Cubic smoothing spline
Topic of Smooth means, part of Smoothing
Cubic smoothing spline
| Type | Topic |
| Taught in | Smooth means |
| Part | Smoothing |
| Questions using it | 1 |
The function that maximises a log-likelihood penalised by \lambda\int g''(x)^2\,dx. It is a piecewise cubic with a knot at every distinct x_i, with the function and its first two derivatives continuous at each knot. \lambda=0 interpolates; \lambda\to\infty gives the straight line.
In Smooth means
From Smooth means
The maximiser, for any fixed \lambda, is a cubic spline with a knot at each distinct x_i. Inside each interval the function is a cubic. At each knot the function, the first derivative, and the second derivative agree from the left and the right. Your eye can see those three. It cannot see a jump in the third derivative, which is why a cubic, rather than a higher piecewise polynomial, is the one this penalty produces.
Used in
- Taught in Smooth means
- Questions that use it (1) — C31
- All 5 questions on Smooth means
See also
Roughness penalty · Gaussian process · Likelihood function · Regularisation · Knot · Spline · Lidar data · Kyphosis data