Regressionbasisbaserede

Regressionbasisbaserede refers to approaches in statistics and machine learning that model the regression function as a linear combination of basis functions. In this framework, the target variable is predicted by a linear model whose features are transformations of the input variables, rather than the raw variables themselves. The key idea is to represent potentially non-linear relationships through an expanded feature space, turning a nonlinear problem into a linear one in the new coordinates.

A typical formulation expresses the response y as y ≈ α + Σ_j β_j φ_j(x) where φ_j are basis functions chosen to capture the underlying structure. The coefficients β_j are estimated from data, usually by least squares or penalized regression to control complexity. Basis selection and the amount of smoothing are crucial: too few bases may miss patterns, while too many can overfit.

Common basis families include polynomial bases, spline bases (such as B-splines and natural splines) for smooth, flexible curves; radial basis functions for localized deviations; Fourier bases for periodic patterns; and wavelets for multiscale structure. Splines offer local support and easy control of smoothness; polynomial bases provide simple global fits; radial bases are versatile for scattered data.

Regularization methods like ridge, lasso, and elastic net are often applied to basis expansions to prevent overfitting and aid feature selection. This approach is related to generalized additive models, where each predictor can have its own basis expansion, allowing complex, interpretable nonlinear effects. Regressionbasisbaserede methods also connect to kernel methods, which implicitly map inputs into high-dimensional bases.