Definition

Let be a arrangement of observations, modeled by the affiliation . The cutting spline appraisal of the action is authentic to be the minimizer (over the chic of alert differentiable functions) of1

Remarks:

is a cutting parameter, authoritative the accommodation amid allegiance to the abstracts and acerbity of the action estimate.

The basic is evaluated over the ambit of the .

As (no smoothing), the cutting spline converges to the interpolating spline.

As (infinite smoothing), the acerbity amends becomes ascendant and the appraisal converges to a beeline atomic squares estimate.

The acerbity amends based on the additional acquired is the a lot of accepted in avant-garde statistics literature, although the adjustment can calmly be acclimatized to penalties based on added derivatives.

In aboriginal literature, with equally-spaced , additional or third-order differences were acclimated in the penalty, rather than derivatives.

When the sum-of-squares appellation is replaced by a log-likelihood, the consistent appraisal is termed penalized likelihood. The cutting spline is the appropriate case of penalized likelihood consistent from a Gaussian likelihood.