# Ensemble Modelling for Predicting Fish Mortality +
## 4 Materials and Methods
### Decision Tree Models  
Decision Tree (DT) models are integral components of many ensemble methods due to their inherent interpretability, ease of implementation, and capability to handle non-linear data relationships. In contrast to linear models, DTs partition the data space into regions defined by rules based on feature thresholds, allowing them to capture complex patterns through a simple, hierarchical decision structure. This makes them particularly useful in  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Population Size & 5000 \\ \hline Generations & 20 \\ \hline Metric & Mean Absolute Error (MAE) \\ \hline Stopping Criteria & 0.01 (fish deaths) \\ \hline Function Set & add, sub, mul, div, sqrt, cos, sin, log \\ \hline Crossover Probability & 0.7 \\ \hline Sub-tree Mutation Probability & 0.1 \\ \hline Hoist Mutation Probability & 0.05 \\ \hline Point Mutation Probability & 0.1 \\ \hline Parsimony Coefficient & 0.05–0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The setup of hyper-parameters for the gplearn tool.  
diverse settings, ranging from risk assessment and medical diagnosis to financial analysis, where the decisions need to be both accurate and understandable.  
In each ensemble model developed herein, we incorporate a DT regressor, chosen for its ability to model non-linear relationships and interactions between features through successive binary decisions. The architecture and parameters of the utilized DT were meticulously selected to optimize both performance and generalizability, as detailed below:  
* Criterion and Splitting Strategy: The DT utilizes the \"squared error\" criterion to assess the quality of splits within the tree. This measure focuses on minimizing the variance of the target values within each node, aiming to produce homogenized predictions. It uses the \"best\" splitter strategy, which exhaustively considers all possible splits across all features, selecting the one that provides the most significant reduction in squared error, thereby optimizing each decision path for the greatest predictive accuracy.
* Tree Depth and Complexity: The maximum depth of the tree is constrained to 10 levels. This limitation not only prevents the model from overfitting, by curbing the complexity of the decision paths, but also maintains computational efficiency and interpretability. While deeper trees might model the training data with high fidelity, they often fail to generalize well; hence, controlling the depth helps balance bias and variance.
* Leaf Nodes and Samples: To further prevent overfitting, each leaf/terminal node in the tree must contain at least five samples. This parameter helps avoid creating overly specific rules, thus reducing model variance, without a significant increase in bias. The tree also ensures that a node splits only if it contains at least two samples, allowing the tree to expand sufficiently, but avoiding insignificant divisions.
* Node Purity and Randomness: There are no constraints on the maximum number of leaf nodes, and no minimum impurity decrease threshold is enforced, suggesting that the tree's expansion is primarily influenced by its depth and the minimum requirements in leaf samples. No monotonic constraints are applied, offering the flexibility to capture both increasing and decreasing trends in feature relationships.  
Through the aforementioned settings, the DT of each ensemble model is designed to achieve an optimal balance between prediction accuracy and computational efficiency.