Definition
Let be a sequence of observations, modeled by the relation . The smoothing spline estimate of the function is defined to be the minimizer (over the class of twice differentiable functions) of
Remarks:
- is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate.
- The integral is evaluated over the range of the .
- As (no smoothing), the smoothing spline converges to the interpolating spline.
- As (infinite smoothing), the roughness penalty becomes paramount and the estimate converges to a linear least squares estimate.
- The roughness penalty based on the second derivative is the most common in modern statistics literature, although the method can easily be adapted to penalties based on other derivatives.
- In early literature, with equally-spaced, second or third-order differences were used in the penalty, rather than derivatives.
- When the sum-of-squares term is replaced by a log-likelihood, the resulting estimate is termed penalized likelihood. The smoothing spline is the special case of penalized likelihood resulting from a Gaussian likelihood.
Read more about this topic: Smoothing Spline
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