7.4.2 Constraints and Splines

The top left panel of Figure 7.3 looks wrong because the fitted curve is just too flexible. To remedy this problem, we can fit a piecewise polynomial under the constraint that the fitted curve must be continuous. In other words, there cannot be a jump when age=50 . The top right plot in Figure 7.3 shows the resulting fit. This looks better than the top left plot, but the V- shaped join looks unnatural.

In the lower left plot, we have added two additional constraints: now both the first and second derivatives of the piecewise polynomials are continuous derivative at age=50 . In other words, we are requiring that the piecewise polynomial be not only continuous when age=50 , but also very smooth . Each constraint that we impose on the piecewise cubic polynomials effectively frees up one degree of freedom, by reducing the complexity of the resulting piecewise polynomial fit. So in the top left plot, we are using eight degrees of freedom, but in the bottom left plot we imposed three constraints (continuity, continuity of the first derivative, and continuity of the second derivative) and so are left with five degrees of freedom. The curve in the bottom left plot is called a cubic spline .[3] In general, a cubic spline with K knots uses cubic spline a total of 4 + K degrees of freedom.

In Figure 7.3, the lower right plot is a linear spline , which is continuous linear spline at age=50 . The general definition of a degree- d spline is that it is a piecewise degree- d polynomial, with continuity in derivatives up to degree d − 1 at each knot. Therefore, a linear spline is obtained by fitting a line in each region of the predictor space defined by the knots, requiring continuity at each knot.

In Figure 7.3, there is a single knot at age=50 . Of course, we could add more knots, and impose continuity at each.