# Comparative Analysis of Machine Learning Methods for Predicting Energy Recovery from Waste
## 2 Materials and Methods
### ElasticNet  
ElasticNet is a sophisticated regularization technique in regression analysis that mitigates the limitations of ridge and lasso regression by combining their strengths. It is particularly effective in the situations where there are multiple correlated features, where lasso might arbitrarily select one feature among the correlated ones, and ridge might include all but not distinguish the importance. ElasticNet overcomes this by shrinking some coefficients and setting others to zero, thus performing variable selection and complexity regularization. This duality enables it to perform well even under conditions of high dimensionality and multicollinearity among variables, where traditional methods may falter. By incorporating both penalties, it ensures that the model remains sparse yet stable, which is particularly useful when dealing with datasets that exhibit both features of large-scale and high dimensionality. It is also computationally efficient and has been proven to outperform its constituent methods, especially when dealing with data that include numerous features that may influence the response variable [31].  
This method is initiated with a linear system characterized by the state Equation (1)  
\[Y=X\beta+\epsilon \tag{1}\]  
where \(Y\in\ R^{n}\) represents the matrix of output variables, \(X\in\ R^{n\cdot(k+1)}\) is the matrix of input variables, \(\beta\in R^{k+1}\) is a vector of unknown parameters, and \(\epsilon\in R^{n}\) signifies the series of disturbances.  
Elastic Net is a regularization approach that merges L1 and L2 regularization strategies. This technique unifies the L1 norm of LASSO, which can zero out coefficients (thereby performing feature selection), with the L2 norm of ridge regression, which is optimal for multicollinearity. Introduced by [PERSON], the L1 aspect of LASSO offers sparsity, while the L2 component from ridge regression distributes the penalty among all coefficients, which helps when predictors are interdependent [32].  
Elastic Net regularization, as a sophisticated technique, leverages the merits of both L1 and L2 regularization to enhance predictive models. It extends beyond the capacity of LASSO for feature selection by incorporating the ability of ridge regression to handle collinear predictors, a situation where LASSO might falter. The dual regularization approach of Elastic Net allows it to shrink coefficients like LASSO, which can set some coefficients to zero for feature selection, and to distribute penalties across coefficients like ridge regression, which is beneficial when there are multiple highly correlated variables. This duality enables the Elastic Net to perform well in scenarios with numerous features, particularly when there is a mix of relevant and irrelevant features, or when several features are correlated. By adjusting the balance between the L1 and L2 penalties through its parameters, Elastic Net can be fine-tuned to fit the complexity and particularities of the dataset at hand, making it highly adaptable for both prediction accuracy and interpretability.  
The approach to modeling involved the presumption of a \"normal\" distribution within the response variables, adhering to the conventional choice. For model validation and to protect against overfitting, a technique of 3-fold cross-validation was applied to evaluate deviance. The tuning of the model encompassed experimenting with an assortment of alpha values, spanning from 0.1 to 1, in addition to a sequence of lambda values set at predetermined intervals starting from 0.0001 to 1 (0.0001, 0.001, 0.01, 1). This iterative process aimed to pinpoint the optimal parameter mix that would lead to the lowest possible Mean Squared Error (MSE).