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.

Read more about this topic:  Smoothing Spline

Famous quotes containing the word definition:

    I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.”
    Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.
    Ana Castillo (b. 1953)

    The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.
    Ralph Waldo Emerson (1803–1882)