C32. Roughness penalty

Topic: Smooth means · more questions on this topic
‹ C31

Why penalise \int\{f''(x)\}^2\,dx when fitting a smooth? What happens as its penalty weight increases?

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Definitions/notation. For a smooth f, the roughness measure J(f)=\int\{f''(x)\}^2\,dx penalises curvature when fitting by maximising \ell(f)-\lambda J(f)/2, with \lambda\ge0. Larger \lambda favours less wiggle; as \lambda\to\infty the fit approaches the penalty’s linear null space, where an admissible limiting fit exists.