Least-squares Fitting of a Polynomial Constrained to be either Non-negative, Non-decreasing or Convex
Least-squares Fitting of a Polynomial Constrained to be either Non-negative, Non-decreasing or Convex
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约束为非负、非递减或凸的多项式的最小二乘拟合
DOI：
10.1111/j.2517-6161.1969.tb00772.x
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发表时间：
1969
期刊：
影响因子：
--
通讯作者：
D. Hudson
中科院分区：
文献类型：
--
作者：
D. Hudson
提供了一种算法来将（唯一的）最小二乘多项式Y（x）拟合到n个数据点（x，y），所述n个数据点（x，y）受到这样的约束，即它在闭合区间L中的所有点x处是非负的、非正的、非减少的、非增加的、凸的或凹的。可以处理对一般线性模型的一些推广，并且也可以处理某些替代约束。该问题接近于经典的二次规划问题（Q.P.），并且Q.P.中的一些已知结果可用于拟合曲线。该算法最多在三个步骤中终止。步骤的数量取决于多项式Y（x）的次数和数据。当不需要多于两个步骤时，该算法非常容易执行，但当需要三个步骤时，它变得更加困难。一个有用的统计应用是多项式对增长数据的拟合。在这种情况下，我们可能希望拟合曲线在感兴趣的范围内是非递减的。
SUMMARY An algorithm is provided to fit the (unique) least-squares polynomial Y(x) to n data points (x, y) subject to a constraint that it be either non-negative, non-positive, non-decreasing, non-increasing, convex or concave at all points x in a closed interval L Some generalization to a general linear model can be handled, and certain alternative constraints can also be dealt with. The problem is close to the classical problem of Quadratic Programming (Q.P.), and some of the known results in Q.P. may be used in fitting the curve. The algorithm terminates in at most three steps. The number of steps depends on the degree of the polynomial Y(x) and the data. When no more than two steps are required, the algorithm is very easy to carry out but when three steps are necessary, it becomes more difficult. A useful statistical application is the fitting of a polynomial to growth data. In that case, we might like the fitted curve to be non-decreasing over the range of interest.