## 2 Materials and Methods
### WBM Classification Algorithm  
The WBM model uses the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. It is used in optimization techniques and is particularly suited for large-scale problems. L-BFGS is a quasi-Newton method that approximates the Hessian matrix using limited memory, allowing it to handle large datasets without the computational burden associated with BFGS methods.  
The pre-implemented version of the ML_NET tool is named LbfgsMaximumEntropy. This algorithm has been successfully applied in various fields, including ML, image processing, and geophysical inversion. L-BFGS operates by maintaining a fixed number of previous gradient evaluations and updates, which allows it to construct an approximation of the inverse Hessian matrix. This is achieved through a two-loop recursion that computes the search direction based on these curvature pairs, thereby facilitating optimization without requiring the storage of the Hessian matrix [62]. The algorithm's efficiency is particularly notable in applications such as seismic waveform tomography, which has been used to optimize complex models with substantial data [63; 64]. [PERSON] et al. [65] show that L-BFGS can outperform other convergence speed and accuracy optimization methods.  
According to reference [66], the training method LbfgsMaximumEntropy, which uses the L-BFGS, has the following analyzed parameters:  
* The L1 regularization hyperparameter (l1 Regularization) encourages sparsity in the model, pushing many weights toward zero, which can help with feature selection. By default, the ML_NET function is set in the training process to zero for the L1 hyperparameter.
* The L2 regularization hyperparameter (l2 Regularization) helps prevent overfitting by penalizing large weights. The ML_NET training function sets L2 to zero if no specific value is set.  
* The threshold (learning rate) for optimizer convergence determines how slight the gradient should be before the algorithm considers that it has converged and can stop optimizing. This parameter is known as the learning rate.
* The historySize parameter (number of epochs) controls the memory size used by the LbfgsMaximumEntropyMulticlassTrainer. A lower value means faster training but can result in less accurate results. The default value is 20. Increasing this value will increase accuracy but at the cost of speed.
* The enforceNonNegativity hyperparameter (dropout) is true or false, requiring the model to have (or not) only non-negative weights for features. The default value is false.  
Adjusting these parameters will fine-tune the model to fit the specific dataset and classification problem better. The Results section presents the 288 experiments with regularization and optimizer tolerance to improve the WMB model's performance and avoid overfitting.  
The LbfgsMaximumEntropy algorithm was chosen for the WBM because of its suitability for logistic regression-based classification tasks and can handle high-dimensional, multi-class data efficiently. This algorithm was ideal for the provided dataset, where features required classification into well-being levels with precise probability estimates. LbfgsMaximumEntropy optimizes regularization and class balance, both of which are critical given the range of _WBI_ inputs and the need to predict minority well-being states accurately. Maintaining model performance across varied data distributions was a decisive factor, as this consistently delivered high accuracy in predicting well-being across all classes.