Spline
Topic of Smooth means, part of Smoothing
Spline
| Type | Topic |
| Taught in | Smooth means |
| Part | Smoothing |
A piecewise polynomial whose pieces join smoothly at points called knots. A cubic spline has continuous first and second derivatives at each knot. Regression splines use a fixed, smaller set of knots and estimate their coefficients like any linear model.
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.
In Smooth means, continued
From Smooth means
Putting a knot at every x_i is expensive, and \lambda itself changes meaning when you change the likelihood. In practice we use a smaller set of knots, ten equally spaced between the smallest and largest x being the usual default, or knots at the deciles if the x values pile up at one end. More knots are available when the sample is in the thousands. And we do not quote \lambda. Cross-validation hides a fraction of the sample, fits on the rest, and scores the hidden points, rotating which fraction is hidden. The software reports the smoothness as an equivalent degrees of freedom: the polynomial degree the curve is about as flexible as. As \lambda\to\infty that degrees of freedom falls to 1, a straight line, which is the ordinary GLM. As \lambda\to 0 the fit interpolates, and if no x values are tied it interpolates perfectly. Interpolation fits noise. It does not predict the next point.
Used in
- Taught in Smooth means
- All 5 questions on Smooth means
See also
Knot · Cubic smoothing spline · Roughness penalty · Lidar data · Kyphosis data