5.3 Using Splines to Approximate

  A spline is a piecewise polynomial, a function made up of several contiguous polynomial sections. The places where two polynomial pieces meet are called ``knots''. Splines are commonly used to approximate functions that are expected to be fairly smooth, but whose actual form is not completely known. They are especially easy to use because they can be made arbitrarily flexible and are very easy to calculate in terms of a small number of degrees of freedom (typically, one for each knot). At the knots, the function must be continuous in its value, but some of its derivatives might have discontinuities. Usually only it's highest non-trivial derivative is discontinuous, so that one degree of freedom is associated with each knot. Endpoints of the spline need to be dealt with as special cases.

autobk uses something very similar to standard cubic splines (it actually uses b-splines of fourth order, but the distinction is not important for the discussion here, and the functional form of the polynomial pieces is not as important as the amount of freedom at each knot). One free coefficient is associated with each of the knots of the spline. The value of this free coefficient is optimized as discussed in the next section. Good initial values for the free coefficients of the spline turn out to be easy to get for splines by guessing that the spline goes through the values at the knot locations. Aside from giving the initial guesses of the spline coefficients the data is not explicitly used for evaluating the background function. The spline is not forced in any way to go through any points, including either of the endpoints.