6.4 Splines
A general way to write very flexible models is to apply functions Bm to Xm and then multiply them by coefficients βm:
We are free to pick Bm as we wish; for instance, we can pick polynomials. But we can also pick other functions. A popular choice is to use B-splines; we are not going to discuss their definition, but we can think of them as a way to create smooth curves in such a way that we get flexibility, as with polynomials, but less prone to overfitting. We achieve this by using piecewise polynomials, that is, polynomials that are restricted to affect only a portion of the data. Figure 6.6 shows three examples of piecewise polynomials of increasing degrees. The dotted vertical lines show the ”knots,” which are the points used to restrict the regions, the dashed gray line represents the function we want to approximate, and the black lines are the piecewise polynomials.
Figure 6.6: Piecewise polynomials of increasing degrees
Figure 6.7 shows...
