Smoothing with Cubic Splines

A spline function is a curve constructed from polynomial segments that are subject to conditions or continuity at their joints. In this paper, we shall present the algorithm of the cubic smoothing spline and we shall justify its use in estimating trends in time series. Considerable effort has been devoted over several decades to developing the mathematics of spline functions. Much of the interest is due to the importance of splines in industrial design. In statistics, smoothing splines have been used in fitting curves to data ever since workable algorithms first became available in the late sixties—see Schoenberg [9] and Reinsch [8]. However, many statisticians have felt concern at the apparently arbitrary nature in this device. The difficulty is in finding an objective criterion for choosing the value of the parameter which governs the trade-off between smoothness of the curve and its closeness to the data points. At one extreme, where the smoothness is all that matters, the spline degenerates to the straight line of an ordinary linear least-squares regression. At the other extreme, it becomes the interpolating spline which passes through each of the data points. It appears to be a matter of judgment where in the spectrum between these two extremes the most appropriate curve should lie. One attempt at overcoming this arbitrariness has led to the criterion of cross-validation which was first proposed by Craven and Wahba [6]. Here the underlying notion is that the degree of smoothness should be chosen so as to make the spline the best possible predictor of any points in the data set to which it has not been fitted. Instead of reserving a collection of data points for the sole purpose of making this choice, it has been proposed that each of the available points should be left out in turn while the spline is fitted to the remainder. For each omitted point, there is then a corresponding error of prediction; and the optimal degree of smoothing is that which results in the minimum sum of squares of the prediction errors. To find the optimal degree of smoothing by the criterion of cross-validation can require an enormous amount of computing. An alternative procedure, which has emerged more recently, is based on the notion that the spline with an appropriate smoothing parameter represents the optimal predictor of the