Chapter 12 Spline Regression

This chapter is based on CMU stat

Definition: a k-th order spline is a piecewise polynomial function of degree k, that is continuous and has continuous derivatives of orders 1,…, k -1, at its knot points

Equivalently, a function f is a k-th order spline with knot points at \(t_1 < ...< t_m\) if

  • f is a polynomial of degree k on each of the intervals \((-\infty, t_1], [t_1,t_2],...,[t_m, \infty)\)
  • \(f^{(j)}\), the j-th derivative of f, is continuous at \(t_1,...,t_m\) for each j = 0,1,…,k-1

A common case is when k = 3, called cubic splines. (piecewise cubic functions are continuous, and also continuous at its first and second derivatives)

To parameterize the set of splines, we could use truncated power basis, defined as

\[ g_1(x) = 1 \\ g_2(x) = x \\ ... \\ g_{k+1}(x) = x^k \\ g_{k+1+j}(x) = (x-t_j)^k_+ \]

where j = 1,…,m and \(x_+\) = max{x,0}

However, now software typically use B-spline basis.