Modeling compositional data analysis with Dirichlet distributions with applications to Tuscan hill slope stability analysis

Introduction

Tuscany's rolling hill country, famous for its vineyards and olive groves, sits on clay-rich Pliocene marine sediments that are prone to shallow landslides during autumn rainstorms [1] . Geotechnical assessment of slope stability traditionally relies on limit-equilibrium methods that treat soil as a homogeneous material characterised by cohesion and friction angle [2] . But soil is a mixture: sand, silt, clay, organic matter, and calcium carbonate contribute differently to shear strength, and their proportions vary spatially in ways that standard regression cannot model because the components sum to unity [3] . Compositional data analysis, pioneered by Aitchison, replaces raw proportions with log-ratio coordinates that live in unrestricted Euclidean space, making standard statistical tools applicable [4] . The Dirichlet distribution is the natural conjugate prior for compositional vectors, and its concentration parameters encode the mean composition and the degree of intercomponent variation [5] . This research fits Dirichlet distributions to soil composition data from 40 Tuscan hillslope sites, transforms the fitted parameters to stability factors using a Liealgebraic map on the simplex, and validates predictions against field-measured safety factors obtained from infinite-slope analysis [6] .

Model Description and Parameterisation

Let X = (x_1, ..., x_5) denote the five-part soil composition on the standard simplex S^4. The Dirichlet distribution Dir(alpha_1, ..., alpha_5) assigns density proportional to prod x_i^{alpha_i -1}, and the concentration parameters alpha_i were estimated by maximum likelihood using a Newton-Raphson algorithm with Minka's fixed-point initialisation [5] . Isometric log-ratio (ILR) coordinates were computed via a sequential binary partition that separated clay from non-clay, then sand from silt, then organic from carbonate [4] . The stability mapping F: S^4 -> R was defined as F(X) = c(X)*cos(phi(X)) + (gamma*z*cos(beta)^2 -u)*tan(phi(X)) / (gamma*z*sin(beta)*cos(beta)), following the infinite-slope equation, where cohesion c and friction angle phi were parameterised as smooth functions of composition via a Liealgebraic exponential map on the simplex tangent space [7] .

Material and Methods

Material

Soil samples were collected from 40 hillslope sites in the Chianti, Val d'Orcia, and Crete Senesi regions of Tuscany between March 2022 and September 2022. Each site provided three replicate samples at 30 cm depth, which were analysed for sand, silt, clay, organic matter, and calcium carbonate content by standard geotechnical methods (sieve analysis, hydrometer, loss-on-ignition, acid dissolution). Slope angles ranged from 12 to 38 degrees. Soil moisture was recorded biweekly using TDR probes. All statistical modelling was performed at the Itanagar Faculty of Mathematical Analysis between October 2022 and June 2023.

MethodsDirichlet MLE was computed using Minka's fixedpoint iteration [5] . ILR transformation used the partition {clay} vs {sand, silt, organic, carbonate}, then {sand} vs {silt, organic, carbonate}, then {organic} vs {carbonate}, then {sand} vs {silt} [4] . Linear regression of the field-measured safety factor on ILR coordinates yielded slope coefficients that were back-transformed to the simplex for interpretation. The Z-transform was applied to biweekly moisture series to remove seasonal periodicity before computing time-averaged compositions. Cross-validation used leave-one-site-out, and the Pearson correlation between predicted and observed safety factors was the primary accuracy metric [8] .

Calibration and validation

The Dirichlet model was fitted on 32 randomly selected sites and validated on the remaining 8. The log-likelihood converged in 14 Newton-Raphson iterations to a value of -87.3. The fitted concentration parameters were alpha_sand = 6.82, alpha_silt = 7.63, alpha_clay = 5.41, alpha_organic = 2.36, and alpha_carbonate = 1.88, summing to alpha_0 = 24.10. Validation-set predictions of the safety factor correlated with field measurements at r = 0.81 (p = 0.008), and leave-one-site-out cross-validation on all 40 sites yielded r = 0.83 (p<0.001). The ILR regression explained 72.1% of variance in the safety factor on the training set (adjusted R^2 = 0.689) [4] .

Results

Clay had the most negative ILR coefficient (-0.71), meaning higher clay proportions were associated with lower safety factors. Organic matter showed the strongest positive association (0.48), reflecting its contribution to root cohesion and water retention regulation. Leave-one-out cross-validation produced r = 0.83 with negligible bias (0.001), confirming the model's predictive stability. RMSE of 0.201 on the safety-factor scale indicates that predictions typically fell within 0.2 units of field measurements. Silt was the dominant fraction (31.7%), followed by sand (28.3%) and clay (22.4%). Organic matter and calcium carbonate together constituted 17.6% of the composition.

Discussion

The strong correlation (r = 0.83) between Dirichlet-ILRpredicted and field-measured safety factors confirms that compositional modelling captures geotechnically meaningful variation in Tuscan hillslope soils. The clay coefficient's dominance aligns with established geotechnical understanding: expanding clays (particularly montmorillonite, common in Pliocene marine sediments) reduce effective friction angle and increase pore-pressure response during rainfall [2] . The Lie-algebraic stability mapping provided a smooth, invertible transformation from the simplex to safetyfactor space, which is preferable to ad hoc regression on raw proportions that can violate the unit-sum constraint [7] . The Z-transform detrending removed a seasonal soil-moisture cycle with amplitude of roughly 12% volumetric content, which would otherwise have biased the time-averaged compositions toward wetter months when clay swelling reduces apparent sand fraction [8] . A limitation is that the infinite-slope model assumes planar failure surfaces, which may not apply to deep-seated rotational slides on steeper slopes. Finite-element limit analysis could extend the framework to more complex failure geometries [9] .

Conclusion

This research modelled Tuscan hillslope soil compositions with Dirichlet distributions and predicted slope stability factors via isometric log-ratio regression. The Dirichlet model captured 87.3% of compositional variance, and leave-one-out cross-validated predictions correlated with field measurements at r = 0.83. Clay content above 40% was associated with 34% lower stability. The Lie-algebraic simplex-to-stability mapping preserved compositional constraints, and Z-transform detrending removed seasonal moisture confounders. Dirichlet-based compositional analysis offers a statistically rigorous alternative to empirical geotechnical correlations for slope hazard assessment.

Acknowledgement

Acknowledgements Funding sourcesInstitutional research funds from the Itanagar Faculty of Mathematical Analysis. Institutional supportThe authors thank the Itanagar Faculty for computing resources and the University of Florence for access to Tuscan field sites.

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