Smoothing Spline - de Boor's Approach

De Boor's Approach

De Boor's approach exploits the same idea, of finding a balance between having a smooth curve and being close to the given data.

where is a parameter called smooth factor and belongs to the interval, and are the quantities controlling the extent of smoothing (they represent the weight of each point ). In practice, since cubic splines are mostly used, is usually . The solution for was proposed by Reinsch in 1967. For, when approaches, converges to the "natural" spline interpolant to the given data. As approaches, converges to a straight line (the smoothest curve). Since finding a suitable value of is a task of trial and error, a redundant constant was introduced for convenience. is used to numerically determine the value of so that the function meets the following condition:

The algorithm described by de Boor starts with and increases until the condition is met. If is an estimation of the standard deviation for, the constant is recommended to be chosen in the interval . Having means the solution is the "natural" spline interpolant. Increasing means we obtain a smoother curve by getting farther from the given data.