OIKAN v0.0.3: 🧠 Brainstorm -> Handling non-linear complex relationships beyond the current limitation of polynomial functions [Improve Symbolic Regression]

[silvermete0r]

Current limitation:

• Polynomial-Only Basis: The reliance on PolynomialFeatures restricts the model to polynomial relationships, limiting its ability to capture highly non-linear patterns like exponential growth, logarithmic decay, or periodic behavior.
• Lasso Constraints: Lasso regression struggles with correlated features, often arbitrarily selecting one from a correlated group, which can reduce interpretability and accuracy.
• Complexity Explosion: Higher-degree polynomials increase the feature space exponentially, risking overfitting and computational inefficiency despite Lasso’s sparsity.

IDEAS:

• Replace Lasso with Elastic Net for better feature selection.
  • Why? Elastic Net combines L1 (Lasso) and L2 (Ridge) regularization, making it better suited for datasets with correlated features. Unlike Lasso, which may select one feature from a correlated group, Elastic Net can select the group, improving stability and interpretability.
  • Benefits: Improved handling of multicollinearity, potentially leading to more robust feature selection.
  • How to implement?

from sklearn.linear_model import ElasticNet

model = ElasticNet(alpha=self.alpha, l1_ratio=0.5, fit_intercept=False)
model.fit(X_poly, y)

• Implement hierarchical symbolic regression to efficiently introduce non-linear terms. Two-stage hierarchical approach:

  • Stage 1: Coarse Model
    • Fit a low-degree polynomial model (e.g., degree=2) using Elastic Net to identify main trends and key features.
    • Select the top K features based on coefficient magnitudes (e.g., top_k=5).
  • Stage 2: Refined Model
    • For the top K features, generate higher-degree polynomials (e.g., x_i^3) and non-linear transformations (e.g., log(1 + x_i), exp(x_i), sin(x_i)).
    • Combine these with the original low-degree features and fit another Elastic Net model.
  • Benefits: Captures non-linear patterns efficiently by focusing computational effort on important features, improving accuracy without excessive complexity.

• Add SymPy post-processing for simplified, readable formulas.

  • To enhance human-readability, we can apply symbolic simplification using SymPy after fitting the model;