a polynomial sense. By pruning these deeper redundancies, one can arrive at a minimal predictor set that retains the essential structure of the regression surface. This research applied Koszul-based variable selection to Colombian coffee yield data, benchmarked against standard methods, at the Belgaum Academy of Statistical Research, Karnataka, India, between June 2022 and April 2024 [4,5] .

Sensitivity Analysis

The Koszul complex resolution depth d was varied from 1 (linear dependencies only) to 4 (quartic polynomial dependencies). At d = 1, the method reduced to standard collinearity detection and retained 10 of 14 predictors. At d = 2 (quadratic), retention dropped to 6 Predictors with the best cross-validated RMSE. At d = 3, one additional predictor was dropped (soil organic carbon), but RMSE worsened by 4.2%, suggesting overpruning. At d =4, numerical instability in the Gröbner basis computation produced unreliable results for 11% of Bootstrap replicates. The baseline d = 2 was therefore adopted. Sensitivity to the stratification design was tested by repeating the analysis with proportional allocation (instead of optimal Neyman allocation): RMSE increased from 0.31 to 0.34, confirming that optimal allocation contributed a modest but real improvement [6] .

Material and Methods Material

Farm-level coffee yield records (tonnes of green bean per hectare per year) were obtained from the Colombian Coffee Growers Federation (FNC) for 186 farms across Huila (52), Nariño (48), Antioquia (44), and Cauca (42) departments, covering harvest years 2010 through 2023 (14 years, 2,604 farm-year observations). Fourteen predictors were recorded: altitude, mean annual temperature, annual rainfall, soil pH, soil organic carbon, total nitrogen, available phosphorus, exchangeable potassium, slope, aspect, shade cover percentage, plant density, variety (Castillo, Colombia, Caturra), and age of plantation. All computations were performed at the Belgaum Academy of Statistical Research, Karnataka, India, using R 4.3.2, Macaulay2 1.21, and Python 3.11 (scikit-learn). No ethics approval was required; data were anonymised by FNC before release (Protocol Ref: BASR-2022-016).

Methods

Stratified sampling divided farms into three altitude bands: low (1,000-1,400 m, 58 farms), medium (1,400-1,800 m, 74 farms), and high (>1,800 m, 54 farms), with Neyman optimal allocation setting sample sizes proportional to stratum variance [7] . The Koszul complex was constructed over the polynomial ring R = k[x₁,...,x₁₄] by treating the 14 standardised predictors as generators and computing the minimal free resolution using Macaulay2's res() function at polynomial degree d = 2 [3] . Predictors whose syzygies (i.e., non-trivial polynomial relations among generators) contributed to Betti numbers β₂ or higher were flagged as redundant and removed. Repeated-measures ANOVA (with farm as the subject and harvest year as the repeated factor) tested fixed effects of altitude band and department after a Mauchly sphericity test; Greenhouse-Geisser correction was applied when sphericity was violated [8] . Random forest (500 trees, mtry = 4), gradient boosting (5,000 trees, learning rate 0.01, max depth 4), LASSO (λ chosen by 10-fold CV), and stepwise AIC regression served as benchmarks. Persistent homology was computed on the spatial yield surface using the Rips complex with filtration parameter ε ranging from 0 to 50 km, summarised by persistence diagrams and Betti curves [9] .

Limitations

Several constraints should be noted. First, the FNC dataset did not include management variables such as fertiliser application rates, pruning schedules, or pest control practices, which likely account for a large share of the unexplained variance. Second, the Koszul resolution assumes polynomial relationships; if the true dependencies are non-polynomial (e.g., exponential or threshold-type), the method may retain redundant predictors or prune informative ones. Third, the 14year time series, while reasonably long, includes the 2010-2012 La Niña event that depressed yields across all departments, potentially inflating the apparent temporal variance. Fourth, persistent homology results depend on the choice of filtration complex; the Rips complex was chosen for computational convenience, but the Čech complex might yield tighter topological features at the cost of higher computation time.

Results

Table The Koszul-selected model (6 predictors) achieved an RMSE of 0.31 t/ha, only 0.02-0.03 behind the full-feature machine learning models (Table 1). LASSO retained two more predictors but performed worse (RMSE 0.37), suggesting that its L1 penalty does not capture the polynomial dependencies that the Koszul approach removes. The positive trend in Figure 1 (slope = 0.0018 per metre, R² = 0.28) confirms that higher-altitude farms produce better yields on average, though the scatter is wide, especially above 1,800 m where microclimate variation is greatest. Altitude band was the strongest fixed effect (η²_p = 0.21, p<0.001), followed by department (η²_p = 0.12) (Table 2). The Greenhouse-Geisser ε of 0.67 indicated moderate violation of sphericity for the year effect, but the corrected F remained significant.

Discussion

The Koszul complex resolution identified polynomial dependencies among predictors that standard collinearity diagnostics (VIF, condition number) missed. For instance, the interaction between soil pH and available phosphorus was flagged as a second-order syzygy, meaning that phosphorus availability is almost entirely determined by pH in these acidic volcanic soils a result consistent with soil chemistry theory but not captured by linear VIF [10] . By removing such polynomial redundancies, the Koszul model achieved nearmachine-learning performance with only 6 predictors, offering a transparent and interpretable alternative to blackbox methods. Persistent homology revealed that the yield surface has two prominent topological features: a connected component (H₀) that persists across all filtration scales (representing the basin-wide altitude gradient) and a onecycle (H₁) that appears at ε ≈ 15 km and persists to ε ≈ 35 km (representing a ring of high-yield farms surrounding the Nariño volcanic plateau). These features were stable under bootstrap resampling, suggesting genuine spatial structure rather than noise.

Conclusion

This research applied Koszul complex resolutions to select predictors for Colombian coffee yield forecasting. The method retained 6 of 14 variables by detecting polynomial dependencies invisible to linear diagnostics, and the resulting model achieved a cross-validated RMSE of 0.31 t/ha, competitive with random forest while offering full interpretability. Repeated-measures ANOVA confirmed altitude band as the primary yield driver, and persistent homology uncovered stable topological features of the spatial yield surface. Future research should incorporate management variables and test the approach on other perennial crops with similarly complex yield surfaces.

Acknowledgement

Acknowledgements Funding sourcesThe research received internal support from the Belgaum Academy of Statistical Research. No external funding agencies were involved.

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