Smoothing Spline - Definition

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

\sum_{i=1}^n (Y_i - \hat\mu(x_i))^2 + \lambda \int_{x_1}^{x_n} \hat\mu''(x)^2 \,dx.

Remarks:

  1. is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate.
  2. The integral is evaluated over the range of the .
  3. As (no smoothing), the smoothing spline converges to the interpolating spline.
  4. As (infinite smoothing), the roughness penalty becomes paramount and the estimate converges to a linear least squares estimate.
  5. 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.
  6. In early literature, with equally-spaced, second or third-order differences were used in the penalty, rather than derivatives.
  7. 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.

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