Cubic Splines.

Table 8.6: Axial stiffness index data.

Plate ASI Plate ASI Plate ASI Plate ASI Plate ASI 4 309.2 6 402.1 8 392.4 10 346.7 12 407.4 4 409.5 6 347.2 8 366.2 10 452.9 12 441.8 4 311.0 6 361.0 8 351.0 10 461.4 12 419.9 4 326.5 6 404.5 8 357.1 10 433.1 12 410.7 4 316.8 6 331.0 8 409.9 10 410.6 12 473.4 4 349.8 6 348.9 8 367.3 10 384.2 12 441.2 4 309.7 6 381.7 8 382.0 10 362.6 12 465.8 To fit two cubic polynomials on the Hooker partition sets, we can fit the regression function m(x) = β0+β1x+β2x2+β3x3+γ0h(x)+γ1xh(x)+γ2x2h(x)+γ3x3h(x)
= 
β0+β1x+β2x2+β3x3
+h(x)

γ0 +γ1x+γ2x2+γ3x3
, where the polynomial coefficients below the knot are the β js and above the knot are the (β j +γ j)s.
Define the change polynomial as C(x) ≡γ0 +γ1x+γ2x2 +γ3x3.
To turn the two polynomials into cubic splines, we require that the two cubic polynomials be equal at the knot but also that their first and second derivatives be equal at the knot. It is not hard to see that this is equivalent to requiring that the change polynomial have 0 =C(191) = dC(x)
dx 
x=191 = d2C(x)
dx2 
x=191 , where our one knot for the Hooker data is at x = 191. Show that imposing these three conditions leads to the model m(x) = β0+β1x+β2x2 +β3x3 +γ3(x−191)3h(x)
= β0+β1x+β2x2 +β3x3 +γ3[(x−191)+]3.
(It is easy to showthat C(x) =γ3(x−191)3 satisfies the three conditions. It is a little harder to show that satisfying the three conditions implies that C(x) =γ3(x−191)3.)