polynomregression

Polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modeled as an nth-degree polynomial. The standard form is y = β0 + β1 x + β2 x^2 + ... + βp x^p + ε, where p is the degree and ε is the error term assumed to have zero mean and constant variance. The model is linear in the coefficients β, so parameters are typically estimated using ordinary least squares based on the design matrix X with columns [1, x, x^2, ..., x^p].

Benefits: polynomial regression can capture nonlinear patterns while remaining a linear model in parameters. Limitations: high-degree polynomials can exhibit erratic extrapolation and severe multicollinearity among terms; overfitting can occur if p is too large relative to sample size; sensitivity to the scale and range of x.

Degree selection: choose p via cross-validation, information criteria (AIC/BIC), or out-of-sample prediction error. Practical improvements include centering and scaling x, using orthogonal polynomials, or applying ridge or lasso regularization to stabilize estimates. Alternatives include spline regression or kernel methods that fit nonlinear relationships without high-degree polynomials.

Applications: for data exhibiting smooth, nonlinear trends where a simple linear model is inadequate, polynomial regression is used in sciences, economics, and engineering. Where extrapolation is required beyond the observed x-range, caution is advised due to potential poor predictions.