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Show 8 1.3.2 Knot Selection and Placement of Corners The knots used In computing a 8-spline approximation strongly influence the shape of the resulting curve. Enough knots must be used for the curve to fit the data closely, but too many may permit unwanted oscillation. Discon-tinuities resulting from multiple knots are most dramatic, and must be carefully placed to reflect corners or edges in the data." The influence of the placement of simple knots is more subtle; by choosing them carefully, many data points can be approximated by a single section of the approximation, and a more com-pact representation is possible. To reduce the number of degrees of freedom of a piecewise cubic curve that is C 1 at the knots, Schoenberg [22] and Reinsch [2 11 independently proposed "smoothing splines," based on earlier work by Whittaker [23]. Instead of solving the traditional least squares problem and determining appropriate knot locations, they proposed finding an explicit spline function f of order 2k with a knot for every data point, minimizing n (f(x.)-y.)2 xn P ~ I I + ( 1 -p) f (f(k)(t))2 dt, i=1 (ayi)2 x1 ( 1. 1) where [xi,y) are the data points, the ayi are the predicted standard devia-tions for each of the data points, and p is a user-specified parameter controlling the tradeoff between smoothness and the closeness of the approximation to the data. The first term is a least squares metric, so that as p approaches 1, the resulting curve becomes more like a pure least squares approximation. As p approaches 0, the smoothing metric forces the kth derivative of the approximation to 0; the limiting case is a spline of order k-1 or less. The smoothness constraint effectively reduces the number of degrees of freedom of the approximation, and by decreasing p, one can make the approximation arbitrarily smooth. There is not, however, an analytic method for determining an ap- " A corner in a B-spline can also result from e multiple control point. |
