4.2 Generalized additive models

• splines

  • A cubic spline is a piecewise polynomial with the property of being continuously differentiable until second order.
  • The number of knots act as an smoothing parameter in unpenalized splines.
  • Knots defined based on the quantiles of the data make the spline flexible in dense areas and less flexible in sparse areas, which is desirable.
  • In general, it is more appropiate to select more knots than expected and use a penalty term to control for smothness avoiding the need to select number of knots.
  • For \(l\) knots and degree \(r\), the space of polynomial splines is a vector space with dimension equals to the number of free parameters. \(l-1\) polynomial functions of degree \(r\) have \((r+1)(l-1)\) parameters. The condition of \(r-1\) times continuously differentiable generate \(r\) constrains for all the \(l-2\) inside knots. Then, the number of free parameters is \((r+1)(l-1) - r(l-2) = r+l-1\).
  • Natural splines assumes that the curvature, the second derivative, at the first and last knot is zero. Then, a natural cubic spline will have \(l\) free parameters.

  Simon Wood 2016

  B-splines, whose construction from polynomial pieces gives them many attractive computational properties, as described by de Boor (1978). 2016 donnell