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Polynomial Regression
Non-linear regression technique that models the relationship between variables using a polynomial function of degree greater than one. It allows capturing complex relationships while remaining in the linear framework for the coefficients.
Vandermonde Matrix
Matrix structure used in polynomial regression where each column represents an increasing power of the independent variable. It transforms the non-linear problem into a linear problem in the coefficients.
Polynomial Degree
Parameter determining the complexity of the polynomial model, corresponding to the highest exponent in the equation. A high degree increases flexibility but risks overfitting.
Polynomial Overfitting
Phenomenon where a polynomial of too high a degree excessively adapts to training data, capturing noise rather than the underlying trend. It manifests as excellent training performance but poor generalization.
Ridge Regularization
L2 penalty method applied to polynomial coefficients to control their magnitude and prevent overfitting. It adds a penalty term proportional to the square of the coefficients to the cost function.
Lasso Regularization
L1 penalty technique that forces certain polynomial coefficients toward zero, thus performing automatic variable selection. It is particularly useful for eliminating irrelevant polynomial terms.
Interaction Term
Product of predictor variables in a polynomial model capturing synergistic effects between features. These terms allow modeling relationships where the effect of one variable depends on the level of another.
K-fold Cross-Validation
Robust evaluation technique dividing data into K subsets to estimate polynomial model performance on different partitions. It allows selecting the optimal degree by minimizing validation error.
Polynomial Multicollinearity
High correlation between polynomial terms derived from the same variable, particularly problematic for high-degree polynomials. It can destabilize coefficient estimation and often requires standardization.
Orthogonal Polynomials
Family of polynomials (Legendre, Chebyshev) where terms are mathematically orthogonal over a specific interval. They reduce multicollinearity and improve numerical stability of regression.
Polynomial Transformation
Process of creating new features by raising existing variables to different powers and generating interaction terms. It transforms the feature space to capture nonlinear relationships.
Polynomial Learning Curve
Graph showing the evolution of training and validation errors according to sample size for different polynomial degrees. It helps diagnose overfitting or underfitting.
Weighted Least Squares Method
Variant of least squares regression where each observation receives a weight based on its reliability or variance. It is particularly suitable when heteroscedasticity is present in polynomial data.
Polynomial Feature Scaling
Standardization or normalization of variables before polynomial transformation to avoid numerical instabilities. It prevents scaling problems between different polynomial degrees.